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<46050679+SwayStar123@users.noreply.github.com> Date: Sat, 6 Jul 2024 17:19:13 +0530 Subject: [PATCH 02/49] Update 12_2_Multihead_Self_Attention.ipynb title number is incorrect, its actually 12.2 --- Notebooks/Chap12/12_2_Multihead_Self_Attention.ipynb | 2 +- 1 file changed, 1 insertion(+), 1 deletion(-) diff --git a/Notebooks/Chap12/12_2_Multihead_Self_Attention.ipynb b/Notebooks/Chap12/12_2_Multihead_Self_Attention.ipynb index 681b78c..219a8d0 100644 --- a/Notebooks/Chap12/12_2_Multihead_Self_Attention.ipynb +++ b/Notebooks/Chap12/12_2_Multihead_Self_Attention.ipynb @@ -28,7 +28,7 @@ { "cell_type": "markdown", "source": [ - "# **Notebook 12.1: Multihead Self-Attention**\n", + "# **Notebook 12.2: Multihead Self-Attention**\n", "\n", "This notebook builds a multihead self-attention mechanism as in figure 12.6\n", "\n", From b005cec9c1ddd8b9a51b2c77b7fa209f069b7aa1 Mon Sep 17 00:00:00 2001 From: Chris Endemann Date: Wed, 10 Jul 2024 09:42:38 -0500 Subject: [PATCH 03/49] Update 8_3_Double_Descent.ipynb I added a little code to include a vertical dashed line on the plot representing where total_weights = number of train observations. I also moved n_epochs as an argument to fit_model() so learners can play around with the impact of n_epochs more easily. --- Notebooks/Chap08/8_3_Double_Descent.ipynb | 429 ++++++++++++++++++++-- 1 file changed, 390 insertions(+), 39 deletions(-) diff --git a/Notebooks/Chap08/8_3_Double_Descent.ipynb b/Notebooks/Chap08/8_3_Double_Descent.ipynb index f60f12f..aaac87e 100644 --- a/Notebooks/Chap08/8_3_Double_Descent.ipynb +++ b/Notebooks/Chap08/8_3_Double_Descent.ipynb @@ -4,8 +4,7 @@ "metadata": { "colab": { "provenance": [], - "gpuType": "T4", - "include_colab_link": true + "gpuType": "T4" }, "kernelspec": { "name": "python3", @@ -17,16 +16,6 @@ "accelerator": "GPU" }, "cells": [ - { - "cell_type": "markdown", - "metadata": { - "id": "view-in-github", - "colab_type": "text" - }, - "source": [ - "\"Open" - ] - }, { "cell_type": "markdown", "source": [ @@ -51,10 +40,45 @@ "!pip install git+https://github.com/greydanus/mnist1d" ], "metadata": { - "id": "fn9BP5N5TguP" + "id": "fn9BP5N5TguP", + "outputId": "3ba15b8f-2395-4b1a-8a66-8ed80e9b5138", + "colab": { + "base_uri": "https://localhost:8080/" + } }, - "execution_count": null, - "outputs": [] + "execution_count": 29, + "outputs": [ + { + "output_type": "stream", + "name": "stdout", + "text": [ + "Collecting git+https://github.com/greydanus/mnist1d\n", + " Cloning https://github.com/greydanus/mnist1d to /tmp/pip-req-build-cbhxd1j8\n", + " Running command git clone --filter=blob:none --quiet https://github.com/greydanus/mnist1d /tmp/pip-req-build-cbhxd1j8\n", + " Resolved https://github.com/greydanus/mnist1d to commit 350929d12f4c9a4b7355e0c96604e41b9239bdb4\n", + " Installing build dependencies ... \u001b[?25l\u001b[?25hdone\n", + " Getting requirements to build wheel ... \u001b[?25l\u001b[?25hdone\n", + " Preparing metadata (pyproject.toml) ... \u001b[?25l\u001b[?25hdone\n", + "Requirement already satisfied: requests in /usr/local/lib/python3.10/dist-packages (from mnist1d==0.0.2.post9) (2.31.0)\n", + "Requirement already satisfied: numpy in /usr/local/lib/python3.10/dist-packages (from mnist1d==0.0.2.post9) (1.25.2)\n", + "Requirement already satisfied: matplotlib in /usr/local/lib/python3.10/dist-packages (from mnist1d==0.0.2.post9) (3.7.1)\n", + "Requirement already satisfied: scipy in /usr/local/lib/python3.10/dist-packages (from mnist1d==0.0.2.post9) (1.11.4)\n", + "Requirement already satisfied: contourpy>=1.0.1 in /usr/local/lib/python3.10/dist-packages (from matplotlib->mnist1d==0.0.2.post9) (1.2.1)\n", + "Requirement already satisfied: cycler>=0.10 in /usr/local/lib/python3.10/dist-packages (from matplotlib->mnist1d==0.0.2.post9) (0.12.1)\n", + "Requirement already satisfied: fonttools>=4.22.0 in /usr/local/lib/python3.10/dist-packages (from matplotlib->mnist1d==0.0.2.post9) (4.53.0)\n", + "Requirement already satisfied: kiwisolver>=1.0.1 in /usr/local/lib/python3.10/dist-packages (from matplotlib->mnist1d==0.0.2.post9) (1.4.5)\n", + "Requirement already satisfied: packaging>=20.0 in /usr/local/lib/python3.10/dist-packages (from matplotlib->mnist1d==0.0.2.post9) (24.1)\n", + "Requirement already satisfied: pillow>=6.2.0 in /usr/local/lib/python3.10/dist-packages (from matplotlib->mnist1d==0.0.2.post9) (9.4.0)\n", + "Requirement already satisfied: pyparsing>=2.3.1 in /usr/local/lib/python3.10/dist-packages (from matplotlib->mnist1d==0.0.2.post9) (3.1.2)\n", + "Requirement already satisfied: python-dateutil>=2.7 in /usr/local/lib/python3.10/dist-packages (from matplotlib->mnist1d==0.0.2.post9) (2.8.2)\n", + "Requirement already satisfied: charset-normalizer<4,>=2 in /usr/local/lib/python3.10/dist-packages (from requests->mnist1d==0.0.2.post9) (3.3.2)\n", + "Requirement already satisfied: idna<4,>=2.5 in /usr/local/lib/python3.10/dist-packages (from requests->mnist1d==0.0.2.post9) (3.7)\n", + "Requirement already satisfied: urllib3<3,>=1.21.1 in /usr/local/lib/python3.10/dist-packages (from requests->mnist1d==0.0.2.post9) (2.0.7)\n", + "Requirement already satisfied: certifi>=2017.4.17 in /usr/local/lib/python3.10/dist-packages (from requests->mnist1d==0.0.2.post9) (2024.6.2)\n", + "Requirement already satisfied: six>=1.5 in /usr/local/lib/python3.10/dist-packages (from python-dateutil>=2.7->matplotlib->mnist1d==0.0.2.post9) (1.16.0)\n" + ] + } + ] }, { "cell_type": "code", @@ -73,10 +97,22 @@ "print('Using:', DEVICE)" ], "metadata": { - "id": "hFxuHpRqTgri" + "id": "hFxuHpRqTgri", + "outputId": "2a246d1e-8568-4605-df0a-28dad08935ec", + "colab": { + "base_uri": "https://localhost:8080/" + } }, - "execution_count": null, - "outputs": [] + "execution_count": 30, + "outputs": [ + { + "output_type": "stream", + "name": "stdout", + "text": [ + "Using: cuda\n" + ] + } + ] }, { "cell_type": "code", @@ -99,13 +135,28 @@ "# data['x'], data['y'], data['x_test'], and data['y_test']\n", "print(\"Examples in training set: {}\".format(len(data['y'])))\n", "print(\"Examples in test set: {}\".format(len(data['y_test'])))\n", - "print(\"Length of each example: {}\".format(data['x'].shape[-1]))" + "print(\"Dimensionality of each example: {}\".format(data['x'].shape[-1]))" ], "metadata": { - "id": "PW2gyXL5UkLU" + "id": "PW2gyXL5UkLU", + "outputId": "a4186965-3025-4d00-b780-441baab421de", + "colab": { + "base_uri": "https://localhost:8080/" + } }, - "execution_count": null, - "outputs": [] + "execution_count": 31, + "outputs": [ + { + "output_type": "stream", + "name": "stdout", + "text": [ + "Did or could not load data from ./mnist1d_data.pkl. Rebuilding dataset...\n", + "Examples in training set: 4000\n", + "Examples in test set: 4000\n", + "Dimensionality of each example: 40\n" + ] + } + ] }, { "cell_type": "code", @@ -141,13 +192,13 @@ "metadata": { "id": "hAIvZOAlTnk9" }, - "execution_count": null, + "execution_count": 32, "outputs": [] }, { "cell_type": "code", "source": [ - "def fit_model(model, data):\n", + "def fit_model(model, data, n_epoch):\n", "\n", " # choose cross entropy loss function (equation 5.24)\n", " loss_function = torch.nn.CrossEntropyLoss()\n", @@ -164,9 +215,6 @@ " # load the data into a class that creates the batches\n", " data_loader = DataLoader(TensorDataset(x_train,y_train), batch_size=100, shuffle=True, worker_init_fn=np.random.seed(1))\n", "\n", - " # loop over the dataset n_epoch times\n", - " n_epoch = 1000\n", - "\n", " for epoch in range(n_epoch):\n", " # loop over batches\n", " for i, batch in enumerate(data_loader):\n", @@ -200,7 +248,19 @@ "metadata": { "id": "AazlQhheWmHk" }, - "execution_count": null, + "execution_count": 33, + "outputs": [] + }, + { + "cell_type": "code", + "source": [ + "def count_parameters(model):\n", + " return sum(p.numel() for p in model.parameters() if p.requires_grad)" + ], + "metadata": { + "id": "AQNCmFNV6JpV" + }, + "execution_count": 34, "outputs": [] }, { @@ -226,42 +286,333 @@ "# This code will take a while (~30 mins on GPU) to run! Go and make a cup of coffee!\n", "\n", "hidden_variables = np.array([2,4,6,8,10,14,18,22,26,30,35,40,45,50,55,60,70,80,90,100,120,140,160,180,200,250,300,400]) ;\n", + "\n", "errors_train_all = np.zeros_like(hidden_variables)\n", "errors_test_all = np.zeros_like(hidden_variables)\n", + "total_weights_all = np.zeros_like(hidden_variables)\n", + "\n", + "# loop over the dataset n_epoch times\n", + "n_epoch = 1000\n", "\n", "# For each hidden variable size\n", "for c_hidden in range(len(hidden_variables)):\n", " print(f'Training model with {hidden_variables[c_hidden]:3d} hidden variables')\n", " # Get a model\n", " model = get_model(hidden_variables[c_hidden]) ;\n", + " # Count and store number of weights\n", + " total_weights_all[c_hidden] = count_parameters(model)\n", " # Train the model\n", - " errors_train, errors_test = fit_model(model, data)\n", + " errors_train, errors_test = fit_model(model, data, n_epoch)\n", " # Store the results\n", " errors_train_all[c_hidden] = errors_train\n", - " errors_test_all[c_hidden]= errors_test" + " errors_test_all[c_hidden]= errors_test\n", + "\n", + "" ], "metadata": { - "id": "K4OmBZGHWXpk" + "id": "K4OmBZGHWXpk", + "outputId": "62a7aaf0-793a-4ab2-960e-c1127f15717c", + "colab": { + "base_uri": "https://localhost:8080/" + } }, - "execution_count": null, - "outputs": [] + "execution_count": 35, + "outputs": [ + { + "output_type": "stream", + "name": "stdout", + "text": [ + "Training model with 2 hidden variables\n", + "Epoch 0, train loss 2.280202, train error 89.05, test loss 2.276196, test error 89.18\n", + "Epoch 100, train loss 1.925899, train error 71.75, test loss 1.761014, test error 70.57\n", + "Epoch 200, train loss 1.912995, train error 71.20, test loss 1.747572, test error 69.07\n", + "Epoch 300, train loss 1.908751, train error 71.32, test loss 1.741206, test error 69.07\n", + "Epoch 400, train loss 1.906859, train error 71.05, test loss 1.737478, test error 70.00\n", + "Epoch 500, train loss 1.906671, train error 71.43, test loss 1.732123, test error 68.95\n", + "Epoch 600, train loss 1.903380, train error 71.28, test loss 1.741086, test error 69.65\n", + "Epoch 700, train loss 1.902236, train error 70.88, test loss 1.733666, test error 69.22\n", + "Epoch 800, train loss 1.903633, train error 71.88, test loss 1.734591, test error 70.57\n", + "Epoch 900, train loss 1.901804, train error 71.03, test loss 1.742573, test error 68.10\n", + "Training model with 4 hidden variables\n", + "Epoch 0, train loss 2.267358, train error 86.97, test loss 2.269902, test error 87.62\n", + "Epoch 100, train loss 1.716064, train error 62.22, test loss 1.519437, test error 58.25\n", + "Epoch 200, train loss 1.705652, train error 61.88, test loss 1.507422, test error 58.40\n", + "Epoch 300, train loss 1.696660, train error 60.62, test loss 1.507704, test error 57.58\n", + "Epoch 400, train loss 1.691092, train error 61.60, test loss 1.499592, test error 56.85\n", + "Epoch 500, train loss 1.684060, train error 61.05, test loss 1.513196, test error 57.95\n", + "Epoch 600, train loss 1.677890, train error 61.10, test loss 1.504343, test error 56.85\n", + "Epoch 700, train loss 1.677930, train error 61.10, test loss 1.509796, test error 58.15\n", + "Epoch 800, train loss 1.676683, train error 61.17, test loss 1.506790, test error 58.28\n", + "Epoch 900, train loss 1.676809, train error 61.53, test loss 1.505417, test error 57.72\n", + "Training model with 6 hidden variables\n", + "Epoch 0, train loss 2.290775, train error 89.18, test loss 2.284365, test error 87.57\n", + "Epoch 100, train loss 1.634348, train error 58.42, test loss 1.467763, test error 55.67\n", + "Epoch 200, train loss 1.602023, train error 57.25, test loss 1.456007, test error 55.38\n", + "Epoch 300, train loss 1.587963, train error 56.50, test loss 1.453522, test error 55.33\n", + "Epoch 400, train loss 1.587952, train error 56.55, test loss 1.453863, test error 54.92\n", + "Epoch 500, train loss 1.579495, train error 56.42, test loss 1.452398, test error 55.40\n", + "Epoch 600, train loss 1.590881, train error 56.67, test loss 1.473001, test error 56.28\n", + "Epoch 700, train loss 1.581218, train error 56.58, test loss 1.470745, test error 55.90\n", + "Epoch 800, train loss 1.575105, train error 56.30, test loss 1.455353, test error 55.92\n", + "Epoch 900, train loss 1.579539, train error 56.15, test loss 1.449462, test error 55.67\n", + "Training model with 8 hidden variables\n", + "Epoch 0, train loss 2.278947, train error 85.18, test loss 2.266557, test error 84.68\n", + "Epoch 100, train loss 1.593652, train error 55.60, test loss 1.449531, test error 56.00\n", + "Epoch 200, train loss 1.557044, train error 53.92, test loss 1.453447, test error 55.03\n", + "Epoch 300, train loss 1.544829, train error 53.38, test loss 1.457726, test error 55.22\n", + "Epoch 400, train loss 1.538902, train error 54.05, test loss 1.456941, test error 55.33\n", + "Epoch 500, train loss 1.538077, train error 54.28, test loss 1.469433, test error 56.20\n", + "Epoch 600, train loss 1.529778, train error 53.60, test loss 1.470659, test error 55.72\n", + "Epoch 700, train loss 1.522470, train error 53.42, test loss 1.476947, test error 55.95\n", + "Epoch 800, train loss 1.520377, train error 53.22, test loss 1.481796, test error 56.55\n", + "Epoch 900, train loss 1.522042, train error 53.42, test loss 1.489359, test error 56.15\n", + "Training model with 10 hidden variables\n", + "Epoch 0, train loss 2.250067, train error 83.80, test loss 2.238750, test error 81.28\n", + "Epoch 100, train loss 1.570056, train error 55.80, test loss 1.444564, test error 54.62\n", + "Epoch 200, train loss 1.512308, train error 53.33, test loss 1.417375, test error 53.70\n", + "Epoch 300, train loss 1.485546, train error 52.62, test loss 1.415674, test error 53.62\n", + "Epoch 400, train loss 1.465965, train error 50.22, test loss 1.415631, test error 52.95\n", + "Epoch 500, train loss 1.450910, train error 50.50, test loss 1.412722, test error 52.70\n", + "Epoch 600, train loss 1.437163, train error 49.65, test loss 1.409063, test error 52.92\n", + "Epoch 700, train loss 1.430005, train error 49.38, test loss 1.402299, test error 52.83\n", + "Epoch 800, train loss 1.420938, train error 49.17, test loss 1.398151, test error 52.22\n", + "Epoch 900, train loss 1.417372, train error 49.25, test loss 1.407553, test error 52.58\n", + "Training model with 14 hidden variables\n", + "Epoch 0, train loss 2.250147, train error 84.45, test loss 2.224496, test error 83.50\n", + "Epoch 100, train loss 1.467325, train error 49.22, test loss 1.371005, test error 51.35\n", + "Epoch 200, train loss 1.371697, train error 46.47, test loss 1.341893, test error 49.60\n", + "Epoch 300, train loss 1.338310, train error 45.25, test loss 1.345958, test error 49.47\n", + "Epoch 400, train loss 1.329984, train error 44.95, test loss 1.364456, test error 49.85\n", + "Epoch 500, train loss 1.312067, train error 44.70, test loss 1.368620, test error 49.67\n", + "Epoch 600, train loss 1.307369, train error 44.00, test loss 1.383248, test error 49.97\n", + "Epoch 700, train loss 1.302358, train error 44.10, test loss 1.384645, test error 49.78\n", + "Epoch 800, train loss 1.309552, train error 44.42, test loss 1.399194, test error 50.60\n", + "Epoch 900, train loss 1.305113, train error 43.70, test loss 1.416064, test error 51.28\n", + "Training model with 18 hidden variables\n", + "Epoch 0, train loss 2.159715, train error 80.35, test loss 2.139665, test error 80.20\n", + "Epoch 100, train loss 1.385202, train error 47.75, test loss 1.362450, test error 51.58\n", + "Epoch 200, train loss 1.265389, train error 43.55, test loss 1.335479, test error 49.20\n", + "Epoch 300, train loss 1.224858, train error 41.67, test loss 1.384820, test error 48.47\n", + "Epoch 400, train loss 1.186068, train error 40.70, test loss 1.428581, test error 49.33\n", + "Epoch 500, train loss 1.167748, train error 39.30, test loss 1.472766, test error 49.95\n", + "Epoch 600, train loss 1.160923, train error 38.45, test loss 1.499211, test error 50.65\n", + "Epoch 700, train loss 1.145295, train error 38.40, test loss 1.528852, test error 50.67\n", + "Epoch 800, train loss 1.141608, train error 38.08, test loss 1.569496, test error 51.33\n", + "Epoch 900, train loss 1.132156, train error 38.22, test loss 1.602549, test error 51.35\n", + "Training model with 22 hidden variables\n", + "Epoch 0, train loss 2.211529, train error 79.80, test loss 2.206877, test error 79.95\n", + "Epoch 100, train loss 1.282252, train error 42.90, test loss 1.323739, test error 48.38\n", + "Epoch 200, train loss 1.161285, train error 38.65, test loss 1.423607, test error 50.78\n", + "Epoch 300, train loss 1.103865, train error 36.65, test loss 1.482603, test error 50.70\n", + "Epoch 400, train loss 1.070859, train error 36.42, test loss 1.543965, test error 51.38\n", + "Epoch 500, train loss 1.044615, train error 34.82, test loss 1.574064, test error 51.38\n", + "Epoch 600, train loss 1.030322, train error 34.82, test loss 1.633871, test error 51.47\n", + "Epoch 700, train loss 1.016668, train error 33.93, test loss 1.711600, test error 52.30\n", + "Epoch 800, train loss 0.997430, train error 32.82, test loss 1.763077, test error 52.95\n", + "Epoch 900, train loss 1.001320, train error 33.82, test loss 1.814692, test error 52.95\n", + "Training model with 26 hidden variables\n", + "Epoch 0, train loss 2.200904, train error 81.43, test loss 2.190686, test error 80.72\n", + "Epoch 100, train loss 1.189043, train error 39.80, test loss 1.398701, test error 51.97\n", + "Epoch 200, train loss 1.032893, train error 34.72, test loss 1.504426, test error 52.15\n", + "Epoch 300, train loss 0.975272, train error 33.55, test loss 1.621815, test error 53.30\n", + "Epoch 400, train loss 0.939880, train error 32.05, test loss 1.739801, test error 54.03\n", + "Epoch 500, train loss 0.907610, train error 30.85, test loss 1.822161, test error 54.30\n", + "Epoch 600, train loss 0.893433, train error 30.55, test loss 1.929679, test error 54.80\n", + "Epoch 700, train loss 0.864689, train error 29.20, test loss 1.968643, test error 54.75\n", + "Epoch 800, train loss 0.847545, train error 28.82, test loss 2.045258, test error 54.97\n", + "Epoch 900, train loss 0.843597, train error 28.97, test loss 2.113636, test error 55.55\n", + "Training model with 30 hidden variables\n", + "Epoch 0, train loss 2.105095, train error 78.75, test loss 2.046595, test error 77.50\n", + "Epoch 100, train loss 1.139368, train error 36.85, test loss 1.384092, test error 50.33\n", + "Epoch 200, train loss 0.948281, train error 31.70, test loss 1.521346, test error 50.53\n", + "Epoch 300, train loss 0.866517, train error 28.10, test loss 1.689052, test error 51.30\n", + "Epoch 400, train loss 0.822552, train error 26.95, test loss 1.871116, test error 53.58\n", + "Epoch 500, train loss 0.793155, train error 26.53, test loss 2.016083, test error 53.45\n", + "Epoch 600, train loss 0.747390, train error 25.15, test loss 2.175583, test error 54.60\n", + "Epoch 700, train loss 0.739243, train error 24.82, test loss 2.375611, test error 55.90\n", + "Epoch 800, train loss 0.705187, train error 23.97, test loss 2.521607, test error 56.12\n", + "Epoch 900, train loss 0.678345, train error 22.30, test loss 2.623145, test error 56.05\n", + "Training model with 35 hidden variables\n", + "Epoch 0, train loss 2.133684, train error 77.95, test loss 2.105965, test error 78.03\n", + "Epoch 100, train loss 1.025999, train error 34.07, test loss 1.387318, test error 49.42\n", + "Epoch 200, train loss 0.801432, train error 27.22, test loss 1.734866, test error 52.33\n", + "Epoch 300, train loss 0.665120, train error 22.30, test loss 2.111001, test error 54.28\n", + "Epoch 400, train loss 0.603412, train error 20.30, test loss 2.516824, test error 55.53\n", + "Epoch 500, train loss 0.557454, train error 19.50, test loss 2.875285, test error 55.90\n", + "Epoch 600, train loss 0.541823, train error 18.70, test loss 3.230060, test error 56.78\n", + "Epoch 700, train loss 0.489021, train error 16.43, test loss 3.542463, test error 56.67\n", + "Epoch 800, train loss 0.487397, train error 16.50, test loss 3.868127, test error 56.67\n", + "Epoch 900, train loss 0.452621, train error 15.62, test loss 4.107164, test error 57.90\n", + "Training model with 40 hidden variables\n", + "Epoch 0, train loss 2.131664, train error 78.45, test loss 2.104819, test error 79.57\n", + "Epoch 100, train loss 0.916092, train error 29.90, test loss 1.443106, test error 50.62\n", + "Epoch 200, train loss 0.637736, train error 20.70, test loss 1.931214, test error 52.22\n", + "Epoch 300, train loss 0.477490, train error 15.45, test loss 2.592646, test error 54.35\n", + "Epoch 400, train loss 0.375117, train error 11.97, test loss 3.317954, test error 55.75\n", + "Epoch 500, train loss 0.320583, train error 10.82, test loss 4.105999, test error 56.25\n", + "Epoch 600, train loss 0.282808, train error 9.62, test loss 4.787386, test error 56.17\n", + "Epoch 700, train loss 0.270115, train error 9.78, test loss 5.615366, test error 57.00\n", + "Epoch 800, train loss 0.271310, train error 10.05, test loss 6.246390, test error 56.17\n", + "Epoch 900, train loss 0.298141, train error 11.35, test loss 6.893893, test error 56.62\n", + "Training model with 45 hidden variables\n", + "Epoch 0, train loss 2.124081, train error 78.05, test loss 2.105962, test error 77.93\n", + "Epoch 100, train loss 0.826444, train error 26.68, test loss 1.503941, test error 49.47\n", + "Epoch 200, train loss 0.488764, train error 15.75, test loss 2.213665, test error 52.97\n", + "Epoch 300, train loss 0.339090, train error 11.00, test loss 3.221352, test error 54.25\n", + "Epoch 400, train loss 0.239335, train error 8.18, test loss 4.424680, test error 55.75\n", + "Epoch 500, train loss 0.163451, train error 4.93, test loss 5.617220, test error 55.38\n", + "Epoch 600, train loss 0.099984, train error 3.05, test loss 6.670816, test error 55.85\n", + "Epoch 700, train loss 0.015253, train error 0.00, test loss 7.652980, test error 55.70\n", + "Epoch 800, train loss 0.008259, train error 0.00, test loss 8.460521, test error 55.85\n", + "Epoch 900, train loss 0.005923, train error 0.00, test loss 8.991735, test error 55.92\n", + "Training model with 50 hidden variables\n", + "Epoch 0, train loss 2.164114, train error 78.28, test loss 2.153555, test error 78.55\n", + "Epoch 100, train loss 0.760776, train error 23.62, test loss 1.521621, test error 49.42\n", + "Epoch 200, train loss 0.388589, train error 11.07, test loss 2.433003, test error 52.42\n", + "Epoch 300, train loss 0.212128, train error 5.60, test loss 3.755425, test error 54.70\n", + "Epoch 400, train loss 0.101783, train error 2.12, test loss 5.258374, test error 55.00\n", + "Epoch 500, train loss 0.024106, train error 0.03, test loss 6.485045, test error 56.25\n", + "Epoch 600, train loss 0.010166, train error 0.00, test loss 7.247528, test error 56.03\n", + "Epoch 700, train loss 0.007013, train error 0.00, test loss 7.764050, test error 56.17\n", + "Epoch 800, train loss 0.005290, train error 0.00, test loss 8.180995, test error 56.25\n", + "Epoch 900, train loss 0.004218, train error 0.00, test loss 8.507399, test error 56.30\n", + "Training model with 60 hidden variables\n", + "Epoch 0, train loss 2.054841, train error 76.82, test loss 2.010036, test error 77.35\n", + "Epoch 100, train loss 0.533140, train error 15.78, test loss 1.685822, test error 50.55\n", + "Epoch 200, train loss 0.092306, train error 0.62, test loss 3.330975, test error 53.12\n", + "Epoch 300, train loss 0.018351, train error 0.00, test loss 4.631015, test error 54.05\n", + "Epoch 400, train loss 0.008906, train error 0.00, test loss 5.276160, test error 54.12\n", + "Epoch 500, train loss 0.005695, train error 0.00, test loss 5.702274, test error 54.15\n", + "Epoch 600, train loss 0.004078, train error 0.00, test loss 6.026574, test error 54.28\n", + "Epoch 700, train loss 0.003147, train error 0.00, test loss 6.281788, test error 54.22\n", + "Epoch 800, train loss 0.002537, train error 0.00, test loss 6.493565, test error 54.30\n", + "Epoch 900, train loss 0.002114, train error 0.00, test loss 6.669685, test error 54.45\n", + "Training model with 80 hidden variables\n", + "Epoch 0, train loss 2.028571, train error 74.40, test loss 1.985185, test error 75.28\n", + "Epoch 100, train loss 0.244489, train error 4.47, test loss 2.115000, test error 51.50\n", + "Epoch 200, train loss 0.020814, train error 0.00, test loss 3.415434, test error 51.40\n", + "Epoch 300, train loss 0.008094, train error 0.00, test loss 3.970142, test error 51.72\n", + "Epoch 400, train loss 0.004785, train error 0.00, test loss 4.299542, test error 51.83\n", + "Epoch 500, train loss 0.003310, train error 0.00, test loss 4.534029, test error 51.70\n", + "Epoch 600, train loss 0.002490, train error 0.00, test loss 4.716822, test error 51.78\n", + "Epoch 700, train loss 0.001978, train error 0.00, test loss 4.864742, test error 51.75\n", + "Epoch 800, train loss 0.001629, train error 0.00, test loss 4.988203, test error 51.80\n", + "Epoch 900, train loss 0.001378, train error 0.00, test loss 5.096081, test error 51.78\n", + "Training model with 120 hidden variables\n", + "Epoch 0, train loss 1.962679, train error 72.30, test loss 1.879106, test error 71.68\n", + "Epoch 100, train loss 0.048203, train error 0.00, test loss 2.235627, test error 49.70\n", + "Epoch 200, train loss 0.009333, train error 0.00, test loss 2.851372, test error 49.85\n", + "Epoch 300, train loss 0.004629, train error 0.00, test loss 3.129520, test error 49.95\n", + "Epoch 400, train loss 0.002970, train error 0.00, test loss 3.309089, test error 49.83\n", + "Epoch 500, train loss 0.002143, train error 0.00, test loss 3.442119, test error 49.92\n", + "Epoch 600, train loss 0.001658, train error 0.00, test loss 3.548133, test error 49.80\n", + "Epoch 700, train loss 0.001341, train error 0.00, test loss 3.635033, test error 49.72\n", + "Epoch 800, train loss 0.001120, train error 0.00, test loss 3.709674, test error 49.72\n", + "Epoch 900, train loss 0.000957, train error 0.00, test loss 3.774692, test error 49.65\n", + "Training model with 200 hidden variables\n", + "Epoch 0, train loss 1.913651, train error 70.00, test loss 1.842174, test error 72.20\n", + "Epoch 100, train loss 0.021130, train error 0.00, test loss 1.918857, test error 46.60\n", + "Epoch 200, train loss 0.006089, train error 0.00, test loss 2.209828, test error 46.88\n", + "Epoch 300, train loss 0.003314, train error 0.00, test loss 2.356240, test error 46.92\n", + "Epoch 400, train loss 0.002209, train error 0.00, test loss 2.455862, test error 46.85\n", + "Epoch 500, train loss 0.001633, train error 0.00, test loss 2.532098, test error 46.70\n", + "Epoch 600, train loss 0.001284, train error 0.00, test loss 2.592178, test error 46.67\n", + "Epoch 700, train loss 0.001051, train error 0.00, test loss 2.644237, test error 46.80\n", + "Epoch 800, train loss 0.000885, train error 0.00, test loss 2.687477, test error 46.78\n", + "Epoch 900, train loss 0.000762, train error 0.00, test loss 2.726535, test error 46.72\n", + "Training model with 360 hidden variables\n", + "Epoch 0, train loss 1.881302, train error 67.10, test loss 1.826783, test error 69.07\n", + "Epoch 100, train loss 0.012637, train error 0.00, test loss 1.789455, test error 46.08\n", + "Epoch 200, train loss 0.004398, train error 0.00, test loss 1.978857, test error 45.70\n", + "Epoch 300, train loss 0.002526, train error 0.00, test loss 2.081702, test error 45.45\n", + "Epoch 400, train loss 0.001732, train error 0.00, test loss 2.153348, test error 45.22\n", + "Epoch 500, train loss 0.001301, train error 0.00, test loss 2.208247, test error 45.28\n", + "Epoch 600, train loss 0.001034, train error 0.00, test loss 2.252916, test error 45.28\n", + "Epoch 700, train loss 0.000853, train error 0.00, test loss 2.290226, test error 45.40\n", + "Epoch 800, train loss 0.000724, train error 0.00, test loss 2.322630, test error 45.42\n", + "Epoch 900, train loss 0.000626, train error 0.00, test loss 2.351381, test error 45.33\n", + "Training model with 680 hidden variables\n", + "Epoch 0, train loss 1.808186, train error 63.28, test loss 1.783419, test error 66.82\n", + "Epoch 100, train loss 0.008044, train error 0.00, test loss 1.678780, test error 44.55\n", + "Epoch 200, train loss 0.003155, train error 0.00, test loss 1.810078, test error 44.47\n", + "Epoch 300, train loss 0.001885, train error 0.00, test loss 1.886664, test error 44.42\n", + "Epoch 400, train loss 0.001322, train error 0.00, test loss 1.939946, test error 44.35\n", + "Epoch 500, train loss 0.001008, train error 0.00, test loss 1.982137, test error 44.45\n", + "Epoch 600, train loss 0.000809, train error 0.00, test loss 2.015580, test error 44.45\n", + "Epoch 700, train loss 0.000673, train error 0.00, test loss 2.044045, test error 44.28\n", + "Epoch 800, train loss 0.000574, train error 0.00, test loss 2.068916, test error 44.28\n", + "Epoch 900, train loss 0.000499, train error 0.00, test loss 2.090607, test error 44.28\n" + ] + } + ] }, { "cell_type": "code", "source": [ + "import matplotlib.pyplot as plt\n", + "import numpy as np\n", + "\n", + "# Assuming data['y'] is available and contains the training examples\n", + "num_training_examples = len(data['y'])\n", + "\n", + "# Find the index where total_weights_all is closest to num_training_examples\n", + "closest_index = np.argmin(np.abs(np.array(total_weights_all) - num_training_examples))\n", + "\n", + "# Get the corresponding value of hidden variables\n", + "hidden_variable_at_num_training_examples = hidden_variables[closest_index]\n", + "\n", "# Plot the results\n", "fig, ax = plt.subplots()\n", - "ax.plot(hidden_variables, errors_train_all,'r-',label='train')\n", - "ax.plot(hidden_variables, errors_test_all,'b-',label='test')\n", - "ax.set_ylim(0,100);\n", - "ax.set_xlabel('No hidden variables'); ax.set_ylabel('Error')\n", + "ax.plot(hidden_variables, errors_train_all, 'r-', label='train')\n", + "ax.plot(hidden_variables, errors_test_all, 'b-', label='test')\n", + "\n", + "# Add a vertical line at the point where total weights equal the number of training examples\n", + "ax.axvline(x=hidden_variable_at_num_training_examples, color='g', linestyle='--', label='N(weights) = N(train)')\n", + "\n", + "ax.set_ylim(0, 100)\n", + "ax.set_xlabel('No. hidden variables')\n", + "ax.set_ylabel('Error')\n", "ax.legend()\n", "plt.show()\n" ], "metadata": { - "id": "Rw-iRboTXbck" + "id": "Rw-iRboTXbck", + "outputId": "4b2ec111-9a74-4e10-821c-6f5129c4e109", + "colab": { + "base_uri": "https://localhost:8080/", + "height": 455 + } }, - "execution_count": null, + "execution_count": 36, + "outputs": [ + { + "output_type": "display_data", + "data": { + "text/plain": [ + "

" + ], + "image/png": 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\n" + }, + "metadata": {} + } + ] + }, + { + "cell_type": "code", + "source": [], + "metadata": { + "id": "KT4X8_hE5NFb" + }, + "execution_count": 36, + "outputs": [] + }, + { + "cell_type": "code", + "source": [], + "metadata": { + "id": "iGKZSfVF2r4z" + }, + "execution_count": 36, "outputs": [] } ] From ccbbc4126e6e507d49e248f169a8aaa5fe153557 Mon Sep 17 00:00:00 2001 From: Chris Endemann Date: Wed, 10 Jul 2024 14:15:21 -0500 Subject: [PATCH 04/49] Update 8_3_Double_Descent.ipynb Apologies, accidentally removed the "open in colab" button in the pull request you accepted earlier today. This corrects the mistake! --- Notebooks/Chap08/8_3_Double_Descent.ipynb | 367 ++-------------------- 1 file changed, 34 insertions(+), 333 deletions(-) diff --git a/Notebooks/Chap08/8_3_Double_Descent.ipynb b/Notebooks/Chap08/8_3_Double_Descent.ipynb index aaac87e..d75e15f 100644 --- a/Notebooks/Chap08/8_3_Double_Descent.ipynb +++ b/Notebooks/Chap08/8_3_Double_Descent.ipynb @@ -4,7 +4,8 @@ "metadata": { "colab": { "provenance": [], - "gpuType": "T4" + "gpuType": "T4", + "include_colab_link": true }, "kernelspec": { "name": "python3", @@ -16,6 +17,16 @@ "accelerator": "GPU" }, "cells": [ + { + "cell_type": "markdown", + "metadata": { + "id": "view-in-github", + "colab_type": "text" + }, + "source": [ + "\"Open" + ] + }, { "cell_type": "markdown", "source": [ @@ -40,45 +51,10 @@ "!pip install git+https://github.com/greydanus/mnist1d" ], "metadata": { - "id": "fn9BP5N5TguP", - "outputId": "3ba15b8f-2395-4b1a-8a66-8ed80e9b5138", - "colab": { - "base_uri": "https://localhost:8080/" - } + "id": "fn9BP5N5TguP" }, - "execution_count": 29, - "outputs": [ - { - "output_type": "stream", - "name": "stdout", - "text": [ - "Collecting git+https://github.com/greydanus/mnist1d\n", - " Cloning https://github.com/greydanus/mnist1d to /tmp/pip-req-build-cbhxd1j8\n", - " Running command git clone --filter=blob:none --quiet https://github.com/greydanus/mnist1d /tmp/pip-req-build-cbhxd1j8\n", - " Resolved https://github.com/greydanus/mnist1d to commit 350929d12f4c9a4b7355e0c96604e41b9239bdb4\n", - " Installing build dependencies ... \u001b[?25l\u001b[?25hdone\n", - " Getting requirements to build wheel ... \u001b[?25l\u001b[?25hdone\n", - " Preparing metadata (pyproject.toml) ... \u001b[?25l\u001b[?25hdone\n", - "Requirement already satisfied: requests in /usr/local/lib/python3.10/dist-packages (from mnist1d==0.0.2.post9) (2.31.0)\n", - "Requirement already satisfied: numpy in /usr/local/lib/python3.10/dist-packages (from mnist1d==0.0.2.post9) (1.25.2)\n", - "Requirement already satisfied: matplotlib in /usr/local/lib/python3.10/dist-packages (from mnist1d==0.0.2.post9) (3.7.1)\n", - "Requirement already satisfied: scipy in /usr/local/lib/python3.10/dist-packages (from mnist1d==0.0.2.post9) (1.11.4)\n", - "Requirement already satisfied: contourpy>=1.0.1 in /usr/local/lib/python3.10/dist-packages (from matplotlib->mnist1d==0.0.2.post9) (1.2.1)\n", - "Requirement already satisfied: cycler>=0.10 in /usr/local/lib/python3.10/dist-packages (from matplotlib->mnist1d==0.0.2.post9) (0.12.1)\n", - "Requirement already satisfied: fonttools>=4.22.0 in /usr/local/lib/python3.10/dist-packages (from matplotlib->mnist1d==0.0.2.post9) (4.53.0)\n", - "Requirement already satisfied: kiwisolver>=1.0.1 in /usr/local/lib/python3.10/dist-packages (from matplotlib->mnist1d==0.0.2.post9) (1.4.5)\n", - "Requirement already satisfied: packaging>=20.0 in /usr/local/lib/python3.10/dist-packages (from matplotlib->mnist1d==0.0.2.post9) (24.1)\n", - "Requirement already satisfied: pillow>=6.2.0 in /usr/local/lib/python3.10/dist-packages (from matplotlib->mnist1d==0.0.2.post9) (9.4.0)\n", - "Requirement already satisfied: pyparsing>=2.3.1 in /usr/local/lib/python3.10/dist-packages (from matplotlib->mnist1d==0.0.2.post9) (3.1.2)\n", - "Requirement already satisfied: python-dateutil>=2.7 in /usr/local/lib/python3.10/dist-packages (from matplotlib->mnist1d==0.0.2.post9) (2.8.2)\n", - "Requirement already satisfied: charset-normalizer<4,>=2 in /usr/local/lib/python3.10/dist-packages (from requests->mnist1d==0.0.2.post9) (3.3.2)\n", - "Requirement already satisfied: idna<4,>=2.5 in /usr/local/lib/python3.10/dist-packages (from requests->mnist1d==0.0.2.post9) (3.7)\n", - "Requirement already satisfied: urllib3<3,>=1.21.1 in /usr/local/lib/python3.10/dist-packages (from requests->mnist1d==0.0.2.post9) (2.0.7)\n", - "Requirement already satisfied: certifi>=2017.4.17 in /usr/local/lib/python3.10/dist-packages (from requests->mnist1d==0.0.2.post9) (2024.6.2)\n", - "Requirement already satisfied: six>=1.5 in /usr/local/lib/python3.10/dist-packages (from python-dateutil>=2.7->matplotlib->mnist1d==0.0.2.post9) (1.16.0)\n" - ] - } - ] + "execution_count": null, + "outputs": [] }, { "cell_type": "code", @@ -97,22 +73,10 @@ "print('Using:', DEVICE)" ], "metadata": { - "id": "hFxuHpRqTgri", - "outputId": "2a246d1e-8568-4605-df0a-28dad08935ec", - "colab": { - "base_uri": "https://localhost:8080/" - } + "id": "hFxuHpRqTgri" }, - "execution_count": 30, - "outputs": [ - { - "output_type": "stream", - "name": "stdout", - "text": [ - "Using: cuda\n" - ] - } - ] + "execution_count": null, + "outputs": [] }, { "cell_type": "code", @@ -138,25 +102,10 @@ "print(\"Dimensionality of each example: {}\".format(data['x'].shape[-1]))" ], "metadata": { - "id": "PW2gyXL5UkLU", - "outputId": "a4186965-3025-4d00-b780-441baab421de", - "colab": { - "base_uri": "https://localhost:8080/" - } + "id": "PW2gyXL5UkLU" }, - "execution_count": 31, - "outputs": [ - { - "output_type": "stream", - "name": "stdout", - "text": [ - "Did or could not load data from ./mnist1d_data.pkl. Rebuilding dataset...\n", - "Examples in training set: 4000\n", - "Examples in test set: 4000\n", - "Dimensionality of each example: 40\n" - ] - } - ] + "execution_count": null, + "outputs": [] }, { "cell_type": "code", @@ -192,7 +141,7 @@ "metadata": { "id": "hAIvZOAlTnk9" }, - "execution_count": 32, + "execution_count": null, "outputs": [] }, { @@ -248,7 +197,7 @@ "metadata": { "id": "AazlQhheWmHk" }, - "execution_count": 33, + "execution_count": null, "outputs": [] }, { @@ -260,7 +209,7 @@ "metadata": { "id": "AQNCmFNV6JpV" }, - "execution_count": 34, + "execution_count": null, "outputs": [] }, { @@ -306,245 +255,13 @@ " # Store the results\n", " errors_train_all[c_hidden] = errors_train\n", " errors_test_all[c_hidden]= errors_test\n", - "\n", - "" + "\n" ], "metadata": { - "id": "K4OmBZGHWXpk", - "outputId": "62a7aaf0-793a-4ab2-960e-c1127f15717c", - "colab": { - "base_uri": "https://localhost:8080/" - } + "id": "K4OmBZGHWXpk" }, - "execution_count": 35, - "outputs": [ - { - "output_type": "stream", - "name": "stdout", - "text": [ - "Training model with 2 hidden variables\n", - "Epoch 0, train loss 2.280202, train error 89.05, test loss 2.276196, test error 89.18\n", - "Epoch 100, train loss 1.925899, train error 71.75, test loss 1.761014, test error 70.57\n", - "Epoch 200, train loss 1.912995, train error 71.20, test loss 1.747572, test error 69.07\n", - "Epoch 300, train loss 1.908751, train error 71.32, test loss 1.741206, test error 69.07\n", - "Epoch 400, train loss 1.906859, train error 71.05, test loss 1.737478, test error 70.00\n", - "Epoch 500, train loss 1.906671, train error 71.43, test loss 1.732123, test error 68.95\n", - "Epoch 600, train loss 1.903380, train error 71.28, test loss 1.741086, test error 69.65\n", - "Epoch 700, train loss 1.902236, train error 70.88, test loss 1.733666, test error 69.22\n", - "Epoch 800, train loss 1.903633, train error 71.88, test loss 1.734591, test error 70.57\n", - "Epoch 900, train loss 1.901804, train error 71.03, test loss 1.742573, test error 68.10\n", - "Training model with 4 hidden variables\n", - "Epoch 0, train loss 2.267358, train error 86.97, test loss 2.269902, test error 87.62\n", - "Epoch 100, train loss 1.716064, train error 62.22, test loss 1.519437, test error 58.25\n", - "Epoch 200, train loss 1.705652, train error 61.88, test loss 1.507422, test error 58.40\n", - "Epoch 300, train loss 1.696660, train error 60.62, test loss 1.507704, test error 57.58\n", - "Epoch 400, train loss 1.691092, train error 61.60, test loss 1.499592, test error 56.85\n", - "Epoch 500, train loss 1.684060, train error 61.05, test loss 1.513196, test error 57.95\n", - "Epoch 600, train loss 1.677890, train error 61.10, test loss 1.504343, test error 56.85\n", - "Epoch 700, train loss 1.677930, train error 61.10, test loss 1.509796, test error 58.15\n", - "Epoch 800, train loss 1.676683, train error 61.17, test loss 1.506790, test error 58.28\n", - "Epoch 900, train loss 1.676809, train error 61.53, test loss 1.505417, test error 57.72\n", - "Training model with 6 hidden variables\n", - "Epoch 0, train loss 2.290775, train error 89.18, test loss 2.284365, test error 87.57\n", - "Epoch 100, train loss 1.634348, train error 58.42, test loss 1.467763, test error 55.67\n", - "Epoch 200, train loss 1.602023, train error 57.25, test loss 1.456007, test error 55.38\n", - "Epoch 300, train loss 1.587963, train error 56.50, test loss 1.453522, test error 55.33\n", - "Epoch 400, train loss 1.587952, train error 56.55, test loss 1.453863, test error 54.92\n", - "Epoch 500, train loss 1.579495, train error 56.42, test loss 1.452398, test error 55.40\n", - "Epoch 600, train loss 1.590881, train error 56.67, test loss 1.473001, test error 56.28\n", - "Epoch 700, train loss 1.581218, train error 56.58, test loss 1.470745, test error 55.90\n", - "Epoch 800, train loss 1.575105, train error 56.30, test loss 1.455353, test error 55.92\n", - "Epoch 900, train loss 1.579539, train error 56.15, test loss 1.449462, test error 55.67\n", - "Training model with 8 hidden variables\n", - "Epoch 0, train loss 2.278947, train error 85.18, test loss 2.266557, test error 84.68\n", - "Epoch 100, train loss 1.593652, train error 55.60, test loss 1.449531, test error 56.00\n", - "Epoch 200, train loss 1.557044, train error 53.92, test loss 1.453447, test error 55.03\n", - "Epoch 300, train loss 1.544829, train error 53.38, test loss 1.457726, test error 55.22\n", - "Epoch 400, train loss 1.538902, train error 54.05, test loss 1.456941, test error 55.33\n", - "Epoch 500, train loss 1.538077, train error 54.28, test loss 1.469433, test error 56.20\n", - "Epoch 600, train loss 1.529778, train error 53.60, test loss 1.470659, test error 55.72\n", - "Epoch 700, train loss 1.522470, train error 53.42, test loss 1.476947, test error 55.95\n", - "Epoch 800, train loss 1.520377, train error 53.22, test loss 1.481796, test error 56.55\n", - "Epoch 900, train loss 1.522042, train error 53.42, test loss 1.489359, test error 56.15\n", - "Training model with 10 hidden variables\n", - "Epoch 0, train loss 2.250067, train error 83.80, test loss 2.238750, test error 81.28\n", - "Epoch 100, train loss 1.570056, train error 55.80, test loss 1.444564, test error 54.62\n", - "Epoch 200, train loss 1.512308, train error 53.33, test loss 1.417375, test error 53.70\n", - "Epoch 300, train loss 1.485546, train error 52.62, test loss 1.415674, test error 53.62\n", - "Epoch 400, train loss 1.465965, train error 50.22, test loss 1.415631, test error 52.95\n", - "Epoch 500, train loss 1.450910, train error 50.50, test loss 1.412722, test error 52.70\n", - "Epoch 600, train loss 1.437163, train error 49.65, test loss 1.409063, test error 52.92\n", - "Epoch 700, train loss 1.430005, train error 49.38, test loss 1.402299, test error 52.83\n", - "Epoch 800, train loss 1.420938, train error 49.17, test loss 1.398151, test error 52.22\n", - "Epoch 900, train loss 1.417372, train error 49.25, test loss 1.407553, test error 52.58\n", - "Training model with 14 hidden variables\n", - "Epoch 0, train loss 2.250147, train error 84.45, test loss 2.224496, test error 83.50\n", - "Epoch 100, train loss 1.467325, train error 49.22, test loss 1.371005, test error 51.35\n", - "Epoch 200, train loss 1.371697, train error 46.47, test loss 1.341893, test error 49.60\n", - "Epoch 300, train loss 1.338310, train error 45.25, test loss 1.345958, test error 49.47\n", - "Epoch 400, train loss 1.329984, train error 44.95, test loss 1.364456, test error 49.85\n", - "Epoch 500, train loss 1.312067, train error 44.70, test loss 1.368620, test error 49.67\n", - "Epoch 600, train loss 1.307369, train error 44.00, test loss 1.383248, test error 49.97\n", - "Epoch 700, train loss 1.302358, train error 44.10, test loss 1.384645, test error 49.78\n", - "Epoch 800, train loss 1.309552, train error 44.42, test loss 1.399194, test error 50.60\n", - "Epoch 900, train loss 1.305113, train error 43.70, test loss 1.416064, test error 51.28\n", - "Training model with 18 hidden variables\n", - "Epoch 0, train loss 2.159715, train error 80.35, test loss 2.139665, test error 80.20\n", - "Epoch 100, train loss 1.385202, train error 47.75, test loss 1.362450, test error 51.58\n", - "Epoch 200, train loss 1.265389, train error 43.55, test loss 1.335479, test error 49.20\n", - "Epoch 300, train loss 1.224858, train error 41.67, test loss 1.384820, test error 48.47\n", - "Epoch 400, train loss 1.186068, train error 40.70, test loss 1.428581, test error 49.33\n", - "Epoch 500, train loss 1.167748, train error 39.30, test loss 1.472766, test error 49.95\n", - "Epoch 600, train loss 1.160923, train error 38.45, test loss 1.499211, test error 50.65\n", - "Epoch 700, train loss 1.145295, train error 38.40, test loss 1.528852, test error 50.67\n", - "Epoch 800, train loss 1.141608, train error 38.08, test loss 1.569496, test error 51.33\n", - "Epoch 900, train loss 1.132156, train error 38.22, test loss 1.602549, test error 51.35\n", - "Training model with 22 hidden variables\n", - "Epoch 0, train loss 2.211529, train error 79.80, test loss 2.206877, test error 79.95\n", - "Epoch 100, train loss 1.282252, train error 42.90, test loss 1.323739, test error 48.38\n", - "Epoch 200, train loss 1.161285, train error 38.65, test loss 1.423607, test error 50.78\n", - "Epoch 300, train loss 1.103865, train error 36.65, test loss 1.482603, test error 50.70\n", - "Epoch 400, train loss 1.070859, train error 36.42, test loss 1.543965, test error 51.38\n", - "Epoch 500, train loss 1.044615, train error 34.82, test loss 1.574064, test error 51.38\n", - "Epoch 600, train loss 1.030322, train error 34.82, test loss 1.633871, test error 51.47\n", - "Epoch 700, train loss 1.016668, train error 33.93, test loss 1.711600, test error 52.30\n", - "Epoch 800, train loss 0.997430, train error 32.82, test loss 1.763077, test error 52.95\n", - "Epoch 900, train loss 1.001320, train error 33.82, test loss 1.814692, test error 52.95\n", - "Training model with 26 hidden variables\n", - "Epoch 0, train loss 2.200904, train error 81.43, test loss 2.190686, test error 80.72\n", - "Epoch 100, train loss 1.189043, train error 39.80, test loss 1.398701, test error 51.97\n", - "Epoch 200, train loss 1.032893, train error 34.72, test loss 1.504426, test error 52.15\n", - "Epoch 300, train loss 0.975272, train error 33.55, test loss 1.621815, test error 53.30\n", - "Epoch 400, train loss 0.939880, train error 32.05, test loss 1.739801, test error 54.03\n", - "Epoch 500, train loss 0.907610, train error 30.85, test loss 1.822161, test error 54.30\n", - "Epoch 600, train loss 0.893433, train error 30.55, test loss 1.929679, test error 54.80\n", - "Epoch 700, train loss 0.864689, train error 29.20, test loss 1.968643, test error 54.75\n", - "Epoch 800, train loss 0.847545, train error 28.82, test loss 2.045258, test error 54.97\n", - "Epoch 900, train loss 0.843597, train error 28.97, test loss 2.113636, test error 55.55\n", - "Training model with 30 hidden variables\n", - "Epoch 0, train loss 2.105095, train error 78.75, test loss 2.046595, test error 77.50\n", - "Epoch 100, train loss 1.139368, train error 36.85, test loss 1.384092, test error 50.33\n", - "Epoch 200, train loss 0.948281, train error 31.70, test loss 1.521346, test error 50.53\n", - "Epoch 300, train loss 0.866517, train error 28.10, test loss 1.689052, test error 51.30\n", - "Epoch 400, train loss 0.822552, train error 26.95, test loss 1.871116, test error 53.58\n", - "Epoch 500, train loss 0.793155, train error 26.53, test loss 2.016083, test error 53.45\n", - "Epoch 600, train loss 0.747390, train error 25.15, test loss 2.175583, test error 54.60\n", - "Epoch 700, train loss 0.739243, train error 24.82, test loss 2.375611, test error 55.90\n", - "Epoch 800, train loss 0.705187, train error 23.97, test loss 2.521607, test error 56.12\n", - "Epoch 900, train loss 0.678345, train error 22.30, test loss 2.623145, test error 56.05\n", - "Training model with 35 hidden variables\n", - "Epoch 0, train loss 2.133684, train error 77.95, test loss 2.105965, test error 78.03\n", - "Epoch 100, train loss 1.025999, train error 34.07, test loss 1.387318, test error 49.42\n", - "Epoch 200, train loss 0.801432, train error 27.22, test loss 1.734866, test error 52.33\n", - "Epoch 300, train loss 0.665120, train error 22.30, test loss 2.111001, test error 54.28\n", - "Epoch 400, train loss 0.603412, train error 20.30, test loss 2.516824, test error 55.53\n", - "Epoch 500, train loss 0.557454, train error 19.50, test loss 2.875285, test error 55.90\n", - "Epoch 600, train loss 0.541823, train error 18.70, test loss 3.230060, test error 56.78\n", - "Epoch 700, train loss 0.489021, train error 16.43, test loss 3.542463, test error 56.67\n", - "Epoch 800, train loss 0.487397, train error 16.50, test loss 3.868127, test error 56.67\n", - "Epoch 900, train loss 0.452621, train error 15.62, test loss 4.107164, test error 57.90\n", - "Training model with 40 hidden variables\n", - "Epoch 0, train loss 2.131664, train error 78.45, test loss 2.104819, test error 79.57\n", - "Epoch 100, train loss 0.916092, train error 29.90, test loss 1.443106, test error 50.62\n", - "Epoch 200, train loss 0.637736, train error 20.70, test loss 1.931214, test error 52.22\n", - "Epoch 300, train loss 0.477490, train error 15.45, test loss 2.592646, test error 54.35\n", - "Epoch 400, train loss 0.375117, train error 11.97, test loss 3.317954, test error 55.75\n", - "Epoch 500, train loss 0.320583, train error 10.82, test loss 4.105999, test error 56.25\n", - "Epoch 600, train loss 0.282808, train error 9.62, test loss 4.787386, test error 56.17\n", - "Epoch 700, train loss 0.270115, train error 9.78, test loss 5.615366, test error 57.00\n", - "Epoch 800, train loss 0.271310, train error 10.05, test loss 6.246390, test error 56.17\n", - "Epoch 900, train loss 0.298141, train error 11.35, test loss 6.893893, test error 56.62\n", - "Training model with 45 hidden variables\n", - "Epoch 0, train loss 2.124081, train error 78.05, test loss 2.105962, test error 77.93\n", - "Epoch 100, train loss 0.826444, train error 26.68, test loss 1.503941, test error 49.47\n", - "Epoch 200, train loss 0.488764, train error 15.75, test loss 2.213665, test error 52.97\n", - "Epoch 300, train loss 0.339090, train error 11.00, test loss 3.221352, test error 54.25\n", - "Epoch 400, train loss 0.239335, train error 8.18, test loss 4.424680, test error 55.75\n", - "Epoch 500, train loss 0.163451, train error 4.93, test loss 5.617220, test error 55.38\n", - "Epoch 600, train loss 0.099984, train error 3.05, test loss 6.670816, test error 55.85\n", - "Epoch 700, train loss 0.015253, train error 0.00, test loss 7.652980, test error 55.70\n", - "Epoch 800, train loss 0.008259, train error 0.00, test loss 8.460521, test error 55.85\n", - "Epoch 900, train loss 0.005923, train error 0.00, test loss 8.991735, test error 55.92\n", - "Training model with 50 hidden variables\n", - "Epoch 0, train loss 2.164114, train error 78.28, test loss 2.153555, test error 78.55\n", - "Epoch 100, train loss 0.760776, train error 23.62, test loss 1.521621, test error 49.42\n", - "Epoch 200, train loss 0.388589, train error 11.07, test loss 2.433003, test error 52.42\n", - "Epoch 300, train loss 0.212128, train error 5.60, test loss 3.755425, test error 54.70\n", - "Epoch 400, train loss 0.101783, train error 2.12, test loss 5.258374, test error 55.00\n", - "Epoch 500, train loss 0.024106, train error 0.03, test loss 6.485045, test error 56.25\n", - "Epoch 600, train loss 0.010166, train error 0.00, test loss 7.247528, test error 56.03\n", - "Epoch 700, train loss 0.007013, train error 0.00, test loss 7.764050, test error 56.17\n", - "Epoch 800, train loss 0.005290, train error 0.00, test loss 8.180995, test error 56.25\n", - "Epoch 900, train loss 0.004218, train error 0.00, test loss 8.507399, test error 56.30\n", - "Training model with 60 hidden variables\n", - "Epoch 0, train loss 2.054841, train error 76.82, test loss 2.010036, test error 77.35\n", - "Epoch 100, train loss 0.533140, train error 15.78, test loss 1.685822, test error 50.55\n", - "Epoch 200, train loss 0.092306, train error 0.62, test loss 3.330975, test error 53.12\n", - "Epoch 300, train loss 0.018351, train error 0.00, test loss 4.631015, test error 54.05\n", - "Epoch 400, train loss 0.008906, train error 0.00, test loss 5.276160, test error 54.12\n", - "Epoch 500, train loss 0.005695, train error 0.00, test loss 5.702274, test error 54.15\n", - "Epoch 600, train loss 0.004078, train error 0.00, test loss 6.026574, test error 54.28\n", - "Epoch 700, train loss 0.003147, train error 0.00, test loss 6.281788, test error 54.22\n", - "Epoch 800, train loss 0.002537, train error 0.00, test loss 6.493565, test error 54.30\n", - "Epoch 900, train loss 0.002114, train error 0.00, test loss 6.669685, test error 54.45\n", - "Training model with 80 hidden variables\n", - "Epoch 0, train loss 2.028571, train error 74.40, test loss 1.985185, test error 75.28\n", - "Epoch 100, train loss 0.244489, train error 4.47, test loss 2.115000, test error 51.50\n", - "Epoch 200, train loss 0.020814, train error 0.00, test loss 3.415434, test error 51.40\n", - "Epoch 300, train loss 0.008094, train error 0.00, test loss 3.970142, test error 51.72\n", - "Epoch 400, train loss 0.004785, train error 0.00, test loss 4.299542, test error 51.83\n", - "Epoch 500, train loss 0.003310, train error 0.00, test loss 4.534029, test error 51.70\n", - "Epoch 600, train loss 0.002490, train error 0.00, test loss 4.716822, test error 51.78\n", - "Epoch 700, train loss 0.001978, train error 0.00, test loss 4.864742, test error 51.75\n", - "Epoch 800, train loss 0.001629, train error 0.00, test loss 4.988203, test error 51.80\n", - "Epoch 900, train loss 0.001378, train error 0.00, test loss 5.096081, test error 51.78\n", - "Training model with 120 hidden variables\n", - "Epoch 0, train loss 1.962679, train error 72.30, test loss 1.879106, test error 71.68\n", - "Epoch 100, train loss 0.048203, train error 0.00, test loss 2.235627, test error 49.70\n", - "Epoch 200, train loss 0.009333, train error 0.00, test loss 2.851372, test error 49.85\n", - "Epoch 300, train loss 0.004629, train error 0.00, test loss 3.129520, test error 49.95\n", - "Epoch 400, train loss 0.002970, train error 0.00, test loss 3.309089, test error 49.83\n", - "Epoch 500, train loss 0.002143, train error 0.00, test loss 3.442119, test error 49.92\n", - "Epoch 600, train loss 0.001658, train error 0.00, test loss 3.548133, test error 49.80\n", - "Epoch 700, train loss 0.001341, train error 0.00, test loss 3.635033, test error 49.72\n", - "Epoch 800, train loss 0.001120, train error 0.00, test loss 3.709674, test error 49.72\n", - "Epoch 900, train loss 0.000957, train error 0.00, test loss 3.774692, test error 49.65\n", - "Training model with 200 hidden variables\n", - "Epoch 0, train loss 1.913651, train error 70.00, test loss 1.842174, test error 72.20\n", - "Epoch 100, train loss 0.021130, train error 0.00, test loss 1.918857, test error 46.60\n", - "Epoch 200, train loss 0.006089, train error 0.00, test loss 2.209828, test error 46.88\n", - "Epoch 300, train loss 0.003314, train error 0.00, test loss 2.356240, test error 46.92\n", - "Epoch 400, train loss 0.002209, train error 0.00, test loss 2.455862, test error 46.85\n", - "Epoch 500, train loss 0.001633, train error 0.00, test loss 2.532098, test error 46.70\n", - "Epoch 600, train loss 0.001284, train error 0.00, test loss 2.592178, test error 46.67\n", - "Epoch 700, train loss 0.001051, train error 0.00, test loss 2.644237, test error 46.80\n", - "Epoch 800, train loss 0.000885, train error 0.00, test loss 2.687477, test error 46.78\n", - "Epoch 900, train loss 0.000762, train error 0.00, test loss 2.726535, test error 46.72\n", - "Training model with 360 hidden variables\n", - "Epoch 0, train loss 1.881302, train error 67.10, test loss 1.826783, test error 69.07\n", - "Epoch 100, train loss 0.012637, train error 0.00, test loss 1.789455, test error 46.08\n", - "Epoch 200, train loss 0.004398, train error 0.00, test loss 1.978857, test error 45.70\n", - "Epoch 300, train loss 0.002526, train error 0.00, test loss 2.081702, test error 45.45\n", - "Epoch 400, train loss 0.001732, train error 0.00, test loss 2.153348, test error 45.22\n", - "Epoch 500, train loss 0.001301, train error 0.00, test loss 2.208247, test error 45.28\n", - "Epoch 600, train loss 0.001034, train error 0.00, test loss 2.252916, test error 45.28\n", - "Epoch 700, train loss 0.000853, train error 0.00, test loss 2.290226, test error 45.40\n", - "Epoch 800, train loss 0.000724, train error 0.00, test loss 2.322630, test error 45.42\n", - "Epoch 900, train loss 0.000626, train error 0.00, test loss 2.351381, test error 45.33\n", - "Training model with 680 hidden variables\n", - "Epoch 0, train loss 1.808186, train error 63.28, test loss 1.783419, test error 66.82\n", - "Epoch 100, train loss 0.008044, train error 0.00, test loss 1.678780, test error 44.55\n", - "Epoch 200, train loss 0.003155, train error 0.00, test loss 1.810078, test error 44.47\n", - "Epoch 300, train loss 0.001885, train error 0.00, test loss 1.886664, test error 44.42\n", - "Epoch 400, train loss 0.001322, train error 0.00, test loss 1.939946, test error 44.35\n", - "Epoch 500, train loss 0.001008, train error 0.00, test loss 1.982137, test error 44.45\n", - "Epoch 600, train loss 0.000809, train error 0.00, test loss 2.015580, test error 44.45\n", - "Epoch 700, train loss 0.000673, train error 0.00, test loss 2.044045, test error 44.28\n", - "Epoch 800, train loss 0.000574, train error 0.00, test loss 2.068916, test error 44.28\n", - "Epoch 900, train loss 0.000499, train error 0.00, test loss 2.090607, test error 44.28\n" - ] - } - ] + "execution_count": null, + "outputs": [] }, { "cell_type": "code", @@ -576,26 +293,10 @@ "plt.show()\n" ], "metadata": { - "id": "Rw-iRboTXbck", - "outputId": "4b2ec111-9a74-4e10-821c-6f5129c4e109", - "colab": { - "base_uri": "https://localhost:8080/", - "height": 455 - } + "id": "Rw-iRboTXbck" }, - "execution_count": 36, - "outputs": [ - { - "output_type": "display_data", - "data": { - "text/plain": [ - "
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\n" - }, - "metadata": {} - } - ] + "execution_count": null, + "outputs": [] }, { "cell_type": "code", @@ -603,7 +304,7 @@ "metadata": { "id": "KT4X8_hE5NFb" }, - "execution_count": 36, + "execution_count": null, "outputs": [] }, { @@ -612,8 +313,8 @@ "metadata": { "id": "iGKZSfVF2r4z" }, - "execution_count": 36, + "execution_count": null, "outputs": [] } ] -} \ No newline at end of file +} From 0d135f1ee7fc6452b1afede82ee4f73f8683266f Mon Sep 17 00:00:00 2001 From: udlbook <110402648+udlbook@users.noreply.github.com> Date: Fri, 19 Jul 2024 15:55:44 -0400 Subject: [PATCH 05/49] Fixed problems with MNIST1D --- Notebooks/Chap20/20_3_Lottery_Tickets.ipynb | 17 ++++++++++++----- 1 file changed, 12 insertions(+), 5 deletions(-) diff --git a/Notebooks/Chap20/20_3_Lottery_Tickets.ipynb b/Notebooks/Chap20/20_3_Lottery_Tickets.ipynb index 182cfa3..aafad0f 100644 --- a/Notebooks/Chap20/20_3_Lottery_Tickets.ipynb +++ b/Notebooks/Chap20/20_3_Lottery_Tickets.ipynb @@ -44,7 +44,8 @@ }, "source": [ "# Run this if you're in a Colab to install MNIST 1D repository\n", - "!pip install git+https://github.com/greydanus/mnist1d" + "!pip install git+https://github.com/greydanus/mnist1d\n", + "!git clone https://github.com/greydanus/mnist1d" ], "execution_count": null, "outputs": [] @@ -95,6 +96,12 @@ "id": "I-vm_gh5xTJs" }, "source": [ + "from mnist1d.data import get_dataset, get_dataset_args\n", + "from mnist1d.utils import set_seed, to_pickle, from_pickle\n", + "\n", + "import sys ; sys.path.append('./mnist1d/notebooks')\n", + "from train import get_model_args, train_model\n", + "\n", "args = mnist1d.get_dataset_args()\n", "data = mnist1d.get_dataset(args=args) # by default, this will download a pre-made dataset from the GitHub repo\n", "\n", @@ -210,7 +217,7 @@ " # we would return [1,1,0,0,1]\n", " # Remember that these are torch tensors and not numpy arrays\n", " # Replace this function:\n", - " mask = torch.ones_like(scores)\n", + " mask = torch.ones_like(absolute_weights)\n", "\n", "\n", " return mask" @@ -237,7 +244,6 @@ "def find_lottery_ticket(model, dataset, args, sparsity_schedule, criteria_fn=None, **kwargs):\n", "\n", " criteria_fn = lambda init_params, final_params: final_params.abs()\n", - "\n", " init_params = model.get_layer_vecs()\n", " stats = {'train_losses':[], 'test_losses':[], 'train_accs':[], 'test_accs':[]}\n", " models = []\n", @@ -253,7 +259,7 @@ " model.set_layer_masks(masks)\n", "\n", " # training process\n", - " results = mnist1d.train_model(dataset, model, args)\n", + " results = train_model(dataset, model, args)\n", " model = results['checkpoints'][-1]\n", "\n", " # store stats\n", @@ -291,7 +297,8 @@ }, "source": [ "# train settings\n", - "model_args = mnist1d.get_model_args()\n", + "from train import get_model_args, train_model\n", + "model_args = get_model_args()\n", "model_args.total_steps = 1501\n", "model_args.hidden_size = 500\n", "model_args.print_every = 5000 # print never\n", From afd20d03640a61178555a79fa90bd923210538c7 Mon Sep 17 00:00:00 2001 From: udlbook <110402648+udlbook@users.noreply.github.com> Date: Mon, 22 Jul 2024 15:03:17 -0400 Subject: [PATCH 06/49] Update 17_1_Latent_Variable_Models.ipynb --- Notebooks/Chap17/17_1_Latent_Variable_Models.ipynb | 2 +- 1 file changed, 1 insertion(+), 1 deletion(-) diff --git a/Notebooks/Chap17/17_1_Latent_Variable_Models.ipynb b/Notebooks/Chap17/17_1_Latent_Variable_Models.ipynb index b79df19..a8cba0a 100644 --- a/Notebooks/Chap17/17_1_Latent_Variable_Models.ipynb +++ b/Notebooks/Chap17/17_1_Latent_Variable_Models.ipynb @@ -55,7 +55,7 @@ "Pr(z) = \\text{Norm}_{z}[0,1]\n", "\\end{equation}\n", "\n", - "As in figure 17.2, we'll assume that the output is two dimensional, we we need to define a function that maps from the 1D latent variable to two dimensions. Usually, we would use a neural network, but in this case, we'll just define an arbitrary relationship.\n", + "As in figure 17.2, we'll assume that the output is two dimensional, we need to define a function that maps from the 1D latent variable to two dimensions. Usually, we would use a neural network, but in this case, we'll just define an arbitrary relationship.\n", "\n", "\\begin{align}\n", "x_{1} &=& 0.5\\cdot\\exp\\Bigl[\\sin\\bigl[2+ 3.675 z \\bigr]\\Bigr]\\\\\n", From 37011065d7e758c012efadbd75bed0913348a39a Mon Sep 17 00:00:00 2001 From: udlbook <110402648+udlbook@users.noreply.github.com> Date: Mon, 22 Jul 2024 17:09:15 -0400 Subject: [PATCH 07/49] Add files via upload --- UDL_Errata.pdf | Bin 1171121 -> 1184552 bytes 1 file changed, 0 insertions(+), 0 deletions(-) diff --git a/UDL_Errata.pdf b/UDL_Errata.pdf index 04ffe5ce29ad140791839d3e64b3b5b28e576339..743353922f4c6ff639345cda80f3a53e662f5898 100644 GIT binary patch delta 166073 zcmZs>b8zQR&@~#{+}QRvwr$(CjZd)2h8r6jb7R}KHqOS`U}N*{?|q+Jx1Opyf6bif z>Y39seQKtv>$|1POJ&fhR3xOB*qAvHsCpj@KM+8mM08?N9u7`U5GjicsXi%)m6eB- zo1K%?kd#H46a?ZRWsxN1;^bvxBV|z{1@V%yXppk8@ql=_SV@J15GP72PM4$caJn~VW2CIDh3W#eW0&y}o`l`|pa&cR*nsT#o@vxbhnzLG&@tSgQaIpyq 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zBk(Fq19KA;J&@n^Vtm$f@m-+gg66!%l< q- Date: Mon, 22 Jul 2024 18:42:22 -0400 Subject: [PATCH 09/49] Update index.html --- index.html | 1 + 1 file changed, 1 insertion(+) diff --git a/index.html b/index.html index 523f9c4..37be10d 100644 --- a/index.html +++ b/index.html @@ -10,6 +10,7 @@ href="https://fonts.googleapis.com/css2?family=Encode+Sans+Expanded:wght@400;700&display=swap" rel="stylesheet" /> + Understanding Deep Learning From 5f16e0f9bcae7f3047c924428a025ad29b55b9ae Mon Sep 17 00:00:00 2001 From: udlbook <110402648+udlbook@users.noreply.github.com> Date: Mon, 29 Jul 2024 18:52:49 -0400 Subject: [PATCH 10/49] Fixed problem with example label. --- Notebooks/Chap03/3_1_Shallow_Networks_I.ipynb | 2 +- 1 file changed, 1 insertion(+), 1 deletion(-) diff --git a/Notebooks/Chap03/3_1_Shallow_Networks_I.ipynb b/Notebooks/Chap03/3_1_Shallow_Networks_I.ipynb index 80047d3..1c4d9e6 100644 --- a/Notebooks/Chap03/3_1_Shallow_Networks_I.ipynb +++ b/Notebooks/Chap03/3_1_Shallow_Networks_I.ipynb @@ -202,7 +202,7 @@ }, "source": [ "If your code is correct, then the final output should look like this:\n", - "" + "" ] }, { From c5fafbca972a91a36daf861f508ce1f619f426e9 Mon Sep 17 00:00:00 2001 From: =?UTF-8?q?Jan=20Hr=C4=8Dek?= Date: Tue, 30 Jul 2024 11:16:30 +0200 Subject: [PATCH 11/49] Fix duplicate word occurrences in notebooks --- Blogs/BorealisGradientFlow.ipynb | 4 ++-- Blogs/BorealisNTK.ipynb | 4 ++-- CM20315/CM20315_Coursework_IV.ipynb | 2 +- CM20315/CM20315_Loss_II.ipynb | 2 +- CM20315/CM20315_Loss_III.ipynb | 2 +- CM20315_2023/CM20315_Coursework_IV.ipynb | 4 ++-- CM20315_2023/CM20315_Coursework_V_2023.ipynb | 4 ++-- Notebooks/Chap05/5_3_Multiclass_Cross_entropy_Loss.ipynb | 4 ++-- Notebooks/Chap08/8_4_High_Dimensional_Spaces.ipynb | 2 +- Notebooks/Chap19/19_2_Dynamic_Programming.ipynb | 4 ++-- Notebooks/Chap21/21_1_Bias_Mitigation.ipynb | 4 ++-- 11 files changed, 18 insertions(+), 18 deletions(-) diff --git a/Blogs/BorealisGradientFlow.ipynb b/Blogs/BorealisGradientFlow.ipynb index 483bd0e..53cb3cb 100644 --- a/Blogs/BorealisGradientFlow.ipynb +++ b/Blogs/BorealisGradientFlow.ipynb @@ -31,7 +31,7 @@ "source": [ "# Gradient flow\n", "\n", - "This notebook replicates some of the results in the the Borealis AI [blog](https://www.borealisai.com/research-blogs/gradient-flow/) on gradient flow. \n" + "This notebook replicates some of the results in the Borealis AI [blog](https://www.borealisai.com/research-blogs/gradient-flow/) on gradient flow. \n" ], "metadata": { "id": "ucrRRJ4dq8_d" @@ -398,4 +398,4 @@ "outputs": [] } ] -} \ No newline at end of file +} diff --git a/Blogs/BorealisNTK.ipynb b/Blogs/BorealisNTK.ipynb index d2062a8..ee697d8 100644 --- a/Blogs/BorealisNTK.ipynb +++ b/Blogs/BorealisNTK.ipynb @@ -166,7 +166,7 @@ { "cell_type": "markdown", "source": [ - "Routines to calculate the empirical and analytical NTK (i.e. the NTK with infinite hidden units) for the the shallow network" + "Routines to calculate the empirical and analytical NTK (i.e. the NTK with infinite hidden units) for the shallow network" ], "metadata": { "id": "mxW8E5kYIzlj" @@ -1106,4 +1106,4 @@ }, "nbformat": 4, "nbformat_minor": 0 -} \ No newline at end of file +} diff --git a/CM20315/CM20315_Coursework_IV.ipynb b/CM20315/CM20315_Coursework_IV.ipynb index dfdb5e9..e3e759e 100644 --- a/CM20315/CM20315_Coursework_IV.ipynb +++ b/CM20315/CM20315_Coursework_IV.ipynb @@ -128,7 +128,7 @@ "\n", "In part (b) of the practical we calculate the volume of a hypersphere of radius 0.5 (i.e., of diameter 1) as a function of the radius. You will find that the volume decreases to almost nothing in high dimensions. All of the volume is in the corners of the unit hypercube (which always has volume 1). Double weird.\n", "\n", - "Note that you you can check your answer by doing the calculation for 2D using the standard formula for the area of a circle and making sure it matches." + "Note that you can check your answer by doing the calculation for 2D using the standard formula for the area of a circle and making sure it matches." ], "metadata": { "id": "b2FYKV1SL4Z7" diff --git a/CM20315/CM20315_Loss_II.ipynb b/CM20315/CM20315_Loss_II.ipynb index 9846d61..d51cd70 100644 --- a/CM20315/CM20315_Loss_II.ipynb +++ b/CM20315/CM20315_Loss_II.ipynb @@ -199,7 +199,7 @@ { "cell_type": "markdown", "source": [ - "The left is model output and the right is the model output after the sigmoid has been applied, so it now lies in the range [0,1] and represents the probability, that y=1. The black dots show the training data. We'll compute the the likelihood and the negative log likelihood." + "The left is model output and the right is the model output after the sigmoid has been applied, so it now lies in the range [0,1] and represents the probability, that y=1. The black dots show the training data. We'll compute the likelihood and the negative log likelihood." ], "metadata": { "id": "MvVX6tl9AEXF" diff --git a/CM20315/CM20315_Loss_III.ipynb b/CM20315/CM20315_Loss_III.ipynb index ef914c9..dae3ae4 100644 --- a/CM20315/CM20315_Loss_III.ipynb +++ b/CM20315/CM20315_Loss_III.ipynb @@ -218,7 +218,7 @@ { "cell_type": "markdown", "source": [ - "The left is model output and the right is the model output after the softmax has been applied, so it now lies in the range [0,1] and represents the probability, that y=0 (red), 1 (green) and 2 (blue) The dots at the bottom show the training data with the same color scheme. So we want the red curve to be high where there are red dots, the green curve to be high where there are green dotsmand the blue curve to be high where there are blue dots We'll compute the the likelihood and the negative log likelihood." + "The left is model output and the right is the model output after the softmax has been applied, so it now lies in the range [0,1] and represents the probability, that y=0 (red), 1 (green) and 2 (blue) The dots at the bottom show the training data with the same color scheme. So we want the red curve to be high where there are red dots, the green curve to be high where there are green dotsmand the blue curve to be high where there are blue dots We'll compute the likelihood and the negative log likelihood." ], "metadata": { "id": "MvVX6tl9AEXF" diff --git a/CM20315_2023/CM20315_Coursework_IV.ipynb b/CM20315_2023/CM20315_Coursework_IV.ipynb index 8f98ad5..e44f330 100644 --- a/CM20315_2023/CM20315_Coursework_IV.ipynb +++ b/CM20315_2023/CM20315_Coursework_IV.ipynb @@ -128,7 +128,7 @@ "\n", "In part (b) of the practical we calculate the volume of a hypersphere of radius 0.5 (i.e., of diameter 1) as a function of the radius. You will find that the volume decreases to almost nothing in high dimensions. All of the volume is in the corners of the unit hypercube (which always has volume 1). Double weird.\n", "\n", - "Note that you you can check your answer by doing the calculation for 2D using the standard formula for the area of a circle and making sure it matches." + "Note that you can check your answer by doing the calculation for 2D using the standard formula for the area of a circle and making sure it matches." ], "metadata": { "id": "b2FYKV1SL4Z7" @@ -209,4 +209,4 @@ "outputs": [] } ] -} \ No newline at end of file +} diff --git a/CM20315_2023/CM20315_Coursework_V_2023.ipynb b/CM20315_2023/CM20315_Coursework_V_2023.ipynb index 483ea94..fb1dfde 100644 --- a/CM20315_2023/CM20315_Coursework_V_2023.ipynb +++ b/CM20315_2023/CM20315_Coursework_V_2023.ipynb @@ -214,7 +214,7 @@ { "cell_type": "code", "source": [ - "# Compute the derivative of the the loss with respect to the function output f_val\n", + "# Compute the derivative of the loss with respect to the function output f_val\n", "def dl_df(f_val,y):\n", " # Compute sigmoid of network output\n", " sig_f_val = sig(f_val)\n", @@ -522,4 +522,4 @@ "outputs": [] } ] -} \ No newline at end of file +} diff --git a/Notebooks/Chap05/5_3_Multiclass_Cross_entropy_Loss.ipynb b/Notebooks/Chap05/5_3_Multiclass_Cross_entropy_Loss.ipynb index dbab0a1..1021d74 100644 --- a/Notebooks/Chap05/5_3_Multiclass_Cross_entropy_Loss.ipynb +++ b/Notebooks/Chap05/5_3_Multiclass_Cross_entropy_Loss.ipynb @@ -211,7 +211,7 @@ "id": "MvVX6tl9AEXF" }, "source": [ - "The left is model output and the right is the model output after the softmax has been applied, so it now lies in the range [0,1] and represents the probability, that y=0 (red), 1 (green) and 2 (blue). The dots at the bottom show the training data with the same color scheme. So we want the red curve to be high where there are red dots, the green curve to be high where there are green dots, and the blue curve to be high where there are blue dots We'll compute the the likelihood and the negative log likelihood." + "The left is model output and the right is the model output after the softmax has been applied, so it now lies in the range [0,1] and represents the probability, that y=0 (red), 1 (green) and 2 (blue). The dots at the bottom show the training data with the same color scheme. So we want the red curve to be high where there are red dots, the green curve to be high where there are green dots, and the blue curve to be high where there are blue dots We'll compute the likelihood and the negative log likelihood." ] }, { @@ -460,4 +460,4 @@ }, "nbformat": 4, "nbformat_minor": 0 -} \ No newline at end of file +} diff --git a/Notebooks/Chap08/8_4_High_Dimensional_Spaces.ipynb b/Notebooks/Chap08/8_4_High_Dimensional_Spaces.ipynb index 80b27f0..f9f914d 100644 --- a/Notebooks/Chap08/8_4_High_Dimensional_Spaces.ipynb +++ b/Notebooks/Chap08/8_4_High_Dimensional_Spaces.ipynb @@ -134,7 +134,7 @@ "source": [ "# Volume of a hypersphere\n", "\n", - "In the second part of this notebook we calculate the volume of a hypersphere of radius 0.5 (i.e., of diameter 1) as a function of the radius. Note that you you can check your answer by doing the calculation for 2D using the standard formula for the area of a circle and making sure it matches." + "In the second part of this notebook we calculate the volume of a hypersphere of radius 0.5 (i.e., of diameter 1) as a function of the radius. Note that you can check your answer by doing the calculation for 2D using the standard formula for the area of a circle and making sure it matches." ], "metadata": { "id": "b2FYKV1SL4Z7" diff --git a/Notebooks/Chap19/19_2_Dynamic_Programming.ipynb b/Notebooks/Chap19/19_2_Dynamic_Programming.ipynb index a1a2859..62a1275 100644 --- a/Notebooks/Chap19/19_2_Dynamic_Programming.ipynb +++ b/Notebooks/Chap19/19_2_Dynamic_Programming.ipynb @@ -393,7 +393,7 @@ { "cell_type": "code", "source": [ - "# Update the state values for the current policy, by making the values at at adjacent\n", + "# Update the state values for the current policy, by making the values at adjacent\n", "# states compatible with the Bellman equation (equation 19.11)\n", "def policy_evaluation(policy, state_values, rewards, transition_probabilities_given_action, gamma):\n", "\n", @@ -527,4 +527,4 @@ } } ] -} \ No newline at end of file +} diff --git a/Notebooks/Chap21/21_1_Bias_Mitigation.ipynb b/Notebooks/Chap21/21_1_Bias_Mitigation.ipynb index dba4b1a..2235dd0 100644 --- a/Notebooks/Chap21/21_1_Bias_Mitigation.ipynb +++ b/Notebooks/Chap21/21_1_Bias_Mitigation.ipynb @@ -137,7 +137,7 @@ "id": "CfZ-srQtmff2" }, "source": [ - "Why might the distributions for blue and yellow populations be different? It could be that the behaviour of the populations is identical, but the credit rating algorithm is biased; it may favor one population over another or simply be more noisy for one group. Alternatively, it could be that that the populations genuinely behave differently. In practice, the differences in blue and yellow distributions are probably attributable to a combination of these factors.\n", + "Why might the distributions for blue and yellow populations be different? It could be that the behaviour of the populations is identical, but the credit rating algorithm is biased; it may favor one population over another or simply be more noisy for one group. Alternatively, it could be that the populations genuinely behave differently. In practice, the differences in blue and yellow distributions are probably attributable to a combination of these factors.\n", "\n", "Let’s assume that we can’t retrain the credit score prediction algorithm; our job is to adjudicate whether each individual is refused the loan ($\\hat{y}=0$)\n", " or granted it ($\\hat{y}=1$). Since we only have the credit score\n", @@ -382,7 +382,7 @@ "source": [ "# Equal opportunity:\n", "\n", - "The thresholds are chosen so that so that the true positive rate is is the same for both population. Of the people who pay back the loan, the same proportion are offered credit in each group. In terms of the two ROC curves, it means choosing thresholds so that the vertical position on each curve is the same without regard for the horizontal position." + "The thresholds are chosen so that so that the true positive rate is the same for both population. Of the people who pay back the loan, the same proportion are offered credit in each group. In terms of the two ROC curves, it means choosing thresholds so that the vertical position on each curve is the same without regard for the horizontal position." ] }, { From fed3962bce2130e027b7fc1f3057853d9660fedd Mon Sep 17 00:00:00 2001 From: =?UTF-8?q?Jan=20Hr=C4=8Dek?= Date: Tue, 30 Jul 2024 11:25:47 +0200 Subject: [PATCH 12/49] Fix markdown headings --- Notebooks/Chap04/4_1_Composing_Networks.ipynb | 4 ++-- Notebooks/Chap04/4_2_Clipping_functions.ipynb | 4 ++-- 2 files changed, 4 insertions(+), 4 deletions(-) diff --git a/Notebooks/Chap04/4_1_Composing_Networks.ipynb b/Notebooks/Chap04/4_1_Composing_Networks.ipynb index 284fdd3..a8261f0 100644 --- a/Notebooks/Chap04/4_1_Composing_Networks.ipynb +++ b/Notebooks/Chap04/4_1_Composing_Networks.ipynb @@ -28,7 +28,7 @@ { "cell_type": "markdown", "source": [ - "#Notebook 4.1 -- Composing networks\n", + "# Notebook 4.1 -- Composing networks\n", "\n", "The purpose of this notebook is to understand what happens when we feed one neural network into another. It works through an example similar to 4.1 and varies both networks\n", "\n", @@ -358,4 +358,4 @@ "outputs": [] } ] -} \ No newline at end of file +} diff --git a/Notebooks/Chap04/4_2_Clipping_functions.ipynb b/Notebooks/Chap04/4_2_Clipping_functions.ipynb index 0dbf46d..95ca408 100644 --- a/Notebooks/Chap04/4_2_Clipping_functions.ipynb +++ b/Notebooks/Chap04/4_2_Clipping_functions.ipynb @@ -29,7 +29,7 @@ { "cell_type": "markdown", "source": [ - "#Notebook 4.2 -- Clipping functions\n", + "# Notebook 4.2 -- Clipping functions\n", "\n", "The purpose of this notebook is to understand how a neural network with two hidden layers build more complicated functions by clipping and recombining the representations at the intermediate hidden variables.\n", "\n", @@ -216,4 +216,4 @@ } } ] -} \ No newline at end of file +} From a9868e6da8fb1285edaf38f9b5bd37c38268f793 Mon Sep 17 00:00:00 2001 From: udlbook <110402648+udlbook@users.noreply.github.com> Date: Tue, 30 Jul 2024 16:01:39 -0400 Subject: 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zr^7bhLkqq234Mwg%b2>9erj+Hm5bU%db4>Cb_o^Oo_&R1LfodBUgr()w(V3-*_b|H wC&8)J^T!nXwA4TUh6NXYTk{)gy-`~VWo~41baG{3Z3<k From 3c8dab14e6ffe1cfbb6215e535afb17d3cff4bed Mon Sep 17 00:00:00 2001 From: muddlebee Date: Sat, 17 Aug 2024 03:48:56 +0530 Subject: [PATCH 15/49] fix(8.1) : error in Chap08\8_1_MNIST_1D_Performance.ipynb --- .../Chap08/8_1_MNIST_1D_Performance.ipynb | 157 ++++++++++-------- 1 file changed, 85 insertions(+), 72 deletions(-) diff --git a/Notebooks/Chap08/8_1_MNIST_1D_Performance.ipynb b/Notebooks/Chap08/8_1_MNIST_1D_Performance.ipynb index fedcf24..1c01370 100644 --- a/Notebooks/Chap08/8_1_MNIST_1D_Performance.ipynb +++ b/Notebooks/Chap08/8_1_MNIST_1D_Performance.ipynb @@ -1,28 +1,10 @@ { - "nbformat": 4, - "nbformat_minor": 0, - "metadata": { - "colab": { - "provenance": [], - "gpuType": "T4", - "authorship_tag": "ABX9TyOuKMUcKfOIhIL2qTX9jJCy", - "include_colab_link": true - }, - "kernelspec": { - "name": "python3", - "display_name": "Python 3" - }, - "language_info": { - "name": "python" - }, - "accelerator": "GPU" - }, "cells": [ { "cell_type": "markdown", "metadata": { - "id": "view-in-github", - "colab_type": "text" + "colab_type": "text", + "id": "view-in-github" }, "source": [ "\"Open" @@ -30,6 +12,9 @@ }, { "cell_type": "markdown", + "metadata": { + "id": "L6chybAVFJW2" + }, "source": [ "# **Notebook 8.1: MNIST_1D_Performance**\n", "\n", @@ -38,25 +23,27 @@ "Work through the cells below, running each cell in turn. In various places you will see the words \"TO DO\". Follow the instructions at these places and make predictions about what is going to happen or write code to complete the functions.\n", "\n", "Contact me at udlbookmail@gmail.com if you find any mistakes or have any suggestions." - ], - "metadata": { - "id": "L6chybAVFJW2" - } + ] }, { "cell_type": "code", - "source": [ - "# Run this if you're in a Colab to install MNIST 1D repository\n", - "%pip install git+https://github.com/greydanus/mnist1d" - ], + "execution_count": null, "metadata": { "id": "ifVjS4cTOqKz" }, - "execution_count": null, - "outputs": [] + "outputs": [], + "source": [ + "# Run this if you're in a Colab to install MNIST 1D repository\n", + "%pip install git+https://github.com/greydanus/mnist1d" + ] }, { "cell_type": "code", + "execution_count": null, + "metadata": { + "id": "qyE7G1StPIqO" + }, + "outputs": [], "source": [ "import torch, torch.nn as nn\n", "from torch.utils.data import TensorDataset, DataLoader\n", @@ -64,26 +51,30 @@ "import numpy as np\n", "import matplotlib.pyplot as plt\n", "import mnist1d" - ], - "metadata": { - "id": "qyE7G1StPIqO" - }, - "execution_count": null, - "outputs": [] + ] }, { "cell_type": "markdown", - "source": [ - "Let's generate a training and test dataset using the MNIST1D code. The dataset gets saved as a .pkl file so it doesn't have to be regenerated each time." - ], "metadata": { "id": "F7LNq72SP6jO" - } + }, + "source": [ + "Let's generate a training and test dataset using the MNIST1D code. The dataset gets saved as a .pkl file so it doesn't have to be regenerated each time." + ] }, { "cell_type": "code", + "execution_count": null, + "metadata": { + "id": "YLxf7dJfPaqw" + }, + "outputs": [], "source": [ - "!mkdir ./sample_data\n", + "import os\n", + "\n", + "# Create directory in a cross-platform way\n", + "os.makedirs('./sample_data', exist_ok=True)\n", + "\n", "\n", "args = mnist1d.data.get_dataset_args()\n", "data = mnist1d.data.get_dataset(args, path='./sample_data/mnist1d_data.pkl', download=False, regenerate=False)\n", @@ -93,15 +84,15 @@ "print(\"Examples in training set: {}\".format(len(data['y'])))\n", "print(\"Examples in test set: {}\".format(len(data['y_test'])))\n", "print(\"Length of each example: {}\".format(data['x'].shape[-1]))" - ], - "metadata": { - "id": "YLxf7dJfPaqw" - }, - "execution_count": null, - "outputs": [] + ] }, { "cell_type": "code", + "execution_count": null, + "metadata": { + "id": "FxaB5vc0uevl" + }, + "outputs": [], "source": [ "D_i = 40 # Input dimensions\n", "D_k = 100 # Hidden dimensions\n", @@ -122,15 +113,24 @@ "\n", "# Call the function you just defined\n", "model.apply(weights_init)\n" - ], - "metadata": { - "id": "FxaB5vc0uevl" - }, - "execution_count": null, - "outputs": [] + ] }, { "cell_type": "code", + "execution_count": 12, + "metadata": { + "id": "_rX6N3VyyQTY" + }, + "outputs": [ + { + "name": "stdout", + "output_type": "stream", + "text": [ + "Epoch 24, train loss 1.584953, train error 62.60, test loss 1.665801, test error 67.90\n", + "Epoch 25, train loss 1.586464, train error 63.05, test loss 1.666717, test error 68.00\n" + ] + } + ], "source": [ "# choose cross entropy loss function (equation 5.24)\n", "loss_function = torch.nn.CrossEntropyLoss()\n", @@ -139,9 +139,9 @@ "# object that decreases learning rate by half every 10 epochs\n", "scheduler = StepLR(optimizer, step_size=10, gamma=0.5)\n", "x_train = torch.tensor(data['x'].astype('float32'))\n", - "y_train = torch.tensor(data['y'].transpose().astype('long'))\n", + "y_train = torch.tensor(data['y'].transpose().astype('int64'))\n", "x_test= torch.tensor(data['x_test'].astype('float32'))\n", - "y_test = torch.tensor(data['y_test'].astype('long'))\n", + "y_test = torch.tensor(data['y_test'].astype('int64'))\n", "\n", "# load the data into a class that creates the batches\n", "data_loader = DataLoader(TensorDataset(x_train,y_train), batch_size=100, shuffle=True, worker_init_fn=np.random.seed(1))\n", @@ -186,15 +186,15 @@ "\n", " # tell scheduler to consider updating learning rate\n", " scheduler.step()" - ], - "metadata": { - "id": "_rX6N3VyyQTY" - }, - "execution_count": null, - "outputs": [] + ] }, { "cell_type": "code", + "execution_count": null, + "metadata": { + "id": "yI-l6kA_EH9G" + }, + "outputs": [], "source": [ "# Plot the results\n", "fig, ax = plt.subplots()\n", @@ -215,25 +215,38 @@ "ax.set_title('Train loss %3.2f, Test loss %3.2f'%(losses_train[-1],losses_test[-1]))\n", "ax.legend()\n", "plt.show()" - ], - "metadata": { - "id": "yI-l6kA_EH9G" - }, - "execution_count": null, - "outputs": [] + ] }, { "cell_type": "markdown", + "metadata": { + "id": "q-yT6re6GZS4" + }, "source": [ "**TO DO**\n", "\n", "Play with the model -- try changing the number of layers, hidden units, learning rate, batch size, momentum or anything else you like. See if you can improve the test results.\n", "\n", "Is it a good idea to optimize the hyperparameters in this way? Will the final result be a good estimate of the true test performance?" - ], - "metadata": { - "id": "q-yT6re6GZS4" - } + ] } - ] + ], + "metadata": { + "accelerator": "GPU", + "colab": { + "authorship_tag": "ABX9TyOuKMUcKfOIhIL2qTX9jJCy", + "gpuType": "T4", + "include_colab_link": true, + "provenance": [] + }, + "kernelspec": { + "display_name": "Python 3", + "name": "python3" + }, + "language_info": { + "name": "python" + } + }, + "nbformat": 4, + "nbformat_minor": 0 } From b423a67855de0874bd823cb5301d8351cd12de13 Mon Sep 17 00:00:00 2001 From: muddlebee Date: Sat, 17 Aug 2024 03:50:15 +0530 Subject: [PATCH 16/49] fix(8.1) : error in Chap08\8_1_MNIST_1D_Performance.ipynb --- Notebooks/Chap08/8_1_MNIST_1D_Performance.ipynb | 13 ++----------- 1 file changed, 2 insertions(+), 11 deletions(-) diff --git a/Notebooks/Chap08/8_1_MNIST_1D_Performance.ipynb b/Notebooks/Chap08/8_1_MNIST_1D_Performance.ipynb index 1c01370..69b4f91 100644 --- a/Notebooks/Chap08/8_1_MNIST_1D_Performance.ipynb +++ b/Notebooks/Chap08/8_1_MNIST_1D_Performance.ipynb @@ -117,20 +117,11 @@ }, { "cell_type": "code", - "execution_count": 12, + "execution_count": null, "metadata": { "id": "_rX6N3VyyQTY" }, - "outputs": [ - { - "name": "stdout", - "output_type": "stream", - "text": [ - "Epoch 24, train loss 1.584953, train error 62.60, test loss 1.665801, test error 67.90\n", - "Epoch 25, train loss 1.586464, train error 63.05, test loss 1.666717, test error 68.00\n" - ] - } - ], + "outputs": [], "source": [ "# choose cross entropy loss function (equation 5.24)\n", "loss_function = torch.nn.CrossEntropyLoss()\n", From ccedbb72e72121ef055ef0ce98fed99eb10852f0 Mon Sep 17 00:00:00 2001 From: muddlebee Date: Sat, 17 Aug 2024 19:20:02 +0530 Subject: [PATCH 17/49] fix(8.1) : error in Chap08\8_1_MNIST_1D_Performance.ipynb --- Notebooks/Chap08/8_1_MNIST_1D_Performance.ipynb | 8 +------- 1 file changed, 1 insertion(+), 7 deletions(-) diff --git a/Notebooks/Chap08/8_1_MNIST_1D_Performance.ipynb b/Notebooks/Chap08/8_1_MNIST_1D_Performance.ipynb index 69b4f91..a6b8208 100644 --- a/Notebooks/Chap08/8_1_MNIST_1D_Performance.ipynb +++ b/Notebooks/Chap08/8_1_MNIST_1D_Performance.ipynb @@ -70,14 +70,8 @@ }, "outputs": [], "source": [ - "import os\n", - "\n", - "# Create directory in a cross-platform way\n", - "os.makedirs('./sample_data', exist_ok=True)\n", - "\n", - "\n", "args = mnist1d.data.get_dataset_args()\n", - "data = mnist1d.data.get_dataset(args, path='./sample_data/mnist1d_data.pkl', download=False, regenerate=False)\n", + "data = mnist1d.data.get_dataset(args, path='./mnist1d_data.pkl', download=False, regenerate=False)\n", "\n", "# The training and test input and outputs are in\n", "# data['x'], data['y'], data['x_test'], and data['y_test']\n", From 87cf590af9dbc9aac3d2d39ceb5ba818956d095b Mon Sep 17 00:00:00 2001 From: aleksandrskoselevs <111278851+aleksandrskoselevs@users.noreply.github.com> Date: Fri, 23 Aug 2024 09:57:38 +0200 Subject: [PATCH 18/49] Remove duplicate weight initialization --- Notebooks/Chap09/9_5_Augmentation.ipynb | 187 ++++++++++++------------ 1 file changed, 92 insertions(+), 95 deletions(-) diff --git a/Notebooks/Chap09/9_5_Augmentation.ipynb b/Notebooks/Chap09/9_5_Augmentation.ipynb index 66819ed..1386674 100644 --- a/Notebooks/Chap09/9_5_Augmentation.ipynb +++ b/Notebooks/Chap09/9_5_Augmentation.ipynb @@ -1,25 +1,10 @@ { - "nbformat": 4, - "nbformat_minor": 0, - "metadata": { - "colab": { - "provenance": [], - "include_colab_link": true - }, - "kernelspec": { - "name": "python3", - "display_name": "Python 3" - }, - "language_info": { - "name": "python" - } - }, "cells": [ { "cell_type": "markdown", "metadata": { - "id": "view-in-github", - "colab_type": "text" + "colab_type": "text", + "id": "view-in-github" }, "source": [ "\"Open" @@ -27,6 +12,9 @@ }, { "cell_type": "markdown", + "metadata": { + "id": "el8l05WQEO46" + }, "source": [ "# **Notebook 9.5: Augmentation**\n", "\n", @@ -35,25 +23,27 @@ "Work through the cells below, running each cell in turn. In various places you will see the words \"TO DO\". Follow the instructions at these places and make predictions about what is going to happen or write code to complete the functions.\n", "\n", "Contact me at udlbookmail@gmail.com if you find any mistakes or have any suggestions.\n" - ], - "metadata": { - "id": "el8l05WQEO46" - } + ] }, { "cell_type": "code", - "source": [ - "# Run this if you're in a Colab to install MNIST 1D repository\n", - "!pip install git+https://github.com/greydanus/mnist1d" - ], + "execution_count": null, "metadata": { "id": "syvgxgRr3myY" }, - "execution_count": null, - "outputs": [] + "outputs": [], + "source": [ + "# Run this if you're in a Colab to install MNIST 1D repository\n", + "!pip install git+https://github.com/greydanus/mnist1d" + ] }, { "cell_type": "code", + "execution_count": null, + "metadata": { + "id": "ckrNsYd13pMe" + }, + "outputs": [], "source": [ "import torch, torch.nn as nn\n", "from torch.utils.data import TensorDataset, DataLoader\n", @@ -62,15 +52,15 @@ "import matplotlib.pyplot as plt\n", "import mnist1d\n", "import random" - ], - "metadata": { - "id": "ckrNsYd13pMe" - }, - "execution_count": null, - "outputs": [] + ] }, { "cell_type": "code", + "execution_count": null, + "metadata": { + "id": "D_Woo9U730lZ" + }, + "outputs": [], "source": [ "args = mnist1d.data.get_dataset_args()\n", "data = mnist1d.data.get_dataset(args, path='./mnist1d_data.pkl', download=False, regenerate=False)\n", @@ -80,15 +70,15 @@ "print(\"Examples in training set: {}\".format(len(data['y'])))\n", "print(\"Examples in test set: {}\".format(len(data['y_test'])))\n", "print(\"Length of each example: {}\".format(data['x'].shape[-1]))" - ], - "metadata": { - "id": "D_Woo9U730lZ" - }, - "execution_count": null, - "outputs": [] + ] }, { "cell_type": "code", + "execution_count": null, + "metadata": { + "id": "JfIFWFIL33eF" + }, + "outputs": [], "source": [ "D_i = 40 # Input dimensions\n", "D_k = 200 # Hidden dimensions\n", @@ -109,17 +99,17 @@ " nn.init.kaiming_uniform_(layer_in.weight)\n", " layer_in.bias.data.fill_(0.0)\n", "\n", - "# Call the function you just defined\n", + "# Initialize model weights\n", "model.apply(weights_init)" - ], - "metadata": { - "id": "JfIFWFIL33eF" - }, - "execution_count": null, - "outputs": [] + ] }, { "cell_type": "code", + "execution_count": null, + "metadata": { + "id": "YFfVbTPE4BkJ" + }, + "outputs": [], "source": [ "# choose cross entropy loss function (equation 5.24)\n", "loss_function = torch.nn.CrossEntropyLoss()\n", @@ -136,9 +126,6 @@ "# load the data into a class that creates the batches\n", "data_loader = DataLoader(TensorDataset(x_train,y_train), batch_size=100, shuffle=True, worker_init_fn=np.random.seed(1))\n", "\n", - "# Initialize model weights\n", - "model.apply(weights_init)\n", - "\n", "# loop over the dataset n_epoch times\n", "n_epoch = 50\n", "# store the loss and the % correct at each epoch\n", @@ -169,15 +156,15 @@ " errors_train[epoch] = 100 - 100 * (predicted_train_class == y_train).float().sum() / len(y_train)\n", " errors_test[epoch]= 100 - 100 * (predicted_test_class == y_test).float().sum() / len(y_test)\n", " print(f'Epoch {epoch:5d}, train error {errors_train[epoch]:3.2f}, test error {errors_test[epoch]:3.2f}')" - ], - "metadata": { - "id": "YFfVbTPE4BkJ" - }, - "execution_count": null, - "outputs": [] + ] }, { "cell_type": "code", + "execution_count": null, + "metadata": { + "id": "FmGDd4vB8LyM" + }, + "outputs": [], "source": [ "# Plot the results\n", "fig, ax = plt.subplots()\n", @@ -188,24 +175,24 @@ "ax.set_title('Train Error %3.2f, Test Error %3.2f'%(errors_train[-1],errors_test[-1]))\n", "ax.legend()\n", "plt.show()" - ], - "metadata": { - "id": "FmGDd4vB8LyM" - }, - "execution_count": null, - "outputs": [] + ] }, { "cell_type": "markdown", - "source": [ - "The best test performance is about 33%. Let's see if we can improve on that by augmenting the data." - ], "metadata": { "id": "55XvoPDO8Qp-" - } + }, + "source": [ + "The best test performance is about 33%. Let's see if we can improve on that by augmenting the data." + ] }, { "cell_type": "code", + "execution_count": null, + "metadata": { + "id": "IP6z2iox8MOF" + }, + "outputs": [], "source": [ "def augment(input_vector):\n", " # Create output vector\n", @@ -221,15 +208,15 @@ " data_out = np.array(data_out)\n", "\n", " return data_out" - ], - "metadata": { - "id": "IP6z2iox8MOF" - }, - "execution_count": null, - "outputs": [] + ] }, { "cell_type": "code", + "execution_count": null, + "metadata": { + "id": "bzN0lu5J95AJ" + }, + "outputs": [], "source": [ "n_data_orig = data['x'].shape[0]\n", "# We'll double the amount of data\n", @@ -247,15 +234,15 @@ " # Augment the point and store\n", " augmented_x[c_augment,:] = augment(data['x'][random_data_index,:])\n", " augmented_y[c_augment] = data['y'][random_data_index]\n" - ], - "metadata": { - "id": "bzN0lu5J95AJ" - }, - "execution_count": null, - "outputs": [] + ] }, { "cell_type": "code", + "execution_count": null, + "metadata": { + "id": "hZUNrXpS_kRs" + }, + "outputs": [], "source": [ "# choose cross entropy loss function (equation 5.24)\n", "loss_function = torch.nn.CrossEntropyLoss()\n", @@ -305,15 +292,15 @@ " errors_train_aug[epoch] = 100 - 100 * (predicted_train_class == y_train).float().sum() / len(y_train)\n", " errors_test_aug[epoch]= 100 - 100 * (predicted_test_class == y_test).float().sum() / len(y_test)\n", " print(f'Epoch {epoch:5d}, train error {errors_train_aug[epoch]:3.2f}, test error {errors_test_aug[epoch]:3.2f}')" - ], - "metadata": { - "id": "hZUNrXpS_kRs" - }, - "execution_count": null, - "outputs": [] + ] }, { "cell_type": "code", + "execution_count": null, + "metadata": { + "id": "IcnAW4ixBnuc" + }, + "outputs": [], "source": [ "# Plot the results\n", "fig, ax = plt.subplots()\n", @@ -325,21 +312,31 @@ "ax.set_title('TrainError %3.2f, Test Error %3.2f'%(errors_train_aug[-1],errors_test_aug[-1]))\n", "ax.legend()\n", "plt.show()" - ], - "metadata": { - "id": "IcnAW4ixBnuc" - }, - "execution_count": null, - "outputs": [] + ] }, { "cell_type": "markdown", - "source": [ - "Hopefully, you should see an improvement in performance when we augment the data." - ], "metadata": { "id": "jgsR7ScJHc9b" - } + }, + "source": [ + "Hopefully, you should see an improvement in performance when we augment the data." + ] } - ] -} \ No newline at end of file + ], + "metadata": { + "colab": { + "include_colab_link": true, + "provenance": [] + }, + "kernelspec": { + "display_name": "Python 3", + "name": "python3" + }, + "language_info": { + "name": "python" + } + }, + "nbformat": 4, + "nbformat_minor": 0 +} From 305a0550793d830b0f2acddf7aa41a01ee498e00 Mon Sep 17 00:00:00 2001 From: aleksandrskoselevs Date: Fri, 23 Aug 2024 10:29:04 +0200 Subject: [PATCH 19/49] Revert "Remove duplicate weight initialization" This reverts commit 87cf590af9dbc9aac3d2d39ceb5ba818956d095b. --- Notebooks/Chap09/9_5_Augmentation.ipynb | 187 ++++++++++++------------ 1 file changed, 95 insertions(+), 92 deletions(-) diff --git a/Notebooks/Chap09/9_5_Augmentation.ipynb b/Notebooks/Chap09/9_5_Augmentation.ipynb index 1386674..66819ed 100644 --- a/Notebooks/Chap09/9_5_Augmentation.ipynb +++ b/Notebooks/Chap09/9_5_Augmentation.ipynb @@ -1,10 +1,25 @@ { + "nbformat": 4, + "nbformat_minor": 0, + "metadata": { + "colab": { + "provenance": [], + "include_colab_link": true + }, + "kernelspec": { + "name": "python3", + "display_name": "Python 3" + }, + "language_info": { + "name": "python" + } + }, "cells": [ { "cell_type": "markdown", "metadata": { - "colab_type": "text", - "id": "view-in-github" + "id": "view-in-github", + "colab_type": "text" }, "source": [ "\"Open" @@ -12,9 +27,6 @@ }, { "cell_type": "markdown", - "metadata": { - "id": "el8l05WQEO46" - }, "source": [ "# **Notebook 9.5: Augmentation**\n", "\n", @@ -23,27 +35,25 @@ "Work through the cells below, running each cell in turn. In various places you will see the words \"TO DO\". Follow the instructions at these places and make predictions about what is going to happen or write code to complete the functions.\n", "\n", "Contact me at udlbookmail@gmail.com if you find any mistakes or have any suggestions.\n" - ] + ], + "metadata": { + "id": "el8l05WQEO46" + } }, { "cell_type": "code", - "execution_count": null, - "metadata": { - "id": "syvgxgRr3myY" - }, - "outputs": [], "source": [ "# Run this if you're in a Colab to install MNIST 1D repository\n", "!pip install git+https://github.com/greydanus/mnist1d" - ] + ], + "metadata": { + "id": "syvgxgRr3myY" + }, + "execution_count": null, + "outputs": [] }, { "cell_type": "code", - "execution_count": null, - "metadata": { - "id": "ckrNsYd13pMe" - }, - "outputs": [], "source": [ "import torch, torch.nn as nn\n", "from torch.utils.data import TensorDataset, DataLoader\n", @@ -52,15 +62,15 @@ "import matplotlib.pyplot as plt\n", "import mnist1d\n", "import random" - ] + ], + "metadata": { + "id": "ckrNsYd13pMe" + }, + "execution_count": null, + "outputs": [] }, { "cell_type": "code", - "execution_count": null, - "metadata": { - "id": "D_Woo9U730lZ" - }, - "outputs": [], "source": [ "args = mnist1d.data.get_dataset_args()\n", "data = mnist1d.data.get_dataset(args, path='./mnist1d_data.pkl', download=False, regenerate=False)\n", @@ -70,15 +80,15 @@ "print(\"Examples in training set: {}\".format(len(data['y'])))\n", "print(\"Examples in test set: {}\".format(len(data['y_test'])))\n", "print(\"Length of each example: {}\".format(data['x'].shape[-1]))" - ] + ], + "metadata": { + "id": "D_Woo9U730lZ" + }, + "execution_count": null, + "outputs": [] }, { "cell_type": "code", - "execution_count": null, - "metadata": { - "id": "JfIFWFIL33eF" - }, - "outputs": [], "source": [ "D_i = 40 # Input dimensions\n", "D_k = 200 # Hidden dimensions\n", @@ -99,17 +109,17 @@ " nn.init.kaiming_uniform_(layer_in.weight)\n", " layer_in.bias.data.fill_(0.0)\n", "\n", - "# Initialize model weights\n", + "# Call the function you just defined\n", "model.apply(weights_init)" - ] + ], + "metadata": { + "id": "JfIFWFIL33eF" + }, + "execution_count": null, + "outputs": [] }, { "cell_type": "code", - "execution_count": null, - "metadata": { - "id": "YFfVbTPE4BkJ" - }, - "outputs": [], "source": [ "# choose cross entropy loss function (equation 5.24)\n", "loss_function = torch.nn.CrossEntropyLoss()\n", @@ -126,6 +136,9 @@ "# load the data into a class that creates the batches\n", "data_loader = DataLoader(TensorDataset(x_train,y_train), batch_size=100, shuffle=True, worker_init_fn=np.random.seed(1))\n", "\n", + "# Initialize model weights\n", + "model.apply(weights_init)\n", + "\n", "# loop over the dataset n_epoch times\n", "n_epoch = 50\n", "# store the loss and the % correct at each epoch\n", @@ -156,15 +169,15 @@ " errors_train[epoch] = 100 - 100 * (predicted_train_class == y_train).float().sum() / len(y_train)\n", " errors_test[epoch]= 100 - 100 * (predicted_test_class == y_test).float().sum() / len(y_test)\n", " print(f'Epoch {epoch:5d}, train error {errors_train[epoch]:3.2f}, test error {errors_test[epoch]:3.2f}')" - ] + ], + "metadata": { + "id": "YFfVbTPE4BkJ" + }, + "execution_count": null, + "outputs": [] }, { "cell_type": "code", - "execution_count": null, - "metadata": { - "id": "FmGDd4vB8LyM" - }, - "outputs": [], "source": [ "# Plot the results\n", "fig, ax = plt.subplots()\n", @@ -175,24 +188,24 @@ "ax.set_title('Train Error %3.2f, Test Error %3.2f'%(errors_train[-1],errors_test[-1]))\n", "ax.legend()\n", "plt.show()" - ] + ], + "metadata": { + "id": "FmGDd4vB8LyM" + }, + "execution_count": null, + "outputs": [] }, { "cell_type": "markdown", - "metadata": { - "id": "55XvoPDO8Qp-" - }, "source": [ "The best test performance is about 33%. Let's see if we can improve on that by augmenting the data." - ] + ], + "metadata": { + "id": "55XvoPDO8Qp-" + } }, { "cell_type": "code", - "execution_count": null, - "metadata": { - "id": "IP6z2iox8MOF" - }, - "outputs": [], "source": [ "def augment(input_vector):\n", " # Create output vector\n", @@ -208,15 +221,15 @@ " data_out = np.array(data_out)\n", "\n", " return data_out" - ] + ], + "metadata": { + "id": "IP6z2iox8MOF" + }, + "execution_count": null, + "outputs": [] }, { "cell_type": "code", - "execution_count": null, - "metadata": { - "id": "bzN0lu5J95AJ" - }, - "outputs": [], "source": [ "n_data_orig = data['x'].shape[0]\n", "# We'll double the amount of data\n", @@ -234,15 +247,15 @@ " # Augment the point and store\n", " augmented_x[c_augment,:] = augment(data['x'][random_data_index,:])\n", " augmented_y[c_augment] = data['y'][random_data_index]\n" - ] + ], + "metadata": { + "id": "bzN0lu5J95AJ" + }, + "execution_count": null, + "outputs": [] }, { "cell_type": "code", - "execution_count": null, - "metadata": { - "id": "hZUNrXpS_kRs" - }, - "outputs": [], "source": [ "# choose cross entropy loss function (equation 5.24)\n", "loss_function = torch.nn.CrossEntropyLoss()\n", @@ -292,15 +305,15 @@ " errors_train_aug[epoch] = 100 - 100 * (predicted_train_class == y_train).float().sum() / len(y_train)\n", " errors_test_aug[epoch]= 100 - 100 * (predicted_test_class == y_test).float().sum() / len(y_test)\n", " print(f'Epoch {epoch:5d}, train error {errors_train_aug[epoch]:3.2f}, test error {errors_test_aug[epoch]:3.2f}')" - ] + ], + "metadata": { + "id": "hZUNrXpS_kRs" + }, + "execution_count": null, + "outputs": [] }, { "cell_type": "code", - "execution_count": null, - "metadata": { - "id": "IcnAW4ixBnuc" - }, - "outputs": [], "source": [ "# Plot the results\n", "fig, ax = plt.subplots()\n", @@ -312,31 +325,21 @@ "ax.set_title('TrainError %3.2f, Test Error %3.2f'%(errors_train_aug[-1],errors_test_aug[-1]))\n", "ax.legend()\n", "plt.show()" - ] + ], + "metadata": { + "id": "IcnAW4ixBnuc" + }, + "execution_count": null, + "outputs": [] }, { "cell_type": "markdown", - "metadata": { - "id": "jgsR7ScJHc9b" - }, "source": [ "Hopefully, you should see an improvement in performance when we augment the data." - ] + ], + "metadata": { + "id": "jgsR7ScJHc9b" + } } - ], - "metadata": { - "colab": { - "include_colab_link": true, - "provenance": [] - }, - "kernelspec": { - "display_name": "Python 3", - "name": "python3" - }, - "language_info": { - "name": "python" - } - }, - "nbformat": 4, - "nbformat_minor": 0 -} + ] +} \ No newline at end of file From 3aeb8db4cd37c179ec1de3f201b8f9c9cfe159f7 Mon Sep 17 00:00:00 2001 From: aleksandrskoselevs Date: Fri, 23 Aug 2024 10:29:52 +0200 Subject: [PATCH 20/49] cleaner diff --- Notebooks/Chap09/9_5_Augmentation.ipynb | 5 +---- 1 file changed, 1 insertion(+), 4 deletions(-) diff --git a/Notebooks/Chap09/9_5_Augmentation.ipynb b/Notebooks/Chap09/9_5_Augmentation.ipynb index 66819ed..bd0c607 100644 --- a/Notebooks/Chap09/9_5_Augmentation.ipynb +++ b/Notebooks/Chap09/9_5_Augmentation.ipynb @@ -107,10 +107,7 @@ " # Initialize the parameters with He initialization\n", " if isinstance(layer_in, nn.Linear):\n", " nn.init.kaiming_uniform_(layer_in.weight)\n", - " layer_in.bias.data.fill_(0.0)\n", - "\n", - "# Call the function you just defined\n", - "model.apply(weights_init)" + " layer_in.bias.data.fill_(0.0)\n" ], "metadata": { "id": "JfIFWFIL33eF" From b7869e8b411fcd9f78111da65f62b122eb384a32 Mon Sep 17 00:00:00 2001 From: udlbook <110402648+udlbook@users.noreply.github.com> Date: Wed, 28 Aug 2024 13:01:31 -0400 Subject: [PATCH 21/49] Add files via upload --- UDL_Errata.pdf | Bin 1185665 -> 1196466 bytes 1 file changed, 0 insertions(+), 0 deletions(-) diff --git a/UDL_Errata.pdf b/UDL_Errata.pdf index 88f5a942e6b551600f724b426b8408d3aeb4e8c8..99dcb1a84179c3590e6d8b9b49cede3f08c28f0e 100644 GIT binary patch delta 143443 zcmZ6xb95$M@GcrB6We%W+qP|UV%y0ZPHdYK+qP}noYz5 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z!*kpO_hWt=Yo17#Xg0I|%NQuye5Bt`Pqw?7pRq~NB`*N6l6G+^%-7`SH)@hA`*mlp b?eWsW*$tvOtgHuT(VzlW1rE#>LBsqX5gDeZ From e9c8d846f22a6d4a425c2fdadf3b8cfacdd02d2b Mon Sep 17 00:00:00 2001 From: udlbook <110402648+udlbook@users.noreply.github.com> Date: Mon, 16 Sep 2024 07:36:27 -0400 Subject: [PATCH 22/49] Created using Colab --- .../Chap10/10_5_Convolution_For_MNIST.ipynb | 16 +++++++++++++++- 1 file changed, 15 insertions(+), 1 deletion(-) diff --git a/Notebooks/Chap10/10_5_Convolution_For_MNIST.ipynb b/Notebooks/Chap10/10_5_Convolution_For_MNIST.ipynb index e50c024..33322a0 100644 --- a/Notebooks/Chap10/10_5_Convolution_For_MNIST.ipynb +++ b/Notebooks/Chap10/10_5_Convolution_For_MNIST.ipynb @@ -4,7 +4,7 @@ "metadata": { "colab": { "provenance": [], - "authorship_tag": "ABX9TyNAcc98STMeyQgh9SbVHWG+", + "authorship_tag": "ABX9TyNELb86uz5qbhEKH81UqFKT", "include_colab_link": true }, "kernelspec": { @@ -65,6 +65,11 @@ "source": [ "# Run this once to load the train and test data straight into a dataloader class\n", "# that will provide the batches\n", + "\n", + "# (It may complain that some files are missing because the files seem to have been\n", + "# reorganized on the underlying website, but it still seems to work). If everything is working\n", + "# properly, then the whole notebook should run to the end without further problems\n", + "# even before you make changes.\n", "batch_size_train = 64\n", "batch_size_test = 1000\n", "train_loader = torch.utils.data.DataLoader(\n", @@ -91,6 +96,15 @@ "execution_count": null, "outputs": [] }, + { + "cell_type": "code", + "source": [], + "metadata": { + "id": "YGwbxJDEm88i" + }, + "execution_count": null, + "outputs": [] + }, { "cell_type": "code", "source": [ From f8878356460d54fbea41565b8070a3c17f24a842 Mon Sep 17 00:00:00 2001 From: udlbook <110402648+udlbook@users.noreply.github.com> Date: Mon, 16 Sep 2024 09:18:12 -0400 Subject: [PATCH 23/49] Created using Colab --- Notebooks/Chap06/6_2_Gradient_Descent.ipynb | 26 +++++++-------------- 1 file changed, 9 insertions(+), 17 deletions(-) diff --git a/Notebooks/Chap06/6_2_Gradient_Descent.ipynb b/Notebooks/Chap06/6_2_Gradient_Descent.ipynb index 21a96bb..c28d51a 100644 --- a/Notebooks/Chap06/6_2_Gradient_Descent.ipynb +++ b/Notebooks/Chap06/6_2_Gradient_Descent.ipynb @@ -1,18 +1,16 @@ { "cells": [ { - "attachments": {}, "cell_type": "markdown", "metadata": { - "colab_type": "text", - "id": "view-in-github" + "id": "view-in-github", + "colab_type": "text" }, "source": [ "\"Open" ] }, { - "attachments": {}, "cell_type": "markdown", "metadata": { "id": "el8l05WQEO46" @@ -111,7 +109,6 @@ ] }, { - "attachments": {}, "cell_type": "markdown", "metadata": { "id": "QU5mdGvpTtEG" @@ -140,7 +137,6 @@ ] }, { - "attachments": {}, "cell_type": "markdown", "metadata": { "id": "eB5DQvU5hYNx" @@ -162,7 +158,6 @@ ] }, { - "attachments": {}, "cell_type": "markdown", "metadata": { "id": "F3trnavPiHpH" @@ -218,7 +213,6 @@ ] }, { - "attachments": {}, "cell_type": "markdown", "metadata": { "id": "s9Duf05WqqSC" @@ -252,7 +246,6 @@ ] }, { - "attachments": {}, "cell_type": "markdown", "metadata": { "id": "RS1nEcYVuEAM" @@ -290,7 +283,6 @@ ] }, { - "attachments": {}, "cell_type": "markdown", "metadata": { "id": "5EIjMM9Fw2eT" @@ -333,11 +325,11 @@ " print('Iter %d, a=%3.3f, b=%3.3f, c=%3.3f, d=%3.3f'%(n_iter, a,b,c,d))\n", " print('a %f, b%f, c%f, d%f'%(lossa,lossb,lossc,lossd))\n", "\n", - " # Rule #1 If point A is less than points B, C, and D then halve points B,C, and D\n", + " # Rule #1 If point A is less than points B, C, and D then halve distance from A to points B,C, and D\n", " if np.argmin((lossa,lossb,lossc,lossd))==0:\n", - " b = b/2\n", - " c = c/2\n", - " d = d/2\n", + " b = a+ (b-a)/2\n", + " c = a+ (c-a)/2\n", + " d = a+ (d-a)/2\n", " continue;\n", "\n", " # Rule #2 If point b is less than point c then\n", @@ -412,8 +404,8 @@ ], "metadata": { "colab": { - "include_colab_link": true, - "provenance": [] + "provenance": [], + "include_colab_link": true }, "kernelspec": { "display_name": "Python 3", @@ -425,4 +417,4 @@ }, "nbformat": 4, "nbformat_minor": 0 -} +} \ No newline at end of file From e1df2156a3e18f2a386cf71a78daffa8e470f952 Mon Sep 17 00:00:00 2001 From: udlbook <110402648+udlbook@users.noreply.github.com> Date: Mon, 16 Sep 2024 09:19:49 -0400 Subject: [PATCH 24/49] Created using Colab --- .../6_3_Stochastic_Gradient_Descent.ipynb | 27 +++++++------------ 1 file changed, 9 insertions(+), 18 deletions(-) diff --git a/Notebooks/Chap06/6_3_Stochastic_Gradient_Descent.ipynb b/Notebooks/Chap06/6_3_Stochastic_Gradient_Descent.ipynb index 51b5055..ce84689 100644 --- a/Notebooks/Chap06/6_3_Stochastic_Gradient_Descent.ipynb +++ b/Notebooks/Chap06/6_3_Stochastic_Gradient_Descent.ipynb @@ -1,18 +1,16 @@ { "cells": [ { - "attachments": {}, "cell_type": "markdown", "metadata": { - "colab_type": "text", - "id": "view-in-github" + "id": "view-in-github", + "colab_type": "text" }, "source": [ "\"Open" ] }, { - "attachments": {}, "cell_type": "markdown", "metadata": { "id": "el8l05WQEO46" @@ -122,7 +120,6 @@ ] }, { - "attachments": {}, "cell_type": "markdown", "metadata": { "id": "QU5mdGvpTtEG" @@ -150,7 +147,6 @@ ] }, { - "attachments": {}, "cell_type": "markdown", "metadata": { "id": "eB5DQvU5hYNx" @@ -172,7 +168,6 @@ ] }, { - "attachments": {}, "cell_type": "markdown", "metadata": { "id": "F3trnavPiHpH" @@ -228,7 +223,6 @@ ] }, { - "attachments": {}, "cell_type": "markdown", "metadata": { "id": "s9Duf05WqqSC" @@ -279,7 +273,6 @@ ] }, { - "attachments": {}, "cell_type": "markdown", "metadata": { "id": "RS1nEcYVuEAM" @@ -316,7 +309,6 @@ ] }, { - "attachments": {}, "cell_type": "markdown", "metadata": { "id": "5EIjMM9Fw2eT" @@ -359,11 +351,11 @@ " print('Iter %d, a=%3.3f, b=%3.3f, c=%3.3f, d=%3.3f'%(n_iter, a,b,c,d))\n", " print('a %f, b%f, c%f, d%f'%(lossa,lossb,lossc,lossd))\n", "\n", - " # Rule #1 If point A is less than points B, C, and D then halve points B,C, and D\n", + " # Rule #1 If point A is less than points B, C, and D then change B,C,D so they are half their current distance from A\n", " if np.argmin((lossa,lossb,lossc,lossd))==0:\n", - " b = b/2\n", - " c = c/2\n", - " d = d/2\n", + " b = a+ (b-a)/2\n", + " c = a+ (c-a)/2\n", + " d = a+ (d-a)/2\n", " continue;\n", "\n", " # Rule #2 If point b is less than point c then\n", @@ -577,9 +569,8 @@ ], "metadata": { "colab": { - "authorship_tag": "ABX9TyNk5FN4qlw3pk8BwDVWw1jN", - "include_colab_link": true, - "provenance": [] + "provenance": [], + "include_colab_link": true }, "kernelspec": { "display_name": "Python 3", @@ -591,4 +582,4 @@ }, "nbformat": 4, "nbformat_minor": 0 -} +} \ No newline at end of file From c3b97af4567c0a26d3e5f50539403619c6118aee Mon Sep 17 00:00:00 2001 From: udlbook <110402648+udlbook@users.noreply.github.com> Date: Mon, 16 Sep 2024 09:21:22 -0400 Subject: [PATCH 25/49] Created using Colab --- Notebooks/Chap06/6_1_Line_Search.ipynb | 3 ++- 1 file changed, 2 insertions(+), 1 deletion(-) diff --git a/Notebooks/Chap06/6_1_Line_Search.ipynb b/Notebooks/Chap06/6_1_Line_Search.ipynb index ec6a288..67e0095 100644 --- a/Notebooks/Chap06/6_1_Line_Search.ipynb +++ b/Notebooks/Chap06/6_1_Line_Search.ipynb @@ -130,7 +130,8 @@ "\n", " print('Iter %d, a=%3.3f, b=%3.3f, c=%3.3f, d=%3.3f'%(n_iter, a,b,c,d))\n", "\n", - " # Rule #1 If the HEIGHT at point A is less than the HEIGHT at points B, C, and D then halve values of B, C, and D\n", + " # Rule #1 If the HEIGHT at point A is less than the HEIGHT at points B, C, and D then move them to they are half\n", + " # as far from A as they start\n", " # i.e. bring them closer to the original point\n", " # TODO REPLACE THE BLOCK OF CODE BELOW WITH THIS RULE\n", " if (0):\n", From ede7247a0c2107e6c7b3aafb36636bc7608efc53 Mon Sep 17 00:00:00 2001 From: udlbook <110402648+udlbook@users.noreply.github.com> Date: Tue, 1 Oct 2024 15:13:14 -0400 Subject: [PATCH 26/49] Add files via upload --- UDL_Answer_Booklet.pdf | Bin 0 -> 1793468 bytes UDL_Errata.pdf | Bin 1196466 -> 1209076 bytes 2 files changed, 0 insertions(+), 0 deletions(-) create mode 100644 UDL_Answer_Booklet.pdf diff --git a/UDL_Answer_Booklet.pdf b/UDL_Answer_Booklet.pdf new file mode 100644 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z@8geQI;CSEr)g5Uy3AyqEu$^A?G9R*N=-7=lp}s~yVYN}vn|-_1MpNK)1g~6=xcS! zB}>(s?!t^%XsH=;7+J{?vwp|j!;S5Dua6;{Qwd;XMHyD zK*i-()P*V3?RW1D3lxJRN?YB(#cqBj;Pdv<2mB-;ySH8p> Date: Fri, 8 Nov 2024 12:31:21 -0500 Subject: [PATCH 38/49] Created using Colab --- Trees/LinearRegression_LeastSquares.ipynb | 51 +++++++++++++++++++++++ 1 file changed, 51 insertions(+) create mode 100644 Trees/LinearRegression_LeastSquares.ipynb diff --git a/Trees/LinearRegression_LeastSquares.ipynb b/Trees/LinearRegression_LeastSquares.ipynb new file mode 100644 index 0000000..e8b69d8 --- /dev/null +++ b/Trees/LinearRegression_LeastSquares.ipynb @@ -0,0 +1,51 @@ +{ + "nbformat": 4, + "nbformat_minor": 0, + "metadata": { + "colab": { + "provenance": [], + "authorship_tag": "ABX9TyM1pe3HkxLrjbeKezq1MlM5", + "include_colab_link": true + }, + "kernelspec": { + "name": "python3", + "display_name": "Python 3" + }, + "language_info": { + "name": "python" + } + }, + "cells": [ + { + "cell_type": "markdown", + "metadata": { + "id": "view-in-github", + "colab_type": "text" + }, + "source": [ + "\"Open" + ] + }, + { + "cell_type": "markdown", + "source": [ + "# Least Squares Loss" + ], + "metadata": { + "id": "uORlKyPv02ge" + } + }, + { + "cell_type": "code", + "execution_count": null, + "metadata": { + "id": "bbF6SE_F0tU8" + }, + "outputs": [], + "source": [ + "import numpy as np\n", + "import matplotlib.pyplot as plt" + ] + } + ] +} \ No newline at end of file From 57c95132d32c70a4449c90be338268381a7ba033 Mon Sep 17 00:00:00 2001 From: udlbook <110402648+udlbook@users.noreply.github.com> Date: Tue, 12 Nov 2024 17:11:44 -0500 Subject: [PATCH 39/49] Created using Colab --- Trees/LinearRegression_LossFunction.ipynb | 51 +++++++++++++++++++++++ 1 file changed, 51 insertions(+) create mode 100644 Trees/LinearRegression_LossFunction.ipynb diff --git a/Trees/LinearRegression_LossFunction.ipynb b/Trees/LinearRegression_LossFunction.ipynb new file mode 100644 index 0000000..03fbfaa --- /dev/null +++ b/Trees/LinearRegression_LossFunction.ipynb @@ -0,0 +1,51 @@ +{ + "nbformat": 4, + "nbformat_minor": 0, + "metadata": { + "colab": { + "provenance": [], + "authorship_tag": "ABX9TyMIJ9DpOBppPZXAJ5wms6s8", + "include_colab_link": true + }, + "kernelspec": { + "name": "python3", + "display_name": "Python 3" + }, + "language_info": { + "name": "python" + } + }, + "cells": [ + { + "cell_type": "markdown", + "metadata": { + "id": "view-in-github", + "colab_type": "text" + }, + "source": [ + "\"Open" + ] + }, + { + "cell_type": "markdown", + "source": [ + "# Loss function" + ], + "metadata": { + "id": "uORlKyPv02ge" + } + }, + { + "cell_type": "code", + "execution_count": null, + "metadata": { + "id": "bbF6SE_F0tU8" + }, + "outputs": [], + "source": [ + "import numpy as np\n", + "import matplotlib.pyplot as plt" + ] + } + ] +} \ No newline at end of file From 6d425c04d497245a910f654032c3ebf98f7521a6 Mon Sep 17 00:00:00 2001 From: udlbook <110402648+udlbook@users.noreply.github.com> Date: Mon, 18 Nov 2024 15:33:42 -0500 Subject: [PATCH 40/49] Update 3_3_Shallow_Network_Regions.ipynb --- Notebooks/Chap03/3_3_Shallow_Network_Regions.ipynb | 4 ++-- 1 file changed, 2 insertions(+), 2 deletions(-) diff --git a/Notebooks/Chap03/3_3_Shallow_Network_Regions.ipynb b/Notebooks/Chap03/3_3_Shallow_Network_Regions.ipynb index 4ca8d8e..607710c 100644 --- a/Notebooks/Chap03/3_3_Shallow_Network_Regions.ipynb +++ b/Notebooks/Chap03/3_3_Shallow_Network_Regions.ipynb @@ -61,7 +61,7 @@ "source": [ "The number of regions $N$ created by a shallow neural network with $D_i$ inputs and $D$ hidden units is given by Zaslavsky's formula:\n", "\n", - "\\begin{equation}N = \\sum_{j=0}^{D_{i}}\\binom{D}{j}=\\sum_{j=0}^{D_{i}} \\frac{D!}{(D-j)!j!} \\end{equation}
\n", + "\\begin{equation}N = \\sum_{j=0}^{D_{i}}\\binom{D}{j}=\\sum_{j=0}^{D_{i}} \\frac{D!}{(D-j)!j!} \\end{equation} \n", "\n" ], "metadata": { @@ -255,4 +255,4 @@ "outputs": [] } ] -} \ No newline at end of file +} From 33197fde3655f800490018f4c9631eb02cf3e8f1 Mon Sep 17 00:00:00 2001 From: udlbook <110402648+udlbook@users.noreply.github.com> Date: Thu, 21 Nov 2024 16:45:29 -0500 Subject: [PATCH 41/49] Add files via upload --- UDL_Answer_Booklet_Students.pdf | Bin 1793443 -> 1795499 bytes UDL_Errata.pdf | Bin 1244945 -> 1250693 bytes 2 files changed, 0 insertions(+), 0 deletions(-) diff --git a/UDL_Answer_Booklet_Students.pdf b/UDL_Answer_Booklet_Students.pdf index a5bc4d51fee753f260f469bcea45cf37316d1fc4..1cf566a6b266970afe3cc806df526a4b0612b80a 100644 GIT binary patch delta 221330 zcmV)wK$O3uyKSq>ZICboH8eRnku)s`IWRdhF(5uZlTiY0k=L~aIWRdhGLw-36+&S% zF=JvfH#aslGC5*7HDzQtHD)$sW@I)rVmCN5W;i}PG&L|`HZwLhIAdXEGGaGkFgP(b zH8(I~G%_?aF*7wWv*-ar34cUL1Vn^D4ibXe$~rN}#Eln0-UO94j1;XHrAHtu2C3oE z92pg*w&6?K>+G0y>X)AHy#wuEXyN9Y)1_a_8sJ3 zg%&exLO~MxOtU@>HE>rNki$aXh>Wq`LkXo-tU(l`K>}1}W1T^I8|yYYVncFN>SkU? zm9gGdv|dx*K-Vk1_^41x6rd0;QqW6mJ`FT?xok74P+Bq(AEBjC87WYiW&S3VZy|=z z+2NocQM=Ro3d%?98-J+Q=50o&8(F(iof|DcRprF$fonqVa{vg!XigM^7#4J3?6rp} z>nq3NzZx$bNGS{2klNxh4(OMc;ly?8FuA9zNHI8|neNOgxJc(y8xm(*f=aeVaHOS58L!-ae> z&IOA@qkaBx>$+9`(ic2fN*=DBJEY^^)oIn$;?#aaQKOftBdYb}WJDRx>57A#0fi~S zD_%kAi)Js>K!Bn%a)*1F!!3$40YI9j7l(K6A#)zX;eRBg3Nk#N$8A#G@j0|o_cuDO zm8K0ER@84C*zt?AsLwEBmWjm1ONTKsB2)R|#OUyqO)L{nqk=`;Lo&|>izL?pGD~#| z?qR2V!%AFldpsl7m~z27WoD}>)J@?=`34HLQw3R8eVQ~|_QIf{h>IchoNTuhTt51; zA4#&MI)AB6SqjXjNi~ls;C2Jt4g>JAre`-P2M2au5ql(2?hp!bX zI-^Ji;~5?`3&yehBW49jT|_ClnDtSuwEhsqbv$t0yM=iLlol5f8GZR|OLQF017JHOdh^lFtw|~>dNs@!O_FLrHOxg2(keVcfl&TxH zbZlXHSem33MQXD`jFEXDQGC|2qfJ|w4$Rrpqaw2Qc zAb;JK9kq)D2yxq0KX66rR#j^%e?SM`eM(%%))Vdf6}<_ns}wwo?-xRw!?OG&w#P zpg;|z)gBBlb=|z{N!|Kx_D6s-Cl3hO%}o7vIoxz`A!C5d zVc2&-s_Cy}uAaLRn3o-puCdM(x_m8)EQDvXc-5`PT%Lu8a$5HYq3aeKvJ4t(_J3Ol zV}QE1ejJh@D;Ia}I8E%g7xS&{t>*cww z0=w#Zar#9bxS&?r5ITf;`OUcsGgxhEQA`US0=+LH)eMcI8XGnc60OyToXy1o%BHFS zlJfc)WY%O7+2n^cYi@W979YZe2gDO|l*Ne z!^U;faXDBO(0)1Dv%$%f83uxARj&QXBI8jx6U_f-h((@rAVsZ z+WuVEo|y2F`%k|*8A<=4V&mDxn~ZyR6w>-qD z$lD@3fgbzNzeec&`G|ax3Y_;ZF0_Pam(iXF6O-x@x*OxPPBMZ-kr0ca&! 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6072ad445051c0710af30697282ec3dcb407f118 Mon Sep 17 00:00:00 2001 From: Kaj van Rijn Date: Fri, 22 Nov 2024 15:34:52 +0100 Subject: [PATCH 42/49] Change all TO DO to TODO --- Notebooks/Chap02/2_1_Supervised_Learning.ipynb | 2 +- Notebooks/Chap06/6_5_Adam.ipynb | 6 +++--- Notebooks/Chap07/7_2_Backpropagation.ipynb | 2 +- .../Chap10/10_4_Downsampling_and_Upsampling.ipynb | 12 ++++++------ Notebooks/Chap21/21_1_Bias_Mitigation.ipynb | 4 ++-- 5 files changed, 13 insertions(+), 13 deletions(-) diff --git a/Notebooks/Chap02/2_1_Supervised_Learning.ipynb b/Notebooks/Chap02/2_1_Supervised_Learning.ipynb index c7be987..42f3344 100644 --- a/Notebooks/Chap02/2_1_Supervised_Learning.ipynb +++ b/Notebooks/Chap02/2_1_Supervised_Learning.ipynb @@ -174,7 +174,7 @@ { "cell_type": "code", "source": [ - "# TO DO -- Change the parameters manually to fit the model\n", + "# TODO -- Change the parameters manually to fit the model\n", "# First fix phi1 and try changing phi0 until you can't make the loss go down any more\n", "# Then fix phi0 and try changing phi1 until you can't make the loss go down any more\n", "# Repeat this process until you find a set of parameters that fit the model as in figure 2.2d\n", diff --git a/Notebooks/Chap06/6_5_Adam.ipynb b/Notebooks/Chap06/6_5_Adam.ipynb index 211fc58..bb975d4 100644 --- a/Notebooks/Chap06/6_5_Adam.ipynb +++ b/Notebooks/Chap06/6_5_Adam.ipynb @@ -185,11 +185,11 @@ " for c_step in range(n_steps):\n", " # Measure the gradient as in equation 6.13 (first line)\n", " m = get_loss_gradient(grad_path[0,c_step], grad_path[1,c_step]);\n", - " # TO DO -- compute the squared gradient as in equation 6.13 (second line)\n", + " # TODO -- compute the squared gradient as in equation 6.13 (second line)\n", " # Replace this line:\n", " v = np.ones_like(grad_path[:,0])\n", "\n", - " # TO DO -- apply the update rule (equation 6.14)\n", + " # TODO -- apply the update rule (equation 6.14)\n", " # Replace this line:\n", " grad_path[:,c_step+1] = grad_path[:,c_step]\n", "\n", @@ -254,7 +254,7 @@ " v_tilde = v\n", "\n", "\n", - " # TO DO -- apply the update rule (equation 6.17)\n", + " # TODO -- apply the update rule (equation 6.17)\n", " # Replace this line:\n", " grad_path[:,c_step+1] = grad_path[:,c_step]\n", "\n", diff --git a/Notebooks/Chap07/7_2_Backpropagation.ipynb b/Notebooks/Chap07/7_2_Backpropagation.ipynb index dac7e89..b3f3cd8 100644 --- a/Notebooks/Chap07/7_2_Backpropagation.ipynb +++ b/Notebooks/Chap07/7_2_Backpropagation.ipynb @@ -148,7 +148,7 @@ " all_h[layer+1] = all_f[layer]\n", "\n", " # Compute the output from the last hidden layer\n", - " # TO DO -- Replace the line below\n", + " # TODO -- Replace the line below\n", " all_f[K] = np.zeros_like(all_biases[-1])\n", "\n", " # Retrieve the output\n", diff --git a/Notebooks/Chap10/10_4_Downsampling_and_Upsampling.ipynb b/Notebooks/Chap10/10_4_Downsampling_and_Upsampling.ipynb index 4010a6c..d0585a5 100644 --- a/Notebooks/Chap10/10_4_Downsampling_and_Upsampling.ipynb +++ b/Notebooks/Chap10/10_4_Downsampling_and_Upsampling.ipynb @@ -73,7 +73,7 @@ "source": [ "def subsample(x_in):\n", " x_out = np.zeros(( int(np.ceil(x_in.shape[0]/2)), int(np.ceil(x_in.shape[1]/2)) ))\n", - " # TO DO -- write the subsampling routine\n", + " # TODO -- write the subsampling routine\n", " # Replace this line\n", " x_out = x_out\n", "\n", @@ -159,7 +159,7 @@ "# Now let's try max-pooling\n", "def maxpool(x_in):\n", " x_out = np.zeros(( int(np.floor(x_in.shape[0]/2)), int(np.floor(x_in.shape[1]/2)) ))\n", - " # TO DO -- write the maxpool routine\n", + " # TODO -- write the maxpool routine\n", " # Replace this line\n", " x_out = x_out\n", "\n", @@ -230,7 +230,7 @@ "# Finally, let's try mean pooling\n", "def meanpool(x_in):\n", " x_out = np.zeros(( int(np.floor(x_in.shape[0]/2)), int(np.floor(x_in.shape[1]/2)) ))\n", - " # TO DO -- write the meanpool routine\n", + " # TODO -- write the meanpool routine\n", " # Replace this line\n", " x_out = x_out\n", "\n", @@ -316,7 +316,7 @@ "# Let's first use the duplication method\n", "def duplicate(x_in):\n", " x_out = np.zeros(( x_in.shape[0]*2, x_in.shape[1]*2 ))\n", - " # TO DO -- write the duplication routine\n", + " # TODO -- write the duplication routine\n", " # Replace this line\n", " x_out = x_out\n", "\n", @@ -388,7 +388,7 @@ "# The input x_high_res is the original high res image, from which you can deduce the position of the maximum index\n", "def max_unpool(x_in, x_high_res):\n", " x_out = np.zeros(( x_in.shape[0]*2, x_in.shape[1]*2 ))\n", - " # TO DO -- write the subsampling routine\n", + " # TODO -- write the subsampling routine\n", " # Replace this line\n", " x_out = x_out\n", "\n", @@ -460,7 +460,7 @@ " x_out = np.zeros(( x_in.shape[0]*2, x_in.shape[1]*2 ))\n", " x_in_pad = np.zeros((x_in.shape[0]+1, x_in.shape[1]+1))\n", " x_in_pad[0:x_in.shape[0],0:x_in.shape[1]] = x_in\n", - " # TO DO -- write the duplication routine\n", + " # TODO -- write the duplication routine\n", " # Replace this line\n", " x_out = x_out\n", "\n", diff --git a/Notebooks/Chap21/21_1_Bias_Mitigation.ipynb b/Notebooks/Chap21/21_1_Bias_Mitigation.ipynb index 2235dd0..8dbcc8b 100644 --- a/Notebooks/Chap21/21_1_Bias_Mitigation.ipynb +++ b/Notebooks/Chap21/21_1_Bias_Mitigation.ipynb @@ -328,7 +328,7 @@ }, "outputs": [], "source": [ - "# TO DO -- try to change the two thresholds so the overall probability of getting the loan is 0.6 for each group\n", + "# TODO -- try to change the two thresholds so the overall probability of getting the loan is 0.6 for each group\n", "# Change the values in these lines\n", "tau0 = 0.3\n", "tau1 = -0.1\n", @@ -393,7 +393,7 @@ }, "outputs": [], "source": [ - "# TO DO -- try to change the two thresholds so the true positive are 0.8 for each group\n", + "# TODO --try to change the two thresholds so the true positive are 0.8 for each group\n", "# Change the values in these lines so that both points on the curves have a height of 0.8\n", "tau0 = -0.1\n", "tau1 = -0.7\n", From eaff933ff7deffac707538781c0fa9981dc25674 Mon Sep 17 00:00:00 2001 From: udlbook <110402648+udlbook@users.noreply.github.com> Date: Mon, 2 Dec 2024 15:43:55 -0500 Subject: [PATCH 43/49] Created using Colab --- Notebooks/Chap04/4_3_Deep_Networks.ipynb | 25 ++++++++++++------------ 1 file changed, 12 insertions(+), 13 deletions(-) diff --git a/Notebooks/Chap04/4_3_Deep_Networks.ipynb b/Notebooks/Chap04/4_3_Deep_Networks.ipynb index 5754e95..55eb037 100644 --- a/Notebooks/Chap04/4_3_Deep_Networks.ipynb +++ b/Notebooks/Chap04/4_3_Deep_Networks.ipynb @@ -4,7 +4,6 @@ "metadata": { "colab": { "provenance": [], - "authorship_tag": "ABX9TyO2DaD75p+LGi7WgvTzjrk1", "include_colab_link": true }, "kernelspec": { @@ -31,7 +30,7 @@ "source": [ "# **Notebook 4.3 Deep neural networks**\n", "\n", - "This network investigates converting neural networks to matrix form.\n", + "This notebook investigates converting neural networks to matrix form.\n", "\n", "Work through the cells below, running each cell in turn. In various places you will see the words \"TO DO\". Follow the instructions at these places and make predictions about what is going to happen or write code to complete the functions.\n", "\n", @@ -150,7 +149,7 @@ { "cell_type": "markdown", "source": [ - "Now we'll define the same neural network, but this time, we will use matrix form. When you get this right, it will draw the same plot as above." + "Now we'll define the same neural network, but this time, we will use matrix form as in equation 4.15. When you get this right, it will draw the same plot as above." ], "metadata": { "id": "XCJqo_AjfAra" @@ -176,8 +175,8 @@ "n1_in_mat = np.reshape(n1_in,(n_dim_in,n_data))\n", "\n", "# This runs the network for ALL of the inputs, x at once so we can draw graph\n", - "h1 = ReLU(np.matmul(beta_0,np.ones((1,n_data))) + np.matmul(Omega_0,n1_in_mat))\n", - "n1_out = np.matmul(beta_1,np.ones((1,n_data))) + np.matmul(Omega_1,h1)\n", + "h1 = ReLU(beta_0 + np.matmul(Omega_0,n1_in_mat))\n", + "n1_out = beta_1 + np.matmul(Omega_1,h1)\n", "\n", "# Draw the network and check that it looks the same as the non-matrix case\n", "plot_neural(n1_in, n1_out)" @@ -247,9 +246,9 @@ "n1_in_mat = np.reshape(n1_in,(n_dim_in,n_data))\n", "\n", "# This runs the network for ALL of the inputs, x at once so we can draw graph (hence extra np.ones term)\n", - "h1 = ReLU(np.matmul(beta_0,np.ones((1,n_data))) + np.matmul(Omega_0,n1_in_mat))\n", - "h2 = ReLU(np.matmul(beta_1,np.ones((1,n_data))) + np.matmul(Omega_1,h1))\n", - "n1_out = np.matmul(beta_2,np.ones((1,n_data))) + np.matmul(Omega_2,h2)\n", + "h1 = ReLU(beta_0 + np.matmul(Omega_0,n1_in_mat))\n", + "h2 = ReLU(beta_1 + np.matmul(Omega_1,h1))\n", + "n1_out = beta_2 + np.matmul(Omega_2,h2)\n", "\n", "# Draw the network and check that it looks the same as the non-matrix version\n", "plot_neural(n1_in, n1_out)" @@ -291,10 +290,10 @@ "\n", "\n", "# If you set the parameters to the correct sizes, the following code will run\n", - "h1 = ReLU(np.matmul(beta_0,np.ones((1,n_data))) + np.matmul(Omega_0,x));\n", - "h2 = ReLU(np.matmul(beta_1,np.ones((1,n_data))) + np.matmul(Omega_1,h1));\n", - "h3 = ReLU(np.matmul(beta_2,np.ones((1,n_data))) + np.matmul(Omega_2,h2));\n", - "y = np.matmul(beta_3,np.ones((1,n_data))) + np.matmul(Omega_3,h3)\n", + "h1 = ReLU(beta_0 + np.matmul(Omega_0,x));\n", + "h2 = ReLU(beta_1 + np.matmul(Omega_1,h1));\n", + "h3 = ReLU(beta_2 + np.matmul(Omega_2,h2));\n", + "y = beta_3 + np.matmul(Omega_3,h3)\n", "\n", "if h1.shape[0] is not D_1 or h1.shape[1] is not n_data:\n", " print(\"h1 is wrong shape\")\n", @@ -318,4 +317,4 @@ "outputs": [] } ] -} +} \ No newline at end of file From dd9a56d96b2bed85cb2948e3088973e4bbe48d72 Mon Sep 17 00:00:00 2001 From: udlbook <110402648+udlbook@users.noreply.github.com> Date: Mon, 16 Dec 2024 16:06:30 -0500 Subject: [PATCH 44/49] Created using Colab --- Notebooks/Chap07/7_2_Backpropagation.ipynb | 8 ++++---- 1 file changed, 4 insertions(+), 4 deletions(-) diff --git a/Notebooks/Chap07/7_2_Backpropagation.ipynb b/Notebooks/Chap07/7_2_Backpropagation.ipynb index b3f3cd8..432cced 100644 --- a/Notebooks/Chap07/7_2_Backpropagation.ipynb +++ b/Notebooks/Chap07/7_2_Backpropagation.ipynb @@ -67,7 +67,7 @@ "# Set seed so we always get the same random numbers\n", "np.random.seed(0)\n", "\n", - "# Number of layers\n", + "# Number of hidden layers\n", "K = 5\n", "# Number of neurons per layer\n", "D = 6\n", @@ -114,7 +114,7 @@ { "cell_type": "markdown", "source": [ - "Now let's run our random network. The weight matrices $\\boldsymbol\\Omega_{1\\ldots K}$ are the entries of the list \"all_weights\" and the biases $\\boldsymbol\\beta_{1\\ldots K}$ are the entries of the list \"all_biases\"\n", + "Now let's run our random network. The weight matrices $\\boldsymbol\\Omega_{0\\ldots K}$ are the entries of the list \"all_weights\" and the biases $\\boldsymbol\\beta_{0\\ldots K}$ are the entries of the list \"all_biases\"\n", "\n", "We know that we will need the preactivations $\\mathbf{f}_{0\\ldots K}$ and the activations $\\mathbf{h}_{1\\ldots K}$ for the forward pass of backpropagation, so we'll store and return these as well.\n" ], @@ -299,7 +299,7 @@ "delta_fd = 0.000001\n", "\n", "# Test the dervatives of the bias vectors\n", - "for layer in range(K):\n", + "for layer in range(K+1):\n", " dl_dbias = np.zeros_like(all_dl_dbiases[layer])\n", " # For every element in the bias\n", " for row in range(all_biases[layer].shape[0]):\n", @@ -323,7 +323,7 @@ "\n", "\n", "# Test the derivatives of the weights matrices\n", - "for layer in range(K):\n", + "for layer in range(K+1):\n", " dl_dweight = np.zeros_like(all_dl_dweights[layer])\n", " # For every element in the bias\n", " for row in range(all_weights[layer].shape[0]):\n", From b5c65665b684a326b086de0f6042d81cafaaee1c Mon Sep 17 00:00:00 2001 From: udlbook <110402648+udlbook@users.noreply.github.com> Date: Wed, 18 Dec 2024 09:06:15 -0500 Subject: [PATCH 45/49] Update 10_4_Downsampling_and_Upsampling.ipynb --- Notebooks/Chap10/10_4_Downsampling_and_Upsampling.ipynb | 2 +- 1 file changed, 1 insertion(+), 1 deletion(-) diff --git a/Notebooks/Chap10/10_4_Downsampling_and_Upsampling.ipynb b/Notebooks/Chap10/10_4_Downsampling_and_Upsampling.ipynb index d0585a5..3e3d96b 100644 --- a/Notebooks/Chap10/10_4_Downsampling_and_Upsampling.ipynb +++ b/Notebooks/Chap10/10_4_Downsampling_and_Upsampling.ipynb @@ -31,7 +31,7 @@ "source": [ "# **Notebook 10.4: Downsampling and Upsampling**\n", "\n", - "This notebook investigates the down sampling and downsampling methods discussed in section 10.4 of the book.\n", + "This notebook investigates the upsampling and downsampling methods discussed in section 10.4 of the book.\n", "\n", "Work through the cells below, running each cell in turn. In various places you will see the words \"TO DO\". Follow the instructions at these places and make predictions about what is going to happen or write code to complete the functions.\n", "\n", From 6d76e47849ef8e296ff6535983e5982b56632d67 Mon Sep 17 00:00:00 2001 From: udlbook <110402648+udlbook@users.noreply.github.com> Date: Sun, 29 Dec 2024 17:13:26 -0500 Subject: [PATCH 46/49] Created using Colab --- Notebooks/Chap07/7_2_Backpropagation.ipynb | 2 +- 1 file changed, 1 insertion(+), 1 deletion(-) diff --git a/Notebooks/Chap07/7_2_Backpropagation.ipynb b/Notebooks/Chap07/7_2_Backpropagation.ipynb index 432cced..be113ec 100644 --- a/Notebooks/Chap07/7_2_Backpropagation.ipynb +++ b/Notebooks/Chap07/7_2_Backpropagation.ipynb @@ -141,7 +141,7 @@ "\n", " # Run through the layers, calculating all_f[0...K-1] and all_h[1...K]\n", " for layer in range(K):\n", - " # Update preactivations and activations at this layer according to eqn 7.16\n", + " # Update preactivations and activations at this layer according to eqn 7.17\n", " # Remember to use np.matmul for matrix multiplications\n", " # TODO -- Replace the lines below\n", " all_f[layer] = all_h[layer]\n", From 9a2039d3927c2b90e587dcb29ab27852dd90ea05 Mon Sep 17 00:00:00 2001 From: udlbook <110402648+udlbook@users.noreply.github.com> Date: Thu, 23 Jan 2025 15:40:43 -0500 Subject: [PATCH 47/49] Add files via upload --- UDL_Equations.pdf | Bin 0 -> 353977 bytes understanding-deep-learning-final.bib | 8672 +++++++++++++++++++++++++ 2 files changed, 8672 insertions(+) create mode 100644 UDL_Equations.pdf create mode 100644 understanding-deep-learning-final.bib diff --git a/UDL_Equations.pdf b/UDL_Equations.pdf new file mode 100644 index 0000000000000000000000000000000000000000..31e7f1d2b26033c32102693bd805ac6e6ef321f9 GIT binary patch literal 353977 zcmeFYV~{3Yw=G(>tIK-Iwr$(CZQHhO+w8J!SC`pkck%XnzrD}h``qtFoIm&cI#0wC znKN_cj9g>QHP*D^O8~S1jBHFS07eEz4uCFzULL^6{~)#RBlp{fb%IxR^Tq z-P#zsn2MMh+nfAt^{=ghp_8pM;2#i#?QL9b?VJHD0D1){Qxi*L7keiFBh%kb0qE8L 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zgv%mqB~eIOu%bNJ7J{;Ypd?XJP-#h9JE)|TtQ{CED}#`dfEJb8Spq`Nj$PwjLx>Ue?kz;7&0 zL^wldk$UJPW36WM%l`cSZTCVv!_b3k=?bmQxo&Y2Y}`Xlv_o`FvBA^Maf&mmpDfAY zX$I3n=Y<|0xA}S{TGPVvQ)({A6&Y>u`4?{HCA3Z97%Iz~HzrQagRy61`!K!z{|zVL he~$lX;NM9Dq;U}f?dgj~+0jTsq-g{N)pTyt{0+ok;bZ^+ literal 0 HcmV?d00001 diff --git a/understanding-deep-learning-final.bib b/understanding-deep-learning-final.bib new file mode 100644 index 0000000..55fff75 --- /dev/null +++ b/understanding-deep-learning-final.bib @@ -0,0 +1,8672 @@ +@article{cauchy1847methode, + author = {Cauchy, A.}, + journal = {Comptes Rendus de l'Académie des Sciences}, + title = {Methode generale pour la resolution des systemes d'equations simultanees}, + volume = 25, + year = 1847 +} +@incollection{whitney1992congruent, + author = {Whitney, Hassler}, + booktitle = {Hassler Whitney Collected Papers}, + pages = {61--79}, + title = {Congruent graphs and the connectivity of graphs}, + year = 1932 +} +@article{thompson1933likelihood, + author = {Thompson, William R}, + journal = {Biometrika}, + number = {3-4}, + pages = {285--294}, + publisher = {Oxford University Press}, + title = {On the likelihood that one unknown probability exceeds another in view of the evidence of two samples}, + volume = 25, + year = 1933 +} +@article{thompson1935theory, + author = {Thompson, William R}, + journal = {American Journal of Mathematics}, + number = 2, + pages = {450--456}, + publisher = {JSTOR}, + title = {On the theory of apportionment}, + volume = 57, + year = 1935 +} +@article{mcculloch1943logical, + author = {McCulloch, Warren S and Pitts, Walter}, + journal = {The Bulletin of Mathematical Biophysics}, + number = 4, + pages = {115--133}, + publisher = {Springer}, + title = {A logical calculus of the ideas immanent in nervous activity}, + volume = 5, + year = 1943 +} +@article{robbins1951stochastic, + abstract = {Let M(x) denote the expected value at level x of the response to a certain experiment. M(x) is assumed to be a monotone function of x but is unknown to the experimenter, and it is desired to find the solution x = θ of the equation M(x) = α, where α is a given constant. We give a method for making successive experiments at levels x1,x2,⋯ in such a way that xn will tend to θ in probability.}, + author = {Herbert Robbins and Sutton Monro}, + issn = {00034851}, + journal = {The Annals of Mathematical Statistics}, + number = 3, + pages = {400--407}, + publisher = {Institute of Mathematical Statistics}, + title = {A Stochastic Approximation Method}, + volume = 22, + year = 1951 +} +@article{rosenblatt1958perceptron, + author = {Rosenblatt, Frank}, + journal = {Psychological review}, + number = 6, + pages = 386, + publisher = {American Psychological Association}, + title = {The perceptron: {A} probabilistic model for information storage and organization in the brain}, + volume = 65, + year = 1958 +} +@book{howard1960dynamic, + author = {Howard, Ronald A}, + publisher = {Wiley}, + title = {Dynamic programming and {N}arkov processes}, + year = 1960 +} +@article{POLYAK19641some, + author = {Polyak, Boris T}, + journal = {USSR Computational Mathematics and Mathematical Physics}, + number = 5, + pages = {1--17}, + publisher = {Elsevier}, + title = {Some methods of speeding up the convergence of iteration methods}, + volume = 4, + year = 1964 +} +@article{bellman1966dynamic, + author = {Bellman, Richard}, + journal = {Science}, + number = 3731, + pages = {34--37}, + publisher = {American Association for the Advancement of Science}, + title = {Dynamic programming}, + volume = 153, + year = 1966 +} +@article{weisfeiler1968reduction, + author = {Weisfeiler, Boris and Leman, Andrei}, + journal = {NTI, Series}, + number = 9, + pages = {12--16}, + title = {The reduction of a graph to canonical form and the algebra which appears therein}, + volume = 2, + year = 1968 +} +@book{minsky1969perceptrons, + author = {Minsky, Marvin and Papert, Seymour A}, + publisher = {MIT {P}ress}, + title = {Perceptrons: An introduction to computational geometry}, + year = 1969 +} +@article{fukushima1969visual, + author = {Fukushima, Kunihiko}, + doi = 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Harvard University}, + title = {Beyond regression: New tools for prediction and analysis in the behavioral sciences}, + year = 1974 +} +@book{Braverman-1974, + author = {Harry Braverman}, + publisher = {Monthly Review Press}, + title = {Labor and monopoly capital: the degradation of work in the twentieth century}, + year = 1974 +} +@book{zaslavsky1975facing, + author = {Zaslavsky, Thomas}, + publisher = {Memoirs of the American Mathematical Society}, + title = {Facing up to arrangements: Face-count formulas for partitions of space by hyperplanes: Face-count formulas for partitions of space by hyperplanes}, + year = 1975 +} +@article{dempster1977maximum, + author = {Dempster, Arthur P and Laird, Nan M and Rubin, Donald B}, + journal = {Journal of the Royal Statistical Society: Series B}, + number = 1, + pages = {1--22}, + publisher = {Wiley Online Library}, + title = {Maximum likelihood from incomplete data via the {EM} algorithm}, + volume = 39, + year = 1977 +} +@article{bryson1979applied, + author = {Bryson, Arthur and Ho, Y.-C and Siouris, George}, + doi = {10.1109/TSMC.1979.4310229}, + journal = {IEEE Transactions on Systems, Man \& Cybernetics}, + month = {07}, + pages = {366--367}, + title = {Applied Optimal Control: Optimization, Estimation, and Control}, + volume = 9, + year = 1979 +} +@incollection{fukushima1982neocognitron, + author = {Fukushima, Kunihiko and Miyake, Sei}, + booktitle = {Competition and Cooperation in Neural Nets}, + pages = {267--285}, + title = {Neocognitron: A self-organizing neural network model for a mechanism of visual pattern recognition}, + year = 1982 +} +@article{rubin1982algorithms, + author = {Rubin, Donald B and Thayer, Dorothy T}, + journal = {Psychometrika}, + number = 1, + pages = {69--76}, + publisher = {Springer}, + title = {{EM} algorithms for {ML} factor analysis}, + volume = 47, + year = 1982 +} +@inproceedings{nesterov1983method, + author = {Nesterov, Yu E}, + booktitle = {Doklady Akademii Nauk SSSR}, + pages = {543--547}, + title = {A method for solving the convex programming problem with convergence rate}, + volume = 269, + year = 1983 +} +@article{rissanen1983universal, + author = {Rissanen, Jorma}, + journal = {The Annals of Statistics}, + number = 2, + pages = {416--431}, + publisher = {Institute of Mathematical Statistics}, + title = {A universal prior for integers and estimation by minimum description length}, + volume = 11, + year = 1983 +} +@book{sutton1984temporal, + author = {Sutton, Richard Stuart}, + publisher = {Ph.D., University of Massachusetts Amherst}, + title = {Temporal credit assignment in reinforcement learning}, + year = 1984 +} +@inproceedings{lecun1985une, + author = {LeCun, Yann}, + booktitle = {Proceedings of Cognitiva}, + language = {English (US)}, + pages = {599--604}, + title = {Une procedure d'apprentissage pour reseau a seuil asymmetrique}, + year = 1985 +} +@book{parker1985learning, + author = {Parker, David B}, + publisher = {Alfred P. 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Murty and Santosh N. Kabadi}, + journal = {Mathematical Programming}, + number = 2, + pages = {117--129}, + title = {Some {NP}-complete problems in quadratic and nonlinear programming}, + volume = 39, + year = 1987 +} +@incollection{Goldberg-1987, + author = {David E. 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Holland}, + booktitle = {European Conference on Artificial Life}, + title = {The Royal Road for Genetic Algorithms: Fitness Landscapes and {GA} Performance}, + year = 1992 +} +@inproceedings{hinton93keeping, + author = {Hinton, GE and van Camp, Drew}, + booktitle = {Computational learning theory}, + pages = {5--13}, + title = {Keeping neural networks simple by minimising the description length of weights}, + year = 1993 +} +@article{martin1993centered, + author = {Martin, Gale L}, + journal = {Neural Computation}, + number = 3, + pages = {419--429}, + publisher = {MIT Press One Rogers Street, Cambridge, MA 02142-1209, USA journals-info~…}, + title = {Centered-object integrated segmentation and recognition of overlapping handprinted characters}, + volume = 5, + year = 1993 +} +@article{bromley1993signature, + author = {Bromley, Jane and Guyon, Isabelle and LeCun, Yann and S{\"a}ckinger, Eduard and Shah, Roopak}, + journal = {Neural Information Processing Systems}, + pages = {737--744}, + title = {Signature verification using a ``{S}iamese'' time delay neural network}, + volume = 6, + year = 1993 +} +@inproceedings{hassibi1993second, + author = {Hassibi, Babak and Stork, David G}, + booktitle = {Neural Information Processing Systems}, + pages = {164--171}, + title = {Second order derivatives for network pruning: Optimal brain surgeon}, + volume = 6, + year = 1993 +} +@inproceedings{nix1994estimating, + author = {Nix, David A and Weigend, Andreas S}, + booktitle = {IEEE International Conference on Neural Networks}, + pages = {55--60}, + title = {Estimating the mean and variance of the target probability distribution}, + year = 1994 +} +@article{bishop1994mixture, + author = {Bishop, Christopher M}, + journal = {Aston University Technical Report}, + title = {Mixture density networks}, + year = 1994 +} +@book{rummery1994line, + author = {Rummery, Gavin A and Niranjan, Mahesan}, + publisher = {Technical Report, University of Cambridge}, + title = {On-line Q-learning using connectionist systems}, + year = 1994 +} +@book{vapnik95, + address = {New York}, + author = {V. 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Longino}, + booktitle = {Feminism, Science, and the Philosophy of Science}, + pages = {39--58}, + title = {Cognitive and Non-Cognitive Values in Science: Rethinking the Dichotomy}, + year = 1996 +} +@article{freund95, + author = {Freund, Yoav and Schapire, Robert E}, + journal = {Journal of Computer and System Sciences}, + number = 1, + pages = {119--139}, + publisher = {Elsevier}, + title = {A decision-theoretic generalization of on-line learning and an application to boosting}, + volume = 55, + year = 1997 +} +@article{friedman1997bias, + author = {Friedman, Jerome H}, + journal = {Data Mining and Knowledge Discovery}, + number = 1, + pages = {55--77}, + publisher = {Springer}, + title = {On bias, variance, 0/1—loss, and the curse-of-dimensionality}, + volume = 1, + year = 1997 +} +@article{barber1997ensemble, + author = {Barber, David and Bishop, Christopher}, + journal = {Neural Information Processing Systems}, + pages = {395--401}, + title = {Ensemble learning for multi-layer networks}, + volume = 10, + year = 1997 +} +@article{lawrence1997face, + author = {Lawrence, Steve and Giles, C Lee and Tsoi, Ah Chung and Back, Andrew D}, + journal = {IEEE Transactions on Neural Networks}, + number = 1, + pages = {98--113}, + title = {Face recognition: A convolutional neural-network approach}, + volume = 8, + year = 1997 +} +@article{hochreiter1997long, + author = {Hochreiter, Sepp and Schmidhuber, J{\"u}rgen}, + journal = {Neural Computation}, + number = 8, + pages = {1735--1780}, + publisher = {MIT Press}, + title = {Long short-term memory}, + volume = 9, + year = 1997 +} +@article{hochreiter1997flat, + author = {Hochreiter, Sepp and Schmidhuber, J{\"u}rgen}, + journal = {Neural Computation}, + number = 1, + pages = {1--42}, + title = {Flat minima}, + volume = 9, + year = 1997 +} +@article{lecun1998gradient, + author = {LeCun, Yann and Bottou, L{\'e}on and Bengio, Yoshua and Haffner, Patrick}, + journal = {Proceedings of the IEEE}, + number = 11, + pages = {2278--2324}, + publisher = {Ieee}, + title = {Gradient-based learning applied to document recognition}, + volume = 86, + year = 1998 +} +@article{gales1998maximum, + author = {Gales, Mark JF}, + journal = {Computer Speech \& Language}, + number = 2, + pages = {75--98}, + publisher = {Elsevier}, + title = {Maximum likelihood linear transformations for {HMM}-based speech recognition}, + volume = 12, + year = 1998 +} +@article{amari1998natural, + author = {Amari, Shun-Ichi}, + journal = {Neural {C}omputation}, + number = 2, + pages = {251--276}, + publisher = {MIT Press}, + title = {Natural gradient works efficiently in learning}, + volume = 10, + year = 1998 +} +@inproceedings{beyer1999nearest, + author = {Beyer, Kevin and Goldstein, Jonathan and Ramakrishnan, Raghu and Shaft, Uri}, + booktitle = {International Conference on Database Theory}, + pages = {217--235}, + title = {When is “nearest neighbor” meaningful?}, + year = 1999 +} +@book{manning1999foundations, + author = {Manning, Christopher and Schutze, Hinrich}, + publisher = {MIT {P}ress}, + title = {Foundations of statistical natural language processing}, + year = 1999 +} +@article{tipping1999probabilistic, + author = {Tipping, Michael E and Bishop, Christopher M}, + journal = {Journal of the Royal Statistical Society: Series B}, + number = 3, + pages = {611--622}, + publisher = {Wiley Online Library}, + title = {Probabilistic principal component analysis}, + volume = 61, + year = 1999 +} +@book{sutton1999reinforcement, + author = {Sutton, Richard S and Barto, Andrew G}, + publisher = {MIT press}, + title = {Reinforcement learning: An introduction}, + year = 1999 +} +@article{sutton1999policy, + author = {Sutton, Richard S and McAllester, David and Singh, Satinder and Mansour, Yishay}, + journal = {Neural Information Processing Systems}, + pages = {1057--1063}, + title = {Policy gradient methods for reinforcement learning with function approximation}, + volume = 12, + year = 1999 +} +@article{konda1999actor, + author = {Konda, Vijay and Tsitsiklis, John}, + journal = {Neural Information Processing Systems}, + pages = {1008--1014}, + title = {Actor-critic algorithms}, + volume = 12, + year = 1999 +} +@book{cristianini00, + author = {M. Cristianini and J. 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Laffont and David Martimort}, + publisher = {Princeton University Press}, + title = {The Theory of Incentives: The Principal-Agent Model}, + year = 2002 +} +@inproceedings{ye2004high, + author = {Ye, Hui and Young, Steve}, + booktitle = {IEEE International Conference on Acoustics, Speech, and Signal Processing}, + pages = {1--9}, + title = {High quality voice morphing}, + year = 2004 +} +@article{greensmith2004variance, + author = {Greensmith, Evan and Bartlett, Peter L and Baxter, Jonathan}, + journal = {Journal of Machine Learning Research}, + number = 9, + pages = {1471--1530}, + title = {Variance Reduction Techniques for Gradient Estimates in Reinforcement Learning}, + volume = 5, + year = 2004 +} +@article{zou2005regularization, + author = {Zou, Hui and Hastie, Trevor}, + journal = {Journal of the Royal Statistical Society: Series B}, + number = 2, + pages = {301--320}, + title = {Regularization and variable selection via the elastic net}, + volume = 67, + year = 2005 +} 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+@book{williams2006gaussian, + author = {Williams, Christopher KI and Rasmussen, Carl Edward}, + publisher = {MIT Press}, + title = {Gaussian processes for machine learning}, + year = 2006 +} +@article{lecun2006tutorial, + author = {LeCun, Yann and Chopra, Sumit and Hadsell, Raia and Ranzato, M and Huang, F}, + editor = {BakIr, G{\"o}khan and Hofmann, Thomas and Smola, Alexander J and Sch{\"o}lkopf, Bernhard and Taskar, Ben}, + journal = {Predicting structured data}, + number = {0}, + title = {A tutorial on energy-based learning}, + volume = 1, + year = 2006 +} +@article{hinton2006reducing, + author = {Hinton, Geoffrey E and Salakhutdinov, Ruslan R}, + journal = {Science}, + number = 5786, + pages = {504--507}, + publisher = {American Association for the Advancement of Science}, + title = {Reducing the dimensionality of data with neural networks}, + volume = 313, + year = 2006 +} +@inproceedings{buciluǎ2006model, + author = {Buciluǎ, Cristian and Caruana, Rich and Niculescu-Mizil, Alexandru}, + booktitle = {ACM SIGKDD International Conference on Knowledge Discovery and Data Mining}, + pages = {535--541}, + title = {Model compression}, + year = 2006 +} +@article{Moor-2006, + author = {James Moor}, + journal = {Intelligence Systems}, + number = 4, + pages = {18--21}, + title = {The Nature, Importance, and Difficulty of Machine Ethics}, + volume = 21, + year = 2006 +} +@inproceedings{cao2007learning, + author = {Cao, Zhe and Qin, Tao and Liu, Tie-Yan and Tsai, Ming-Feng and Li, Hang}, + booktitle = {International Conference on Machine Learning}, + pages = {129--136}, + title = {Learning to rank: {F}rom pairwise approach to listwise approach}, + year = 2007 +} +@inproceedings{xia2008listwise, + author = {Xia, Fen and Liu, Tie-Yan and Wang, Jue and Zhang, Wensheng and Li, Hang}, + booktitle = {International Conference on Machine Learning}, + pages = {1192--1199}, + title = {Listwise approach to learning to rank: theory and algorithm}, + year = 2008 +} 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pages = {682--697}, + publisher = {Elsevier}, + title = {Reinforcement learning of motor skills with policy gradients}, + volume = 21, + year = 2008 +} +@inproceedings{Lehman-Stanley-2008, + author = {Joel Lehman and Kenneth O. Stanley}, + booktitle = {International Conference on Artificial Life}, + pages = {329--336}, + title = {Exploiting Open-endedness to Solve Problems Through the Search for Novelty}, + year = 2008 +} +@inproceedings{Arkin-2008a, + author = {Ronald C. Arkin}, + booktitle = {ACM/IEEE International Conference on Human Robot Interaction}, + pages = {121--128}, + title = {Governing lethal behavior: Embedding ethics in a hybrid deliberative/reactive robot architecture---{P}art {I}: {M}otivation and philosophy}, + year = 2008 +} +@inproceedings{Arkin-2008b, + author = {Ronald C. Arkin}, + booktitle = {Conference on Artificial General Intelligence}, + pages = {51--62}, + title = {Governing lethal behavior: Embedding ethics in a hybrid deliberative/reactive robot architecture---{P}art {II}: {F}ormalization for ethical control}, + year = 2008 +} +@incollection{Anderson-Anderson-2008, + author = {Michael Anderson and Susan Leigh Anderson}, + booktitle = {Advanced Computational Intelligence Paradigms in Healthcare 3. Studies in Computational Intelligence}, + pages = {233--257}, + title = {Ethical healthcare agents}, + volume = 107, + year = 2008 +} +@article{Wallach-et-al-2008, + author = {Wendell Wallach and Colin Allen and Iva Smit}, + journal = {AI \& Society}, + number = 4, + pages = {565--582}, + title = {Machine Morality: Bottom-Up and Top-Down Approaches for Modeling Human Moral Faculties}, + volume = 22, + year = 2008 +} +@incollection{Narayanan-Shmatikov-2008, + author = {Arvind Narayanan and Vitaly Shmatikov}, + booktitle = {IEEE Symposium on Security and Privacy}, + pages = {111--125}, + title = {Robust de-anonymization of large sparse datasets}, + year = 2008 +} +@book{koller2009probabilistic, + author = {Koller, Daphne and Friedman, Nir}, + publisher = {MIT {P}ress}, + title = {Probabilistic graphical models: Principles and techniques}, + year = 2009 +} +@inproceedings{jarrett2009what, + author = {Jarrett, Kevin and Kavukcuoglu, Koray and Ranzato, Marc'Aurelio and LeCun, Yann}, + booktitle = {IEEE International Conference on Computer Vision}, + doi = {10.1109/ICCV.2009.5459469}, + number = {}, + pages = {2146--2153}, + title = {What is the best multi-stage architecture for object recognition?}, + volume = {}, + year = 2009 +} +@article{chen2009ranking, + author = {Chen, Wei and Liu, Tie-Yan and Lan, Yanyan and Ma, Zhi-Ming and Li, Hang}, + journal = {Neural Information Processing Systems}, + pages = {315--323}, + title = {Ranking measures and loss functions in learning to rank}, + volume = 22, + year = 2009 +} +@article{fallah2009nonlinear, + author = {Fallah, Nader and Gu, Hong and Mohammad, Kazem and Seyyedsalehi, Seyyed Ali and Nourijelyani, Keramat and Eshraghian, Mohammad Reza}, + journal = {Neural Computing and Applications}, + number = 8, + pages = {939--943}, + publisher = {Springer}, + title = {Nonlinear {P}oisson regression using neural networks: {A} simulation study}, + volume = 18, + year = 2009 +} +@article{mezard2008constraint, + author = 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+@inproceedings{bergstra2011algorithms, + author = {Bergstra, James S and Bardenet, R{\'e}mi and Bengio, Yoshua and K{\'e}gl, Bal{\'a}zs}, + booktitle = {Neural Information Processing Systems}, + pages = {2546--2554}, + title = {Algorithms for hyper-parameter optimization}, + volume = 24, + year = 2011 +} +@inproceedings{welling2011bayesian, + author = {Welling, Max and Teh, Yee W}, + booktitle = {International Conference on Machine Learning}, + pages = {681--688}, + title = {Bayesian learning via stochastic gradient {L}angevin dynamics}, + year = 2011 +} +@inproceedings{kolomiyets2011model, + author = {Kolomiyets, Oleksandr and Bethard, Steven and Moens, Marie-Francine}, + booktitle = {Meeting of the Association for Computational Linguistics}, + pages = {271--276}, + title = {Model-portability experiments for textual temporal analysis}, + year = 2011 +} +@inproceedings{zeiler2011adaptive, + author = {Zeiler, Matthew D and Taylor, Graham W and Fergus, Rob}, + booktitle = {IEEE International Conference on Computer Vision}, + pages = {2018--2025}, + title = {Adaptive deconvolutional networks for mid and high level feature learning}, + year = 2011 +} +@article{Vallor-2011, + author = {Shannon Vallor}, + journal = {Philosophy and Technology}, + number = 3, + pages = {251--268}, + title = {Carebots and caregivers: {S}ustaining the ethical ideal of care in the 21st century}, + volume = 24, + year = 2011 +} +@book{Kitcher-2011a, + author = {Philip Kitcher}, + publisher = {Harvard University Press}, + title = {The Ethical Project}, + year = 2011 +} +@book{Kitcher-2011b, + author = {Philip Kitcher}, + publisher = {Prometheus Books}, + title = {Science in a Democratic Society}, + year = 2011 +} +@book{prince2012computer, + author = {Prince, Simon Jeremy Damion}, + publisher = {Cambridge University Press}, + title = {Computer vision: Models, learning, and inference}, + year = 2012 +} +@article{krizhevsky2012imagenet, + author = {Krizhevsky, Alex and Sutskever, Ilya and Hinton, Geoffrey E}, + journal = {Neural Information Processing Systems}, + pages = {1097--1105}, + title = {Image{N}et Classification with Deep Convolutional Neural Networks}, + volume = 25, + year = 2012 +} +@inproceedings{Bottou2012stochastic, + abstract = {Chapter 1 strongly advocates the stochastic back-propagation method to train neural networks. This is in fact an instance of a more general technique called stochastic gradient descent (SGD). This chapter provides background material, explains why SGD is a good learning algorithm when the training set is large, and provides useful recommendations.}, + author = {Bottou, L{\'e}on}, + booktitle = {Neural Networks: Tricks of the Trade: Second Edition}, + pages = {421--436}, + title = {Stochastic Gradient Descent Tricks}, + year = 2012 +} +@misc{hinton2012rmsprop, + author = {Geoffrey Hinton and Nitish Srivastava and Kevin Swersky}, + howpublished = {\url{https://www.cs.toronto.edu/~tijmen/csc321/slides/lecture_slides_lec6.pdf}}, + title = {Neural Networks for Machine Learning: {L}ecture 6a -- {O}verview of mini-batch gradient descent}, + year = 2012 +} +@article{zeiler2012adadelta, + author = {Zeiler, Matthew D}, + journal = {arXiv:1212.5701}, + title = {{ADADELTA}: {A}n adaptive learning rate method}, + year = 2012 +} +@incollection{lecun2012efficient, + author = {LeCun, Yann A and Bottou, L{\'e}on and Orr, Genevieve B and M{\"u}ller, Klaus-Robert}, + booktitle = {Neural Networks: Tricks of the trade}, + pages = {9--48}, + publisher = {Springer}, + title = {Efficient backprop}, + year = 2012 +} +@article{moreno2012unifying, + author = {Moreno-Torres, Jose G and Raeder, Troy and Alaiz-Rodr{\'\i}guez, Roc{\'\i}o and Chawla, Nitesh V and Herrera, Francisco}, + journal = {Pattern Recognition}, + number = 1, + pages = {521--530}, + publisher = {Elsevier}, + title = {A unifying view on dataset shift in classification}, + volume = 45, + year = 2012 +} +@article{bergstra2012random, + author = {Bergstra, James and Bengio, Yoshua}, + journal = {Journal of Machine Learning Research}, + number = 10, + pages = {281--305}, + title = {Random search for hyper-parameter optimization}, + volume = 13, + year = 2012 +} +@inproceedings{snoek2012practical, + author = {Snoek, Jasper and Larochelle, Hugo and Adams, Ryan P}, + booktitle = {Neural Information Processing Systems}, + pages = {2951--2959}, + title = {Practical {B}ayesian optimization of machine learning algorithms}, + volume = 25, + year = 2012 +} +@article{hinton2012improving, + author = {Hinton, Geoffrey E and Srivastava, Nitish and Krizhevsky, Alex and Sutskever, Ilya and Salakhutdinov, Ruslan R}, + journal = {arXiv:1207.0580}, + title = {Improving neural networks by preventing co-adaptation of feature detectors}, + year = 2012 +} +@inproceedings{abdel2012applying, + author = {Abdel-Hamid, Ossama and Mohamed, Abdel-rahman and Jiang, Hui and Penn, Gerald}, + booktitle = {IEEE International Conference on Acoustics, Speech and Signal Processing}, + pages = {4277--4280}, + title = {Applying convolutional neural networks concepts to hybrid {NN-HMM} model for speech recognition}, + year = 2012 +} +@article{ji20123d, + author = {Ji, Shuiwang and Xu, Wei and Yang, Ming and Yu, Kai}, + journal = {IEEE Transactions on Pattern Analysis \& Machine Intelligence}, + number = 1, + pages = {221--231}, + title = {{3D} convolutional neural networks for human action recognition}, + volume = 35, + year = 2012 +} +@inproceedings{schuster2012japanese, + author = {Schuster, Mike and Nakajima, Kaisuke}, + booktitle = {IEEE International Conference on Acoustics, Speech and Signal Processing}, + pages = {5149--5152}, + title = {Japanese and {K}orean voice search}, + year = 2012 +} +@article{Sharkey-Sharkey-2012, + author = {Amanda Sharkey and Noel Sharkey}, + journal = {Ethics and Information Technology}, + number = 1, + pages = {27--40}, + title = {Granny and the robots: {E}thical issues in robot care for the elderly}, + volume = 14, + year = 2012 +} +@article{Asaro-2012, + author = {Peter Asaro}, + journal = {International Review of the Red Cross}, + number = 886, + pages = {687--709}, + title = {On banning autonomous weapon systems: human rights, automation, and the dehumanization of lethal decision-making}, + volume = 94, + year = 2012 +} +@inproceedings{maas2013rectifier, + author = {Maas, Andrew L and Hannun, Awni Y and Ng, Andrew Y}, + booktitle = {ICML Workshop on Deep Learning for Audio, Speech, and Language Processing}, + title = {Rectifier nonlinearities improve neural network acoustic models}, + year = 2013 +} +@article{pascanu2014number, + author = {Pascanu, Razvan and Mont{\'u}far, Guido and Bengio, Yoshua}, + journal = {arXiv:1312.6098}, + title = {On the number of response regions of deep feed forward networks with piece-wise linear activations}, + year = 2013 +} +@article{johnson2013accelerating, + author = {Johnson, Rie and Zhang, Tong}, + journal = {Neural Information Processing Systems}, + pages = {315--323}, + publisher = {Citeseer}, + title = {Accelerating stochastic gradient descent using predictive variance reduction}, + volume = 26, + year = 2013 +} +@inproceedings{sutskever2013importance, + author = {Sutskever, Ilya and Martens, James and Dahl, George and Hinton, Geoffrey}, + booktitle = {International Conference on Machine Learning}, + pages = {1139--1147}, + title = {On the importance of initialization and momentum in deep learning}, + year = 2013 +} +@inproceedings{wan2013regularization, + author = {Wan, Li and Zeiler, Matthew and Zhang, Sixin and Le Cun, Yann and Fergus, Rob}, + booktitle = {International Conference on Machine Learning}, + pages = {1058--1066}, + title = {Regularization of neural networks using {D}rop{C}onnect}, + year = 2013 +} +@inproceedings{jaitly2013vocal, + author = {Jaitly, Navdeep and Hinton, Geoffrey E}, + booktitle = {ICML Workshop on Deep Learning for Audio, Speech and Language}, + title = {Vocal tract length perturbation ({VTLP}) improves speech recognition}, + year = 2013 +} +@inproceedings{kanda2013elastic, + author = {Kanda, Naoyuki and Takeda, Ryu and Obuchi, Yasunari}, + booktitle = {IEEE Workshop on Automatic Speech Recognition and Understanding}, + pages = {309--314}, + title = {Elastic spectral distortion for low resource speech recognition with deep neural networks}, + year = 2013 +} +@inproceedings{sainath2013improvements, + author = {Sainath, Tara N and Kingsbury, Brian and Mohamed, Abdel-rahman and Dahl, George E and Saon, George and Soltau, Hagen and Beran, Tomas and Aravkin, Aleksandr Y and Ramabhadran, Bhuvana}, + booktitle = {IEEE Workshop on Automatic Speech Recognition and Understanding}, + pages = {315--320}, + title = {Improvements to deep convolutional neural networks for {LVCSR}}, + year = 2013 +} +@inproceedings{sifre2013rotation, + author = {Sifre, Laurent and Mallat, St{\'e}phane}, + booktitle = {IEEE/CVF Computer Vision \& Pattern Recognition}, + pages = {1233--1240}, + title = {Rotation, scaling and deformation invariant scattering for texture discrimination}, + year = 2013 +} +@article{bruna2013spectral, + author = {Bruna, Joan and Zaremba, Wojciech and Szlam, Arthur and LeCun, Yann}, + journal = {International Conference on Learning Representations}, + title = {Spectral networks and locally connected networks on graphs}, + year = 2013 +} +@article{tabak2013family, + author = {Tabak, Esteban G and Turner, Cristina V}, 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+} +@inproceedings{sharif2014cnn, + author = {Sharif Razavian, Ali and Azizpour, Hossein and Sullivan, Josephine and Carlsson, Stefan}, + booktitle = {IEEE Conference on Computer Vision and Pattern Recognition Workshop}, + pages = {806--813}, + title = {{CNN} features off-the-shelf: {A}n astounding baseline for recognition}, + year = 2014 +} +@article{hannun2014deep, + author = {Awni Y. 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Ng}, + journal = {arXiv:1412.5567}, + title = {Deep speech: Scaling up end-to-end speech recognition}, + year = 2014 +} +@article{simonyan2014very, + author = {Simonyan, Karen and Zisserman, Andrew}, + journal = {International Conference on Learning Representations}, + title = {Very deep convolutional networks for large-scale image recognition}, + year = 2014 +} +@article{lin2013network, + author = {Lin, Min and Chen, Qiang and Yan, Shuicheng}, + journal = {International Conference on Learning Representations}, + title = {Network in network}, + year = 2014 +} +@inproceedings{zeiler2014visualizing, + author = {Zeiler, Matthew D and Fergus, Rob}, + booktitle = {European Conference on Computer Vision}, + pages = {818--833}, + title = {Visualizing and understanding convolutional networks}, + year = 2014 +} +@article{kanazawa2014locally, + author = {Kanazawa, Angjoo and Sharma, Abhishek and Jacobs, David}, + journal = {Neural Information Processing Systems Workshop}, + title = {Locally scale-invariant convolutional neural networks}, + year = 2014 +} +@inproceedings{girshick2014rich, + author = {Girshick, Ross and Donahue, Jeff and Darrell, Trevor and Malik, Jitendra}, + booktitle = {IEEE Computer Vision \& Pattern Recognition}, + pages = {580--587}, + title = {Rich feature hierarchies for accurate object detection and semantic segmentation}, + year = 2014 +} +@inproceedings{cho2014properties, + author = {Kyunghyun Cho and Bart van Merrienboer and Dzmitry Bahdanau and Yoshua Bengio}, + booktitle = {ACL Workshop on Syntax, Semantics and Structure in Statistical Translation}, + pages = {103--111}, + title = {On the Properties of Neural Machine Translation: Encoder-Decoder Approaches}, + year = 2014 +} +@article{chung2014empirical, + author = {Chung, Junyoung and Gulcehre, Caglar and Cho, KyungHyun and Bengio, Yoshua}, + journal = {Deep Learning and Representation Workshop}, + title = {Empirical evaluation of gated recurrent neural networks on sequence modeling}, + year = 2014 +} +@article{goodfellow2014generative, + author = {Goodfellow, Ian and Pouget-Abadie, Jean and Mirza, Mehdi and Xu, Bing and Warde-Farley, David and Ozair, Sherjil and Courville, Aaron and Bengio, Yoshua}, + journal = {Communications of the ACM}, + number = 11, + pages = {139--144}, + publisher = {ACM New York, NY, USA}, + title = {Generative adversarial networks}, + volume = 63, + year = 2014 +} +@article{kingma2013auto, + author = {Kingma, Diederik P and Welling, Max}, + journal = {International Conference on Learning Representations}, + title = {Auto-encoding variational {B}ayes}, + year = 2014 +} +@article{mirza2014conditional, + author = {Mirza, Mehdi and Osindero, Simon}, + journal = {arXiv:1411.1784}, + title = {Conditional generative adversarial nets}, + year = 2014 +} +@inproceedings{rezende2014stochastic, + author = {Rezende, Danilo Jimenez and Mohamed, Shakir and Wierstra, Daan}, + booktitle = {International Conference on Machine Learning}, + pages = {1278--1286}, + title = {Stochastic backpropagation and approximate inference in deep generative models}, + year = 2014 +} +@inproceedings{silver2014deterministic, + author = {Silver, David and Lever, Guy and Heess, Nicolas and Degris, Thomas and Wierstra, Daan and Riedmiller, Martin}, + booktitle = {International Conference on Machine Learning}, + pages = {387--395}, + title = {Deterministic policy gradient algorithms}, + year = 2014 +} +@article{livni2014computational, + author = {Livni, Roi and Shalev-Shwartz, Shai and Shamir, Ohad}, + journal = {Neural Information Processing Systems}, + pages = {855--863}, + title = {On the computational efficiency of training neural networks}, + volume = 27, + year = 2014 +} +@article{pascanu2014saddle, + author = {Pascanu, Razvan and Dauphin, Yann N and Ganguli, Surya and Bengio, Yoshua}, + journal = {arXiv:1405.4604}, + title = {On the saddle point problem for non-convex optimization}, + year = 2014 +} +@inproceedings{dauphin2014identifying, + author = {Yann N. 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Efros}, + booktitle = {European Conference on Computer Vision}, + pages = {597--613}, + title = {Generative Visual Manipulation on the Natural Image Manifold}, + year = 2016 +} +@article{theis2015note, + author = {Theis, Lucas and Oord, A{\"a}ron van den and Bethge, Matthias}, + journal = {International Conference on Learning Representations}, + title = {A note on the evaluation of generative models}, + year = 2016 +} +@article{dinh2016density, + author = {Dinh, Laurent and Sohl-Dickstein, Jascha and Bengio, Samy}, + journal = {International Conference on Learning Representations}, + title = {Density estimation using {R}eal {NVP}}, + year = 2016 +} +@article{tomczak2016improving, + author = {Tomczak, Jakub M and Welling, Max}, + journal = {NIPS Workshop on Bayesian Deep Learning}, + title = {Improving variational auto-encoders using {H}ouseholder flow}, + year = 2016 +} +@article{kingma2016improved, + author = {Kingma, Durk P and Salimans, Tim and Jozefowicz, Rafal and Chen, Xi and 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Yoshua}, + booktitle = {International Conference on Machine Learning}, + pages = {2511--2519}, + title = {Bidirectional {H}elmholtz machines}, + year = 2016 +} +@article{sonderby2016ladder, + author = {S{\o}nderby, Casper Kaae and Raiko, Tapani and Maal{\o}e, Lars and S{\o}nderby, S{\o}ren Kaae and Winther, Ole}, + journal = {Neural Information Processing Systems}, + pages = {738--3746}, + title = {Ladder variational autoencoders}, + volume = 29, + year = 2016 +} +@article{silver2016mastering, + author = {Silver, David and Huang, Aja and Maddison, Chris J and Guez, Arthur and Sifre, Laurent and Van Den Driessche, George and Schrittwieser, Julian and Antonoglou, Ioannis and Panneershelvam, Veda and Lanctot, Marc and others}, + journal = {Nature}, + number = 7587, + pages = {484--489}, + publisher = {Nature Publishing Group}, + title = {Mastering the game of {G}o with deep neural networks and tree search}, + volume = 529, + year = 2016 +} +@article{schaul2015prioritized, + author = 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Christian and Theis, Lucas and Husz{\'a}r, Ferenc and Caballero, Jose and Cunningham, Andrew and Acosta, Alejandro and Aitken, Andrew and Tejani, Alykhan and Totz, Johannes and Wang, Zehan and others}, + booktitle = {IEEE/CVF Computer Vision \& Pattern Recognition}, + pages = {4681--4690}, + title = {Photo-realistic single image super-resolution using a generative adversarial network}, + year = 2017 +} +@inproceedings{zhu2017unpaired, + author = {Zhu, Jun-Yan and Park, Taesung and Isola, Phillip and Efros, Alexei A}, + booktitle = {IEEE/CVF International Conference on Computer Vision}, + pages = {2223--2232}, + title = {Unpaired image-to-image translation using cycle-consistent adversarial networks}, + year = 2017 +} +@inproceedings{susmelj2017abc, + author = {Susmelj, Igor and Agustsson, Eirikur and Timofte, Radu}, + booktitle = {ICML Workshop on Implicit Models}, + title = {{ABC-GAN}: Adaptive blur and control for improved training stability of generative adversarial networks}, + 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Roth}, + journal = {Sociological Methods \& Research}, + number = 1, + pages = {3--44}, + title = {Fairness in Criminal Justice Risk Assessments: the State of the Art}, + volume = 50, + year = 2017 +} +@article{Wachter-et-al-2017, + author = {Sandra Wachter and Brent Mittelstadt and Luciano Floridi}, + journal = {International Data Privacy Law}, + number = 2, + pages = {76--99}, + title = {Why a right to explanation of automated decision-making does not exist in the general Data Protection Regulation}, + volume = 7, + year = 2017 +} +@article{Heyns-2017, + author = {Christof Heyns}, + journal = {South African Journal of Human Rights}, + number = 1, + pages = {46--71}, + title = {Autonomous weapons in armed conflict and the right to a dignified life: {A}n {A}frican perspective}, + volume = 33, + year = 2017 +} +@book{Manyika-et-al-2017, + author = {James Manyika and Susan Lund and Michael Chui and Jacques Bughin and Jonathan Woetzel and Parul Batra and Ryan Ko and Saurabh Sanghvi}, + publisher = {McKinsey Global Institute}, + title = {Jobs Lost, Jobs Gained: Workforce Transitions in a Time of Automation}, + year = 2017 +} +@article{Frey-Osborne-2017, + author = {Carl Benedikt Frey and Michael A. Osborne}, + journal = {Technological forecasting and social change}, + pages = {254--280}, + title = {The future of employment: How susceptible are jobs to computerisation?}, + volume = 114, + year = 2017 +} +@misc{Ahmed-2017, + author = {Tufayel Ahmed}, + howpublished = {Newsweek, 8 Sept 2017. \url{https://www.newsweek.com/ai-can-tell-if-youre-gay-artificial-intelligence-predicts-sexuality-one-photo-661643}}, + title = {{AI} Can Tell If You're Gay: Artificial Intelligence Predicts Sexuality From One Photo with Startling Accuracy}, + year = 2017 +} +@misc{Matsakis-2017, + author = {Louise Matsakis}, + howpublished = {Vice, Sept 8, 2017. \url{https://www.vice.com/en/article/a33xb4/a-frightening-ai-can-determine-a-persons-sexuality-with-91-accuracy}}, + title = {A Frightening {AI} Can Determine Whether a Person Is Gay With 91 Percent Accuracy}, + year = 2017 +} +@misc{Fernandez-2017, + author = {Colin Fernandez}, + note = {Daily Mail, 7 Sept, 2017. \url{https://www.dailymail.co.uk/sciencetech/article-4862676/Artificial-intelligence-tell-gay.html}}, + title = {Can a computer tell if you're GAY? {A}rtificial intelligence system guesses your sexuality with 91\% accuracy just by looking at a photo of your face}, + year = 2017 +} +@incollection{Reiss-Sprenger-2017, + author = {Reiss, Julian and Sprenger, Jan}, + booktitle = {The {Stanford} Encyclopedia of Philosophy}, + title = {{Scientific Objectivity}}, + year = 2017 +} +@article{binns2018algorithmic, + author = {Binns, Reuben}, + journal = {Philosophy \& Technology}, + number = 4, + pages = {543--556}, + publisher = {Springer}, + title = {Algorithmic accountability and public reason}, + volume = 31, + year = 2018 +} +@book{tegmark2018life, + author = {Tegmark, Max}, + publisher = {Vintage}, + title = {Life 3.0: Being human in the age of artificial intelligence}, + year = 2018 +} +@book{sutton2018reinforcement, + author = {Sutton, Richard S and Barto, Andrew G}, + publisher = {MIT Press}, + title = {Reinforcement learning: An introduction, 2nd Edition}, + year = 2018 +} +@book{sejnowski2018deep, + author = {Sejnowski, Terrence J}, + publisher = {MIT press}, + title = {The deep learning revolution}, + year = 2018 +} +@article{elfwing2018sigmoid, + author = {Elfwing, Stefan and Uchibe, Eiji and Doya, Kenji}, + journal = {Neural Networks}, + pages = {3--11}, + publisher = {Elsevier}, + title = {Sigmoid-weighted linear units for neural network function approximation in reinforcement learning}, + volume = 107, + year = 2018 +} +@inproceedings{serra2018bounding, + author = {Serra, Thiago and Tjandraatmadja, Christian and Ramalingam, Srikumar}, + booktitle = {International Conference on Machine Learning}, + pages = {4558--4566}, + title = {Bounding and counting linear regions of deep neural networks}, + year = 2018 +} +@inproceedings{dorta2018structured, + author = {Dorta, Garoe and Vicente, Sara and Agapito, Lourdes and Campbell, Neill DF and Simpson, Ivor}, + booktitle = {IEEE/CVF Computer Vision \& Pattern Recognition}, + pages = {5477--5485}, + title = {Structured uncertainty prediction networks}, + year = 2018 +} +@inproceedings{prokudin2018deep, + author = {Prokudin, Sergey and Gehler, Peter and Nowozin, Sebastian}, + booktitle = {European Conference on Computer Vision}, + pages = {534--551}, + title = {Deep directional statistics: Pose estimation with uncertainty quantification}, + year = 2018 +} +@article{bottou2018optimization, + author = {Bottou, L{\'e}on and Curtis, Frank E and Nocedal, Jorge}, + journal = {SIAM Review}, + number = 2, + pages = {223--311}, + publisher = {SIAM}, + title = {Optimization methods for large-scale machine learning}, + volume = 60, + year = 2018 +} +@article{jastrzębski2018factors, + author = {Jastrz{\k{e}}bski, Stanis{\l}aw and Kenton, Zachary and Arpit, Devansh and Ballas, Nicolas and Fischer, Asja and Bengio, Yoshua and Storkey, Amos}, + journal = {arXiv:1711.04623}, + title = {Three factors influencing minima in {SGD}}, + year = 2018 +} +@inproceedings{smith2018dont, + author = {Samuel L. Smith and Pieter{-}Jan Kindermans and Chris Ying and Quoc V. Le}, + booktitle = {International Conference on Learning Representations}, + title = {Don't Decay the Learning Rate, Increase the Batch Size}, + year = 2018 +} +@article{goyal2018accurate, + author = {Goyal, Priya and Doll{\'a}r, Piotr and Girshick, Ross and Noordhuis, Pieter and Wesolowski, Lukasz and Kyrola, Aapo and Tulloch, Andrew and Jia, Yangqing and He, Kaiming}, + journal = {arXiv:1706.02677}, + title = {Accurate, Large Minibatch {SGD}: {T}raining {I}mage{N}et in 1 Hour}, + year = 2018 +} +@inproceedings{reddi2018on, + author = {Sashank J. Reddi and Satyen Kale and Sanjiv Kumar}, + booktitle = {International Conference on Learning Representations}, + title = {On the Convergence of {A}dam and Beyond}, + year = 2018 +} +@article{zaheer2018adaptive, + author = {Zaheer, Manzil and Reddi, Sashank and Sachan, Devendra and Kale, Satyen and Kumar, Sanjiv}, + journal = {Neural Information Processing Systems}, + pages = {9815--9825}, + title = {Adaptive methods for nonconvex optimization}, + volume = 31, + year = 2018 +} +@article{baydin2018automatic, + author = {Baydin, Atilim Gunes and Pearlmutter, Barak A and Radul, Alexey Andreyevich and Siskind, Jeffrey Mark}, + journal = {Journal of Marchine Learning Research}, + pages = {1--43}, + publisher = {Microtome Publishing}, + title = {Automatic differentiation in machine learning: A survey}, + volume = 18, + year = 2018 +} +@inproceedings{xiao2018dynamical, + author = {Xiao, Lechao and Bahri, Yasaman and Sohl-Dickstein, Jascha and Schoenholz, Samuel and Pennington, Jeffrey}, + booktitle = {International Conference on Machine Learning}, + pages = {5393--5402}, + title = {Dynamical isometry and a mean field theory of {CNN}s: How to train 10,000-layer vanilla convolutional neural networks}, + year = 2018 +} +@inproceedings{falkner2018bohb, + author = {Falkner, Stefan and Klein, Aaron and Hutter, Frank}, + booktitle = {International Conference on Machine Learning}, + pages = {1437--1446}, + title = {{BOHB}: {R}obust and efficient hyperparameter optimization at scale}, + year = 2018 +} +@article{louizos2018learning, + author = {Christos Louizos and Max Welling and Diederik P. Kingma}, + journal = {International Conference on Learning Representations}, + title = {Learning Sparse Neural Networks through $L_0$ Regularization}, + year = 2018 +} +@article{neyshabur2017pac, + author = {Neyshabur, Behnam and Bhojanapalli, Srinadh and Srebro, Nathan}, + journal = {International Conference on Learning Representations}, + title = {A {PAC}-{B}ayesian approach to spectrally-normalized margin bounds for neural networks}, + year = 2018 +} +@article{izmailov2018averaging, + author = {Izmailov, Pavel and Podoprikhin, Dmitrii and Garipov, Timur and Vetrov, Dmitry and Wilson, Andrew Gordon}, + journal = {Uncertainly in Artificial Intelligence}, + pages = {876--885}, + title = {Averaging weights leads to wider optima and better generalization}, + year = 2018 +} +@inproceedings{garipov2018loss, + author = {Garipov, Timur and Izmailov, Pavel and Podoprikhin, Dmitrii and Vetrov, Dmitry and Wilson, Andrew Gordon}, + booktitle = {Neural Information Processing Systems}, + pages = 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Puckette, Miller}, + journal = {International Conference on Learning Representations}, + title = {Adversarial audio synthesis}, + year = 2018 +} +@inproceedings{fang2018high, + author = {Fang, Fuming and Yamagishi, Junichi and Echizen, Isao and Lorenzo-Trueba, Jaime}, + booktitle = {International Conference on Acoustics, Speech and Signal Processing}, + pages = {5279--5283}, + title = {High-quality nonparallel voice conversion based on cycle-consistent adversarial network}, + year = 2018 +} +@article{hartmann2018eeg, + author = {Hartmann, Kay Gregor and Schirrmeister, Robin Tibor and Ball, Tonio}, + journal = {arXiv:1806.01875}, + title = {{EEG-GAN}: Generative adversarial networks for electroencephalograhic ({EEG}) brain signals}, + year = 2018 +} +@article{fedus2018maskgan, + author = {Fedus, William and Goodfellow, Ian and Dai, Andrew M}, + journal = {International Conference on Learning Representations}, + title = {Mask{GAN}: {B}etter text generation via filling in the\_}, + year = 2018 +} +@inproceedings{wang2018video, + author = {Ting{-}Chun Wang and Ming{-}Yu Liu and Jun{-}Yan Zhu and Nikolai Yakovenko and Andrew Tao and Jan Kautz and Bryan Catanzaro}, + booktitle = {Neural Information Processing Systems}, + pages = {1152--1164}, + title = {Video-to-Video Synthesis}, + volume = 31, + year = 2018 +} +@article{miyato2018spectral, + author = {Miyato, Takeru and Kataoka, Toshiki and Koyama, Masanori and Yoshida, Yuichi}, + journal = {International Conference on Learning Representations}, + title = {Spectral normalization for generative adversarial networks}, + year = 2018 +} +@inproceedings{wu2018wasserstein, + author = {Wu, Jiqing and Huang, Zhiwu and Thoma, Janine and Acharya, Dinesh and Van Gool, Luc}, + booktitle = {European Conference on Computer Vision}, + pages = {653--668}, + title = {Wasserstein divergence for {GAN}s}, + year = 2018 +} +@article{adler2018banach, + author = {Adler, Jonas and Lunz, Sebastian}, + journal = {Neural Information Processing Systems}, + pages = {6755--6764}, + title = {Banach {W}asserstein {GAN}}, + volume = 31, + year = 2018 +} +@article{lucic2018gans, + author = {Lucic, Mario and Kurach, Karol and Michalski, Marcin and Gelly, Sylvain and Bousquet, Olivier}, + journal = {Neural Information Processing Systems}, + pages = {698--707}, + title = {Are {GAN}s created equal? {A} large-scale study}, + volume = 31, + year = 2018 +} +@article{pieters2018comparing, + author = {Pieters, Mathijs and Wiering, Marco}, + journal = {arXiv:1803.09093}, + title = {Comparing generative adversarial network techniques for image creation and modification}, + year = 2018 +} +@article{lin2018pacgan, + author = {Lin, Zinan and Khetan, Ashish and Fanti, Giulia and Oh, Sewoong}, + journal = {Neural Information Processing Systems}, + pages = {1505--1514}, + title = {Pac{GAN}: The power of two samples in generative adversarial networks}, + volume = 31, + year = 2018 +} +@inproceedings{ghosh2018multi, + author = {Ghosh, Arnab and Kulharia, Viveka and Namboodiri, Vinay P and Torr, Philip HS and Dokania, Puneet K}, + booktitle = {IEEE/CVF Computer Vision \& Pattern Recognition}, + pages = {8513--8521}, + title = {Multi-agent diverse generative adversarial networks}, + year = 2018 +} +@inproceedings{wang2018high, + author = {Wang, Ting-Chun and Liu, Ming-Yu and Zhu, Jun-Yan and Tao, Andrew and Kautz, Jan and Catanzaro, Bryan}, + booktitle = {IEEE/CVF Computer Vision \& Pattern Recognition}, + pages = {8798--8807}, + title = {High-resolution image synthesis and semantic manipulation with conditional {GAN}s}, + year = 2018 +} +@article{miyato2018cgans, + author = {Miyato, Takeru and Koyama, Masanori}, + journal = {International Conference on Learning Representations}, + title = {c{GAN}s with projection discriminator}, + year = 2018 +} +@article{donahue2018semantically, + author = {Chris Donahue and Zachary C. Lipton and Akshay Balsubramani and Julian McAuley}, + journal = {International Conference on Learning Representations}, + title = {Semantically Decomposing the Latent Spaces of Generative Adversarial Networks}, + year = 2018 +} +@article{xiao2018dnagan, + author = {Taihong Xiao and Jiapeng Hong and Jinwen Ma}, + journal = {International Conference on Learning Representations}, + title = {{DNA-GAN}: Learning Disentangled Representations from Multi-Attribute Images}, + year = 2018 +} +@inproceedings{choi2018stargan, + author = {Choi, Yunjey and Choi, Minje and Kim, Munyoung and Ha, Jung-Woo and Kim, Sunghun and Choo, Jaegul}, + booktitle = {IEEE/CVF Computer Vision \& Pattern Recognition}, + pages = {8789--8797}, + title = {Star{GAN}: Unified generative adversarial networks for multi-domain image-to-image translation}, + year = 2018 +} +@article{creswell2018inverting, + author = {Creswell, Antonia and Bharath, Anil Anthony}, + journal = {IEEE Transactions on Neural Networks and Learning Systems}, + number = 7, + pages = {1967--1974}, + publisher = {IEEE}, + title = {Inverting the generator of a generative adversarial network}, + volume = 30, + year = 2018 +} +@article{kingma2018glow, + author = {Kingma, Durk P and Dhariwal, Prafulla}, + journal = {Neural Information Processing Systems}, + pages = {10236--10245}, + title = {Glow: Generative flow with invertible 1x1 convolutions}, + volume = 31, + year = 2018 +} +@article{jacobsen2018revnet, + author = {Jacobsen, J{\"o}rn-Henrik and Smeulders, Arnold and Oyallon, Edouard}, + journal = {International Conference on Learning Representations}, + title = {i-{R}ev{N}et: Deep invertible networks}, + year = 2018 +} +@article{mackay2018reversible, + author = {MacKay, Matthew and Vicol, Paul and Ba, Jimmy and Grosse, Roger B}, + journal = {Neural Information Processing Systems}, + pages = {9043--9054}, + title = {Reversible recurrent neural networks}, + volume = 31, + year = 2018 +} +@article{kim2018flowavenet, + author = {Kim, Sungwon and Lee, Sang-gil and Song, Jongyoon and Kim, Jaehyeon and Yoon, Sungroh}, + journal = {International Conference on Machine Learning}, + pages = {3370--3378}, + title = {Flo{W}ave{N}et: A generative flow for raw audio}, + year = 2018 +} +@article{he2018unsupervised, + author = {He, Junxian and Neubig, Graham and Berg-Kirkpatrick, Taylor}, + journal = {ACL Empirical Methods in Natural Language Processing}, + pages = {1292--1302}, + title = {Unsupervised learning of syntactic structure with invertible neural projections}, + year = 2018 +} +@inproceedings{haarnoja2018latent, + author = {Haarnoja, Tuomas and Hartikainen, Kristian and Abbeel, Pieter and Levine, Sergey}, + booktitle = {International Conference on Machine Learning}, + pages = {1851--1860}, + title = {Latent space policies for hierarchical reinforcement learning}, + year = 2018 +} +@inproceedings{huang2018neural, + author = {Huang, Chin-Wei and Krueger, David and Lacoste, Alexandre and Courville, Aaron}, + booktitle = {International Conference on Machine Learning}, + pages = {2078--2087}, + title = {Neural autoregressive flows}, + year = 2018 +} +@inproceedings{oord2018parallel, + author = {Van den Oord, Aaron and Li, Yazhe and Babuschkin, Igor and Simonyan, Karen and Vinyals, Oriol and Kavukcuoglu, Koray and Driessche, George and Lockhart, Edward and Cobo, Luis and Stimberg, Florian and others}, + booktitle = {International Conference on Machine Learning}, + pages = {3918--3926}, + title = {Parallel {W}ave{N}et: Fast high-fidelity speech synthesis}, + year = 2018 +} +@inproceedings{chang2018reversible, + author = {Chang, Bo and Meng, Lili and Haber, Eldad and Ruthotto, Lars and Begert, David and Holtham, Elliot}, + booktitle = {AAAI Conference on Artificial Intelligence}, + pages = {2811--2818}, + title = {Reversible architectures for arbitrarily deep residual neural networks}, + year = 2018 +} +@article{chen2018neural, + author = {Chen, Ricky TQ and Rubanova, Yulia and Bettencourt, Jesse and Duvenaud, David K}, + journal = {Neural Information Processing Systems}, + pages = {6572--6583}, + title = {Neural ordinary differential equations}, + volume = 31, + year = 2018 +} +@article{chen2018isolating, + author = {Chen, Ricky TQ and Li, Xuechen and Grosse, Roger B and Duvenaud, David K}, + journal = {Neural Information Processing Systems}, + pages = {2615--2625}, + title = {Isolating sources of disentanglement in variational autoencoders}, + volume = 31, + year = 2018 +} +@article{akuzawa2018expressive, + author = {Akuzawa, Kei and Iwasawa, Yusuke and Matsuo, Yutaka}, + journal = {INTERPSPEECH}, + pages = {3067--3071}, + title = {Expressive speech synthesis via modeling expressions with variational autoencoder}, + year = 2018 +} +@article{gomez2018automatic, + author = {G{\'o}mez-Bombarelli, Rafael and Wei, Jennifer N and Duvenaud, David and Hern{\'a}ndez-Lobato, Jos{\'e} Miguel and S{\'a}nchez-Lengeling, Benjam{\'\i}n and Sheberla, Dennis and Aguilera-Iparraguirre, Jorge and Hirzel, Timothy D and Adams, Ryan P and Aspuru-Guzik, Al{\'a}n}, + journal = {ACS Central Science}, + number = 2, + pages = {268--276}, + publisher = {ACS Publications}, + title = {Automatic chemical design using a data-driven continuous representation of molecules}, + volume = 4, + year = 2018 +} +@article{sultan2018transferable, + author = {Sultan, Mohammad M and Wayment-Steele, Hannah K and Pande, Vijay S}, + journal = {Journal of Chemical Theory and Computation}, + number = 4, + pages = {1887--1894}, + publisher = {ACS Publications}, + title = {Transferable neural networks for enhanced sampling of protein dynamics}, + volume = 14, + year = 2018 +} +@inproceedings{simonovsky2018graphvae, + author = {Simonovsky, Martin and Komodakis, Nikos}, + booktitle = {International Conference on Artificial Neural Networks}, + pages = {412--422}, + title = {Graph{VAE}: Towards generation of small graphs using variational autoencoders}, + year = 2018 +} +@article{hernandez2018variational, + author = {Hern{\'a}ndez, Carlos X and Wayment-Steele, Hannah K and Sultan, Mohammad M and Husic, Brooke E and Pande, Vijay S}, + journal = {Physical Review E}, + number = 6, + pages = {062412}, + publisher = {APS}, + title = {Variational encoding of complex dynamics}, + volume = 97, + year = 2018 +} +@inproceedings{inoue2018transfer, + author = {Inoue, Tadanobu and Choudhury, Subhajit and De Magistris, Giovanni and Dasgupta, Sakyasingha}, + booktitle = {IEEE International Conference on Image Processing}, + pages = {2725--2729}, + title = {Transfer learning from synthetic to real images using variational autoencoders for precise position detection}, + year = 2018 +} +@article{park2018multimodal, + author = {Park, Daehyung and Hoshi, Yuuna and Kemp, Charles C}, + journal = {IEEE Robotics and Automation Letters}, + number = 3, + pages = {1544--1551}, + publisher = {IEEE}, + title = {A multimodal anomaly detector for robot-assisted feeding using an {LSTM}-based variational autoencoder}, + volume = 3, + year = 2018 +} +@article{eslami2018neural, + author = {Eslami, SM Ali and Jimenez Rezende, Danilo and Besse, Frederic and Viola, Fabio and Morcos, Ari S and Garnelo, Marta and Ruderman, Avraham and Rusu, Andrei A and Danihelka, Ivo and Gregor, Karol and others}, + journal = {Science}, + number = 6394, + pages = {1204--1210}, + publisher = {American Association for the Advancement of Science}, + title = {Neural scene representation and rendering}, + volume = 360, + year = 2018 +} +@inproceedings{liang2018variational, + author = {Liang, Dawen and Krishnan, Rahul G and Hoffman, Matthew D and Jebara, Tony}, + booktitle = {World Wide Web Conference}, + pages = {689--698}, + title = {Variational autoencoders for collaborative filtering}, + year = 2018 +} +@inproceedings{zong2018deep, + author = {Zong, Bo and Song, Qi and Min, Martin Renqiang and Cheng, Wei and Lumezanu, Cristian and Cho, Daeki and Chen, Haifeng}, + booktitle = {International Conference on Learning Representations}, + title = {Deep autoencoding {G}aussian mixture model for unsupervised anomaly detection}, + year = 2018 +} +@article{vahdat2018dvae, + author = {Vahdat, Arash and Andriyash, Evgeny and Macready, William}, + journal = {Neural Information Processing Systems}, + pages = {1869--1878}, + title = {{DVAE}\#: Discrete variational autoencoders with relaxed {B}oltzmann priors}, + volume = 31, + year = 2018 +} +@inproceedings{vahdat2018dvae++, + author = {Vahdat, Arash and Macready, William and Bian, Zhengbing and Khoshaman, Amir and Andriyash, Evgeny}, + booktitle = {International Conference on Machine Learning}, + pages = {5035--5044}, + title = {D{VAE}++: Discrete variational autoencoders with overlapping transformations}, + year = 2018 +} +@article{tolstikhin2017wasserstein, + author = {Tolstikhin, Ilya and Bousquet, Olivier and Gelly, Sylvain and Schoelkopf, Bernhard}, + journal = {International Conference on Learning Representations}, + title = {Wasserstein auto-encoders}, + year = 2018 +} +@inproceedings{kim2018disentangling, + author = {Kim, Hyunjik and Mnih, Andriy}, + booktitle = {International Conference on Machine Learning}, + pages = {2649--2658}, + title = {Disentangling by factorising}, + year = 2018 +} +@article{kumar2017variational, + author = {Kumar, Abhishek and Sattigeri, Prasanna and Balakrishnan, Avinash}, + journal = {International Conference on Learning Representations}, + title = {Variational inference of disentangled latent concepts from unlabeled observations}, + year = 2018 +} +@article{franccois2018introduction, + author = {Fran{\c{c}}ois-Lavet, Vincent and Henderson, Peter and Islam, Riashat and Bellemare, Marc G and Pineau, Joelle and others}, + journal = {Foundations and Trends in Machine Learning}, + number = {3-4}, + pages = {219--354}, + publisher = {Now Publishers, Inc.}, + title = {An introduction to deep reinforcement learning}, + volume = 11, + year = 2018 +} +@inproceedings{hessel2018rainbow, + author = {Hessel, Matteo and Modayil, Joseph and van Hasselt, Hado and Schaul, Tom and Ostrovski, Georg and Dabney, Will and Horgan, Dan and Piot, Bilal and Azar, Mohammad and Silver, David}, + booktitle = {AAAI Conference on Artificial Intelligence}, + pages = {3215--3222}, + title = {Rainbow: Combining improvements in deep reinforcement learning}, + year = 2018 +} +@article{fortunato2017noisy, + author = {Fortunato, Meire and Azar, Mohammad Gheshlaghi and Piot, Bilal and Menick, Jacob and Osband, Ian and Graves, Alex and Mnih, Vlad and Munos, Remi and Hassabis, Demis and Pietquin, Olivier and others}, + journal = {International Conference on Learning Representations}, + title = {Noisy networks for exploration}, + year = 2018 +} +@inproceedings{dabney2018distributional, + author = {Dabney, Will and Rowland, Mark and Bellemare, Marc and Munos, R{\'e}mi}, + booktitle = {AAAI Conference on Artificial Intelligence}, + title = {Distributional reinforcement learning with quantile regression}, + year = 2018 +} +@inproceedings{fujimoto2018addressing, + author = {Fujimoto, Scott and Hoof, Herke and Meger, David}, + booktitle = {International Conference on Machine Learning}, + pages = {1587--1596}, + title = {Addressing function approximation error in actor-critic methods}, + year = 2018 +} +@inproceedings{haarnoja2018soft, + author = {Haarnoja, Tuomas and Zhou, Aurick and Abbeel, Pieter and Levine, Sergey}, + booktitle = {International Conference on Machine Learning}, + pages = {1861--1870}, + title = {Soft actor-critic: Off-policy maximum entropy deep reinforcement learning with a stochastic actor}, + year = 2018 +} +@article{christiano2018supervising, + author = {Christiano, Paul and Shlegeris, Buck and Amodei, Dario}, + journal = {arXiv:1810.08575}, + title = {Supervising strong learners by amplifying weak experts}, + year = 2018 +} +@inproceedings{nguyen2018optimization, + author = {Nguyen, Quynh and Hein, Matthias}, + booktitle = {International Conference on Machine Learning}, + pages = {3730--3739}, + title = {Optimization landscape and expressivity of deep {CNN}s}, + year = 2018 +} +@article{li2018measuring, + author = {Chunyuan Li and Heerad Farkhoor and Rosanne Liu and Jason Yosinski}, + journal = {International Conference on Learning Representations}, + title = {Measuring the Intrinsic Dimension of Objective Landscapes}, + year = 2018 +} +@article{li2018learning, + author = {Li, Yuanzhi and Liang, Yingyu}, + journal = {Neural Information Processing Systems}, + pages = {8168--8177}, + title = {Learning overparameterized neural networks via stochastic gradient descent on structured data}, + volume = 31, + year = 2018 +} +@inproceedings{draxler2019essentially, + author = {Felix Draxler and Kambis Veschgini and Manfred Salmhofer and Fred A. Hamprecht}, + booktitle = {International Conference on Machine Learning}, + pages = {1308--1317}, + title = {Essentially No Barriers in Neural Network Energy Landscape}, + year = 2018 +} +@inproceedings{kleinberg2018alternative, + author = {Robert Kleinberg and Yuanzhi Li and Yang Yuan}, + booktitle = {International Conference on Machine Learning}, + pages = {2703--2712}, + title = {An Alternative View: When Does {SGD} Escape Local Minima?}, + year = 2018 +} +@inproceedings{huang2018data, + author = {Huang, Zehao and Wang, Naiyan}, + booktitle = {European Conference on Computer Vision}, + pages = {304--320}, + title = {Data-driven sparse structure selection for deep neural networks}, + year = 2018 +} +@article{nye2018efficient, + author = {Nye, Maxwell and Saxe, Andrew}, + journal = {International Conference on Learning Representations (Workshop)}, + title = {Are efficient deep representations learnable?}, + year = 2018 +} +@inproceedings{ulyanov2018deep, + author = {Ulyanov, Dmitry and Vedaldi, Andrea and Lempitsky, Victor}, + booktitle = {IEEE/CVF Computer Vision \& Pattern Recognition}, + pages = {9446--9454}, + title = {Deep image prior}, + year = 2018 +} +@article{taddeo2018ai, + author = {Taddeo, Mariarosaria and Floridi, Luciano}, + journal = {Science}, + number = 6404, + pages = {751--752}, + publisher = {American Association for the Advancement of Science}, + title = {How {AI} can be a force for good}, + volume = 361, + year = 2018 +} +@inproceedings{McNamara-2018, + author = {Andrew McNamara and Justin Smith and Emerson Murphy-Hill}, + booktitle = {ACM Joint Meeting on European Software Engineering Conference and Symposium on the Foundations of Software Engineering}, + pages = {729--733}, + title = {Does {ACM}'s code of ethics change ethical decision making in software development?}, + year = 2018 +} +@article{Mayson-2018, + author = {Sandra G. Mayson}, + journal = {Yale Law Journal}, + pages = {2122--2473}, + title = {Bias In Bias Out}, + volume = 128, + year = 2018 +} +@article{Buolamwini-Gebru-2018, + author = {Joy Buolamwini and Timnit Gebru}, + journal = {Proceedings of Machine Learning Research}, + title = {Gender Shades: Intersectional Accuracy Disparities in Commercial Gender Classification}, + volume = 81, + year = 2018 +} +@article{Awad-et-al-2018, + author = {Awad, E. and Dsouza, S. and Kim, R. and Schulz, J. and Henrich, J. and Shariff, A. and Bonnefon, J.-F. and Rahwan, I.}, + journal = {Nature}, + pages = {59--64}, + title = {The moral machine experiment}, + volume = 563, + year = 2018 +} +@inproceedings{Noothigattu-et-al-2018, + author = {Ritesh Noothigattu and Snehalkumar (Neil) Gaikwad and Edmond Awad and Sohan Dsouza and Iyad Rahwan and Pradeep Ravikumar and Ariel D. Procaccia}, + booktitle = {AAAI Portuguese Conference on Artificial Intelligence}, + pages = {1587--1594}, + title = {A voting-based system for ethical decision making}, + year = 2018 +} +@article{Akers-et-al-2018, + author = {John Akers and Gagan Bansal and Gabriel Cadamuro and Christine Chen and Quanze Chen and Lucy Lin and Phoebe Mulcaire and Rajalakshmi Nandakumar and Matthew Rockett and Lucy Simko and John Toman and Tongshuang Wu and Eric Zeng and Bill Zorn and Franziska Roesner}, + journal = {arXiv:1812.09383}, + title = {Technology-Enabled Disinformation: Summary, Lessons, and Recommendations}, + year = 2018 +} +@book{Bughin-et-al-2018, + author = {Jacques Bughin and Jeongmin Seong and James Manyika and Michael Chui and Raoul Joshi}, + publisher = {McKinsey Global Institute, Sept 4, 2018.}, + title = {Notes from the {AI} Frontier: Modelling the Impact of {AI} on the World Economy}, + year = 2018 +} +@book{Manyika-Sneader-2018, + author = {James Manyika and Kevin Sneader}, + publisher = {McKinsey Global Institute}, + title = {{AI}, automation, and the future of work: Ten things to solve for}, + year = 2018 +} +@article{Calo-2018, + author = {Ryan Calo}, + journal = {University of Bologna Law Review}, + number = 2, + pages = {180--218}, + title = {Artificial Intelligence Policy: A Primer and Roadmap}, + volume = 3, + year = 2018 +} +@misc{Fei-Fei-Li-2018, + author = {Fei-Fei Li}, + howpublished = {The New York Times, March 7, 2018. \url{https://www.nytimes.com/2018/03/07/opinion/artificial-intelligence-human.html}}, + title = {How to Make {A.I.} That’s Good for People}, + year = 2018 +} +@misc{Knight-2018, + author = {Will Knight}, + howpublished = {MIT Technology Review, Nov 20, 2018. \url{https://www.technologyreview.com/2018/11/17/66372/one-of-the-fathers-of-ai-is-worried-about-its-future/}}, + title = {One of the fathers of {AI} is worried about its future}, + year = 2018 +} +@article{Wang-Kosinski-2018, + author = {Y. Wang and M. Kosinski}, + journal = {Journal of Personality and Social Psychology}, + number = 2, + pages = {246--257}, + title = {Deep neural networks are more accurate than humans at detecting sexual orientation from facial images}, + volume = 114, + year = 2018 +} +@misc{Arcas-et-al-2018, + author = {Ag{\"u}era y Arcas , Blaise and Todorov, Alexander and Mitchell, Margaret}, + howpublished = {Medium, Jan 11, 2018. \url{https://medium.com/@blaisea/do-algorithms-reveal-sexual-orientation-or-just-expose-our-stereotypes-d998fafdf477}}, + title = {Do algorithms reveal sexual orientation or just expose our stereotypes?}, + year = 2018 +} +@book{Noble-2018, + address = {New York}, + author = {Safiya Noble}, + publisher = {NYU Press}, + title = {Algorithms of Oppression}, + year = 2018 +} +@book{Eubanks-2018, + address = {New York}, + author = {Virginia Eubanks}, + publisher = {St. Martin’s Press}, + title = {Automating Inequality: How High-Tech Tools Profile, Police, and Punish the Poor}, + year = 2018 +} +@book{Broussard-2018, + author = {Meredith Broussard}, + publisher = {The MIT Press}, + title = {Artificial Unintelligence: How Computers Misunderstand the World}, + year = 2018 +} +@article{holland2019emotion, + author = {Holland, Catherine AC and Ebner, Natalie C and Lin, Tian and Samanez-Larkin, Gregory R}, + journal = {Cognition and {E}motion}, + number = 2, + pages = {245--257}, + publisher = {Taylor \& Francis}, + title = {Emotion identification across adulthood using the Dynamic FACES database of emotional expressions in younger, middle aged, and older adults}, + volume = 33, + year = 2019 +} +@book{graesser2019foundations, + author = {Graesser, Laura and Keng, Wah Loon}, + publisher = {Addison-Wesley Professional}, + title = {Foundations of deep reinforcement learning}, + year = 2019 +} +@inproceedings{howard2019searching, + author = {Howard, Andrew and Sandler, Mark and Chu, Grace and Chen, Liang-Chieh and Chen, Bo and Tan, Mingxing and Wang, Weijun and Zhu, Yukun and Pang, Ruoming and Vasudevan, Vijay and others}, + booktitle = {IEEE/CVF International Conference on Computer Vision}, + pages = {1314--1324}, + title = {Searching for {M}obile{N}et{V}3}, + year = 2019 +} +@inproceedings{barron2019general, + author = {Barron, Jonathan T}, + booktitle = {IEEE/CVF Computer Vision \& Pattern Recognition}, + pages = {4331--4339}, + title = {A general and adaptive robust loss function}, + year = 2019 +} +@article{he2019control, + author = {He, Fengxiang and Liu, Tongliang and Tao, Dacheng}, + journal = {Neural Information Processing Systems}, + pages = {1143--1152}, + title = {Control batch size and learning rate to generalize well: Theoretical and empirical evidence}, + volume = 32, + year = 2019 +} +@article{loshchilov2017decoupled, + author = {Loshchilov, Ilya and Hutter, Frank}, + journal = {International Conference on Learning Representations}, + title = {Decoupled weight decay regularization}, + year = 2019 +} +@article{choi2019empirical, + author = {Choi, Dami and Shallue, Christopher J and Nado, Zachary and Lee, Jaehoon and Maddison, Chris J and Dahl, George E}, + journal = {arXiv:1910.05446}, + title = {On empirical comparisons of optimizers for deep learning}, + year = 2019 +} +@article{zhang2019fixup, + author = {Zhang, Hongyi and Dauphin, Yann N and Ma, Tengyu}, + journal = {International Conference on Learning Representations}, + title = {Fixup initialization: Residual learning without normalization}, + year = 2019 +} +@article{huang2019gpipe, + author = {Huang, Yanping and Cheng, Youlong and Bapna, Ankur and Firat, Orhan and Chen, Dehao and Chen, Mia and Lee, HyoukJoong and Ngiam, Jiquan and Le, Quoc V and Wu, Yonghui and others}, + journal = {Neural Information Processing Systems}, + pages = {103--112}, + title = {G{P}ipe: {E}fficient Training of Giant Neural Networks using Pipeline Parallelism}, + volume = 32, + year = 2019 +} +@article{sohoni2019lowmemory, + author = {Sohoni, Nimit Sharad and Aberger, Christopher Richard and Leszczynski, Megan and Zhang, Jian and R{\'e}, Christopher}, + journal = {arXiv:1904.10631}, + title = {Low-memory neural network training: A technical report}, + year = 2019 +} +@article{shoeybi2020megatronlm, + author = {Shoeybi, Mohammad and Patwary, Mostofa and Puri, Raul and LeGresley, Patrick and Casper, Jared and Catanzaro, Bryan}, + journal = {arXiv:1909.08053}, + title = {Megatron-{LM}: {T}raining multi-billion parameter language models using model parallelism}, + year = 2019 +} +@article{belkin2019reconciling, + author = {Belkin, Mikhail and Hsu, Daniel and Ma, Siyuan and Mandal, Soumik}, + journal = {Proceedings of the National Academy of Sciences}, + number = 32, + pages = {15849--15854}, + publisher = {National Acad Sciences}, + title = {Reconciling modern machine-learning practice and the classical bias--variance trade-off}, + volume = 116, + year = 2019 +} +@article{bartlett2017nearlytight, + author = {Bartlett, Peter L and Harvey, Nick and Liaw, Christopher and Mehrabian, Abbas}, + journal = {Journal of Machine Learning Research}, + number = 1, + pages = {2285--2301}, + title = {Nearly-tight {VC}-dimension and pseudodimension bounds for piecewise linear neural networks}, + volume = 20, + year = 2019 +} +@article{fort2020deep, + author = {Fort, Stanislav and Hu, Huiyi and Lakshminarayanan, Balaji}, + journal = {arXiv:1912.02757}, + title = {Deep ensembles: A loss landscape perspective}, + year = 2019 +} +@article{liu2019beta, + author = {Liu, Lei and Luo, Yuhao and Shen, Xu and Sun, Mingzhai and Li, Bin}, + journal = {IEEE Access}, + pages = {36140--36153}, + publisher = {IEEE}, + title = {Beta-Dropout: A Unified Dropout}, + volume = 7, + year = 2019 +} +@article{muller2019does, + author = {M{\"u}ller, Rafael and Kornblith, Simon and Hinton, Geoffrey E}, + journal = {Neural Information Processing Systems}, + pages = {4696--4705}, + title = {When does label smoothing help?}, + volume = 32, + year = 2019 +} +@article{chaudhari2017entropysgd, + abstract = {This paper proposes a new optimization algorithm called Entropy-SGD for training deep neural networks that is motivated by the local geometry of the energy landscape. Local extrema with low generalization error have a large proportion of almost-zero eigenvalues in the Hessian with very few positive or negative eigenvalues. We leverage upon this observation to construct a local-entropy-based objective function that favors well-generalizable solutions lying in large flat regions of the energy landscape, while avoiding poorly-generalizable solutions located in the sharp valleys. Conceptually, our algorithm resembles two nested loops of SGD where we use Langevin dynamics in the inner loop to compute the gradient of the local entropy before each update of the weights. We show that the new objective has a smoother energy landscape and show improved generalization over SGD using uniform stability, under certain assumptions. Our experiments on convolutional and recurrent networks demonstrate that Entropy-SGD compares favorably to state-of-the-art techniques in terms of generalization error and training time.}, + author = {Pratik Chaudhari and Anna Choromanska and Stefano Soatto and Yann LeCun and Carlo Baldassi and Christian Borgs and Jennifer Chayes and Levent Sagun and Riccardo Zecchina}, + doi = {10.1088/1742-5468/ab39d9}, + journal = {Journal of Statistical Mechanics: Theory and Experiment}, + pages = 124018, + title = {Entropy-{SGD}: {B}iasing gradient descent into wide valleys}, + volume = 12, + year = 2019 +} +@inproceedings{devlin2018bert, + author = {Jacob Devlin and Ming{-}Wei Chang and Kenton Lee and Kristina Toutanova}, + booktitle = {ACL Human Language Technologies}, + pages = {4171--4186}, + title = {{BERT:} Pre-training of Deep Bidirectional Transformers for Language Understanding}, + year = 2019 +} +@article{radford2019language, + author = {Radford, Alec and Wu, Jeffrey and Child, Rewon and Luan, David and Amodei, Dario and Sutskever, Ilya and others}, + journal = {OpenAI Blog}, + number = 8, + pages = 9, + title = {Language models are unsupervised multitask learners}, + volume = 1, + year = 2019 +} +@inproceedings{schneider2019wav2vec, + author = {Steffen Schneider and Alexei Baevski and Ronan Collobert and Michael Auli}, + booktitle = {INTERSPEECH}, + pages = {3465--3469}, + title = {wav2vec: Unsupervised Pre-Training for Speech Recognition}, + year = 2019 +} +@inproceedings{summers2019improved, + author = {Summers, Cecilia and Dinneen, Michael J}, + booktitle = {Winter Conference on Applications of Computer Vision}, + pages = {1262--1270}, + title = {Improved mixed-example data augmentation}, + year = 2019 +} +@inproceedings{jackson2019style, + author = {Jackson, Philip TG and Abarghouei, Amir Atapour and Bonner, Stephen and Breckon, Toby P and Obara, Boguslaw}, + booktitle = {IEEE Computer Vision and Pattern Recognition Workshops}, + pages = {10--11}, + title = {Style augmentation: {D}ata augmentation via style randomization}, + year = 2019 +} +@article{shorten2019survey, + author = {Shorten, Connor and Khoshgoftaar, Taghi M}, + journal = {Journal of Big Data}, + number = 1, + pages = {1--48}, + title = {A survey on image data augmentation for deep learning}, + volume = 6, + year = 2019 +} +@inproceedings{yun2019cutmix, + author = {Yun, Sangdoo and Han, Dongyoon and Oh, Seong Joon and Chun, Sanghyuk and Choe, Junsuk and Yoo, Youngjoon}, + booktitle = {IEEE/CVF International Conference on Computer Vision}, + pages = {6023--6032}, + title = {Cut{M}ix: Regularization strategy to train strong classifiers with localizable features}, + year = 2019 +} +@article{park2019specaugment, + author = {Park, Daniel S and Chan, William and Zhang, Yu and Chiu, Chung-Cheng and Zoph, Barret and Cubuk, Ekin D and Le, Quoc V}, + journal = {INTERSPEECH}, + title = {Spec{A}ugment: A simple data augmentation method for automatic speech recognition}, + year = 2019 +} +@inproceedings{wei2019eda, + author = {Wei, Jason and Zou, Kai}, + booktitle = {ACL Empirical Methods in Natural Language Processing}, + pages = {6382--6388}, + title = {{EDA}: Easy Data Augmentation Techniques for Boosting Performance on Text Classification Tasks}, + year = 2019 +} +@inproceedings{yu2019free, + author = {Yu, Jiahui and Lin, Zhe and Yang, Jimei and Shen, Xiaohui and Lu, Xin and Huang, Thomas S}, + booktitle = {IEEE/CVF International Conference on Computer Vision}, + pages = {4471--4480}, + title = {Free-form image inpainting with gated convolution}, + year = 2019 +} +@inproceedings{chang2019free, + author = {Chang, Ya-Liang and Liu, Zhe Yu and Lee, Kuan-Ying and Hsu, Winston}, + booktitle = {IEEE/CVF International Conference on Computer Vision}, + pages = {9066--9075}, + title = {Free-form video inpainting with 3{D} gated convolution and temporal {P}atch{GAN}}, + year = 2019 +} +@inproceedings{zhang2019making, + author = {Zhang, Richard}, + booktitle = {International Conference on Machine Learning}, + pages = {7324--7334}, + title = {Making convolutional networks shift-invariant again}, + year = 2019 +} +@article{eren2019generic, + author = {Eren, Levent and Ince, Turker and Kiranyaz, Serkan}, + journal = {Journal of Signal Processing Systems}, + number = 2, + pages = {179--189}, + title = {A generic intelligent bearing fault diagnosis system using compact adaptive {1D CNN} classifier}, + volume = 91, + year = 2019 +} +@inproceedings{su2019pixel, + author = {Su, Hang and Jampani, Varun and Sun, Deqing and Gallo, Orazio and Learned-Miller, Erik and Kautz, Jan}, + booktitle = {IEEE/CVF Computer Vision \& Pattern Recognition}, + pages = {11166--11175}, + title = {Pixel-adaptive convolutional neural networks}, + year = 2019 +} +@article{falk2019u, + author = {Falk, Thorsten and Mai, Dominic and Bensch, Robert and {\c{C}}i{\c{c}}ek, {\"O}zg{\"u}n and Abdulkadir, Ahmed and Marrakchi, Yassine and B{\"o}hm, Anton and Deubner, Jan and J{\"a}ckel, Zoe and Seiwald, Katharina and others}, + journal = {Nature Methods}, + number = 1, + pages = {67--70}, + title = {{U}-{N}et: {D}eep learning for cell counting, detection, and morphometry}, + volume = 16, + year = 2019 +} +@article{yamada2019shakedrop, + author = {Yamada, Yoshihiro and Iwamura, Masakazu and Akiba, Takuya and Kise, Koichi}, + journal = {IEEE Access}, + pages = {186126--186136}, + publisher = {IEEE}, + title = {Shakedrop regularization for deep residual learning}, + volume = 7, + year = 2019 +} +@article{yang2019mean, + author = {Yang, Greg and Pennington, Jeffrey and Rao, Vinay and Sohl-Dickstein, Jascha and Schoenholz, Samuel S}, + journal = {International Conference on Learning Representations}, + title = {A mean field theory of batch normalization}, + year = 2019 +} +@article{li2019exponential, + author = {Li, Zhiyuan and Arora, Sanjeev}, + journal = {International Conference on Learning Representations}, + title = {An exponential learning rate schedule for deep learning}, + year = 2019 +} +@misc{huszar2019exponentially, + author = {Husz{\'a}r, Ferenc}, + howpublished = {\url{https://www.inference.vc/exponentially-growing-learning-rate-implications-of-scale-invariance-induced-by-BatchNorm/}}, + title = {Exponentially Growing Learning Rate? {I}mplications of Scale Invariance induced by Batch Normalization}, + year = 2019 +} +@inproceedings{zeng2019learning, + author = {Zeng, Yanhong and Fu, Jianlong and Chao, Hongyang and Guo, Baining}, + booktitle = {IEEE/CVF Computer Vision \& Pattern Recognition}, + pages = {1486--1494}, + title = {Learning pyramid-context encoder network for high-quality image inpainting}, + year = 2019 +} +@article{velivckovic2017graph, + author = {Veli{\v{c}}kovi{\'c}, Petar and Cucurull, Guillem and Casanova, Arantxa and Romero, Adriana and Lio, Pietro and Bengio, Yoshua}, + journal = {International Conference on Learning Representations}, + title = {Graph attention networks}, + year = 2019 +} +@article{wang2018glue, + author = {Wang, Alex and Singh, Amanpreet and Michael, Julian and Hill, Felix and Levy, Omer and Bowman, Samuel R}, + journal = {International Conference on Learning Representations}, + title = {G{LUE}: A multi-task benchmark and analysis platform for natural language understanding}, + year = 2019 +} +@article{wang2019superglue, + author = {Wang, Alex and Pruksachatkun, Yada and Nangia, Nikita and Singh, Amanpreet and Michael, Julian and Hill, Felix and Levy, Omer and Bowman, Samuel}, + journal = {Neural Information Processing Systems}, + pages = {3261--3275}, + title = {Super{GLUE}: A stickier benchmark for general-purpose language understanding systems}, + volume = 32, + year = 2019 +} +@article{mccoy2019right, + author = {McCoy, R Thomas and Pavlick, Ellie and Linzen, Tal}, + journal = {Meeting of the Association for Computational Linguistics}, + pages = {2428--3448}, + title = {Right for the wrong reasons: Diagnosing syntactic heuristics in natural language inference}, + year = 2019 +} +@article{liu2018rethinking, + author = {Zhuang Liu and Mingjie Sun and Tinghui Zhou and Gao Huang and Trevor Darrell}, + journal = {International Conference on Learning Representations}, + title = {Rethinking the Value of Network Pruning}, + year = 2019 +} +@article{voita2019analyzing, + author = {Voita, Elena and Talbot, David and Moiseev, Fedor and Sennrich, Rico and Titov, Ivan}, + journal = {Meeting of the Association for Computational Linguistics}, + pages = {5797--5808}, + title = {Analyzing multi-head self-attention: Specialized heads do the heavy lifting, the rest can be pruned}, + year = 2019 +} +@article{wu2019pay, + author = {Wu, Felix and Fan, Angela and Baevski, Alexei and Dauphin, Yann N and Auli, Michael}, + journal = {International Conference on Learning Representations}, + title = {Pay less attention with lightweight and dynamic convolutions}, + year = 2019 +} +@article{wei2019nezha, + author = {Wei, Junqiu and Ren, Xiaozhe and Li, Xiaoguang and Huang, Wenyong and Liao, Yi and Wang, Yasheng and Lin, Jiashu and Jiang, Xin and Chen, Xiao and Liu, Qun}, + journal = {arXiv:1909.00204}, + title = {N{EZHA}: Neural contextualized representation for {C}hinese language understanding}, + year = 2019 +} +@article{child2019generating, + author = {Child, Rewon and Gray, Scott and Radford, Alec and Sutskever, Ilya}, + journal = {arXiv:1904.10509}, + title = {Generating long sequences with sparse transformers}, + year = 2019 +} +@inproceedings{ye2019cross, + author = {Ye, Linwei and Rochan, Mrigank and Liu, Zhi and Wang, Yang}, + booktitle = {IEEE/CVF Computer Vision \& Pattern Recognition}, + pages = {10502--10511}, + title = {Cross-modal self-attention network for referring image segmentation}, + year = 2019 +} +@inproceedings{sun2019videobert, + author = {Sun, Chen and Myers, Austin and Vondrick, Carl and Murphy, Kevin and Schmid, Cordelia}, + booktitle = {IEEE/CVF International Conference on Computer Vision}, + pages = {7464--7473}, + title = {Video{BERT}: A joint model for video and language representation learning}, + year = 2019 +} +@inproceedings{girdhar2019video, + author = {Girdhar, Rohit and Carreira, Joao and Doersch, Carl and Zisserman, Andrew}, + booktitle = {IEEE/CVF Computer Vision \& Pattern Recognition}, + pages = {244--253}, + title = {Video action transformer network}, + year = 2019 +} +@article{su2019vl, + author = {Su, Weijie and Zhu, Xizhou and Cao, Yue and Li, Bin and Lu, Lewei and Wei, Furu and Dai, Jifeng}, + journal = {International Conference on Learning Representations}, + title = {V{L}-{BERT}: Pre-training of generic visual-linguistic representations}, + year = 2019 +} +@article{tan2019lxmert, + author = {Tan, Hao and Bansal, Mohit}, + journal = {Empirical Methods in Natural Language Processing}, + pages = {5099--5110}, + title = {{LXMERT}: Learning cross-modality encoder representations from transformers}, + year = 2019 +} +@article{lu2019vilbert, + author = {Lu, Jiasen and Batra, Dhruv and Parikh, Devi and Lee, Stefan}, + journal = {Neural Information Processing Systems}, + pages = {13--23}, + title = {Vil{BERT}: Pretraining task-agnostic visiolinguistic representations for vision-and-language tasks}, + volume = 32, + year = 2019 +} +@article{li2019visualbert, + author = {Li, Liunian Harold and Yatskar, Mark and Yin, Da and Hsieh, Cho-Jui and Chang, Kai-Wei}, + journal = {arXiv:1908.03557}, + title = {Visual{BERT}: A simple and performant baseline for vision and language}, + year = 2019 +} +@inproceedings{hu2019local, + author = {Hu, Han and Zhang, Zheng and Xie, Zhenda and Lin, Stephen}, + booktitle = {IEEE/CVF International Conference on Computer Vision}, + pages = {3464--3473}, + title = {Local relation networks for image recognition}, + year = 2019 +} +@article{ramachandran2019stand, + author = {Niki Parmar and Prajit Ramachandran and Ashish Vaswani and Irwan Bello and Anselm Levskaya and Jonathon Shlens}, + journal = {Neural Information Processing Systems}, + pages = {68--80}, + title = {Stand-alone self-attention in vision models}, + volume = 32, + year = 2019 +} +@article{zhang2019graph, + author = {Zhang, Si and Tong, Hanghang and Xu, Jiejun and Maciejewski, Ross}, + journal = {Computational Social Networks}, + number = 1, + pages = {1--23}, + publisher = {SpringerOpen}, + title = {Graph convolutional networks: A comprehensive review}, + volume = 6, + year = 2019 +} +@article{errica2019fair, + author = {Errica, Federico and Podda, Marco and Bacciu, Davide and Micheli, Alessio}, + journal = {International Conference on Learning Representations}, + title = {A fair comparison of graph neural networks for graph classification}, + year = 2019 +} +@article{selsam2018learning, + author = {Selsam, Daniel and Lamm, Matthew and B{\"u}nz, Benedikt and Liang, Percy and de Moura, Leonardo and Dill, David L}, + journal = {International Conference on Learning Representations}, + title = {Learning a {SAT} solver from single-bit supervision}, + year = 2019 +} +@inproceedings{chiang2019cluster, + author = {Chiang, Wei-Lin and Liu, Xuanqing and Si, Si and Li, Yang and Bengio, Samy and Hsieh, Cho-Jui}, + booktitle = {ACM SIGKDD International Conference on Knowledge Discovery \& Data Mining}, + pages = {257--266}, + title = {Cluster-{GCN}: An efficient algorithm for training deep and large graph convolutional networks}, + year = 2019 +} +@inproceedings{abu2019mixhop, + author = {Abu-El-Haija, Sami and Perozzi, Bryan and Kapoor, Amol and Alipourfard, Nazanin and Lerman, Kristina and Harutyunyan, Hrayr and Ver Steeg, Greg and Galstyan, Aram}, + booktitle = {International Conference on Machine Learning}, + pages = {21--29}, + title = {Mix{H}op: Higher-order graph convolutional architectures via sparsified neighborhood mixing}, + year = 2019 +} +@article{zhang2019gresnet, + author = {Zhang, Jiawei and Meng, Lin}, + journal = {arXiv:1909.05729}, + title = {G{R}es{N}et: Graph residual network for reviving deep gnns from suspended animation}, + year = 2019 +} +@inproceedings{xu2018how, + author = {Keyulu Xu and Weihua Hu and Jure Leskovec and Stefanie Jegelka}, + booktitle = {International Conference on Learning Representations}, + title = {How Powerful are Graph Neural Networks?}, + year = 2019 +} +@article{zou2019layer, + author = {Zou, Difan and Hu, Ziniu and Wang, Yewen and Jiang, Song and Sun, Yizhou and Gu, Quanquan}, + journal = {Neural Information Processing Systems}, + pages = {11247--11256}, + title = {Layer-dependent importance sampling for training deep and large graph convolutional networks}, + volume = 32, + year = 2019 +} +@article{oono2019graph, + author = {Oono, Kenta and Suzuki, Taiji}, + journal = {International Conference on Learning Representations}, + title = {Graph neural networks exponentially lose expressive power for node classification}, + year = 2019 +} +@inproceedings{lee2019self, + author = {Lee, Junhyun and Lee, Inyeop and Kang, Jaewoo}, + booktitle = {International Conference on Machine Learning}, + pages = {3734--3743}, + title = {Self-attention graph pooling}, + year = 2019 +} +@inproceedings{gao2019graph, + author = {Gao, Hongyang and Ji, Shuiwang}, + booktitle = {International Conference on Machine Learning}, + pages = {2083--2092}, + title = {Graph {U}-{N}ets}, + year = 2019 +} +@article{kynkaanniemi2019improved, + author = {Kynk{\"a}{\"a}nniemi, Tuomas and Karras, Tero and Laine, Samuli and Lehtinen, Jaakko and Aila, Timo}, + journal = {Neural Information Processing Systems}, + pages = {3929--3938}, + title = {Improved precision and recall metric for assessing generative models}, + volume = 32, + year = 2019 +} +@article{brock2018large, + author = {Brock, Andrew and Donahue, Jeff and Simonyan, Karen}, + journal = {International Conference on Learning Representations}, + title = {Large scale {GAN} training for high fidelity natural image synthesis}, + year = 2019 +} +@misc{odena2019open, + author = {Odena, Augustus}, + doi = {10.23915/distill.00018}, + howpublished = {Distill, \url{https://distill.pub/2019/gan-open-problems}}, + journal = {Distil}, + title = {Open Questions about Generative Adversarial Networks}, + year = 2019 +} +@inproceedings{karras2019style, + author = {Karras, Tero and Laine, Samuli and Aila, Timo}, + booktitle = {IEEE/CVF Computer Vision \& Pattern Recognition}, + pages = {4401--4410}, + title = {A style-based generator architecture for generative adversarial networks}, + year = 2019 +} +@inproceedings{zhang2019self, + author = {Zhang, Han and Goodfellow, Ian and Metaxas, Dimitris and Odena, Augustus}, + booktitle = {International Conference on Machine Learning}, + pages = {7354--7363}, + title = {Self-attention generative adversarial networks}, + year = 2019 +} +@article{berard2019closer, + author = {Berard, Hugo and Gidel, Gauthier and Almahairi, Amjad and Vincent, Pascal and Lacoste-Julien, Simon}, + journal = {arXiv:1906.04848}, + title = {A closer look at the optimization landscapes of generative adversarial networks}, + year = 2019 +} +@article{jolicoeur2018relativistic, + author = {Jolicoeur-Martineau, Alexia}, + journal = {International Conference on Learning Representations}, + title = {The relativistic discriminator: A key element missing from standard {GAN}}, + year = 2019 +} +@article{peyre2019computational, + author = {Peyr{\'e}, Gabriel and Cuturi, Marco and others}, + journal = {Foundations and Trends in Machine Learning}, + number = {5-6}, + pages = {355--607}, + publisher = {Now Publishers, Inc.}, + title = {Computational optimal transport with applications to data science}, + volume = 11, + year = 2019 +} +@inproceedings{kurach2019large, + author = {Kurach, Karol and Lu{\v{c}}i{\'c}, Mario and Zhai, Xiaohua and Michalski, Marcin and Gelly, Sylvain}, + booktitle = {International Conference on Machine Learning}, + pages = {3581--3590}, + title = {A large-scale study on regularization and normalization in {GAN}s}, + year = 2019 +} +@inproceedings{bau2019seeing, + author = {Bau, David and Zhu, Jun-Yan and Wulff, Jonas and Peebles, William and Strobelt, Hendrik and Zhou, Bolei and Torralba, Antonio}, + booktitle = {IEEE/CVF International Conference on Computer Vision}, + pages = {4502--4511}, + title = {Seeing what a {GAN} cannot generate}, + year = 2019 +} +@inproceedings{mao2019mode, + author = {Mao, Qi and Lee, Hsin-Ying and Tseng, Hung-Yu and Ma, Siwei and Yang, Ming-Hsuan}, + booktitle = {IEEE/CVF Computer Vision \& Pattern Recognition}, + pages = {1429--1437}, + title = {Mode seeking generative adversarial networks for diverse image synthesis}, + year = 2019 +} +@inproceedings{abdal2019image2stylegan, + author = {Abdal, Rameen and Qin, Yipeng and Wonka, Peter}, + booktitle = {IEEE/CVF International Conference on Computer Vision}, + pages = {4432--4441}, + title = {Image2{S}tyle{GAN}: How to embed images into the {S}tyle{GAN} latent space?}, + year = 2019 +} +@inproceedings{behrmann2019invertible, + author = {Behrmann, Jens and Grathwohl, Will and Chen, Ricky TQ and Duvenaud, David and Jacobsen, J{\"o}rn-Henrik}, + booktitle = {International Conference on Machine Learning}, + pages = {573--582}, + title = {Invertible residual networks}, + year = 2019 +} +@article{song2019mintnet, + author = {Song, Yang and Meng, Chenlin and Ermon, Stefano}, + journal = {Neural Information Processing Systems}, + pages = {11002--11012}, + title = {Mint{N}et: Building invertible neural networks with masked convolutions}, + volume = 32, + year = 2019 +} +@inproceedings{brugger2019partially, + author = {Br{\"u}gger, Robin and Baumgartner, Christian F and Konukoglu, Ender}, + booktitle = {International Conference on Medical Image Computing and Computer-Assisted Intervention}, + pages = {429--437}, + title = {A partially reversible {U}-{N}et for memory-efficient volumetric image segmentation}, + year = 2019 +} +@inproceedings{ho2019flow++, + author = {Ho, Jonathan and Chen, Xi and Srinivas, Aravind and Duan, Yan and Abbeel, Pieter}, + booktitle = {International Conference on Machine Learning}, + pages = {2722--2730}, + title = {Flow++: Improving flow-based generative models with variational dequantization and architecture design}, + year = 2019 +} +@inproceedings{abdelhamed2019noise, + author = {Abdelhamed, Abdelrahman and Brubaker, Marcus A and Brown, Michael S}, + booktitle = {IEEE/CVF International Conference on Computer Vision}, + pages = {3165--3173}, + title = {Noise flow: Noise modeling with conditional normalizing flows}, + year = 2019 +} +@article{kumar2019videoflow, + author = {Kumar, Manoj and Babaeizadeh, Mohammad and Erhan, Dumitru and Finn, Chelsea and Levine, Sergey and Dinh, Laurent and Kingma, Durk}, + journal = {ICML Workshop on Invertible Neural Networks and Normalizing Flows}, + title = {Video{F}low: A flow-based generative model for video}, + year = 2019 +} +@article{esling2019universal, + author = {Esling, Philippe and Masuda, Naotake and Bardet, Adrien and Despres, Romeo and others}, + journal = {International Conference on Digital Audio Effects}, + title = {Universal audio synthesizer control with normalizing flows}, + year = 2019 +} +@inproceedings{prenger2019waveglow, + author = {Prenger, Ryan and Valle, Rafael and Catanzaro, Bryan}, + booktitle = {IEEE International Conference on Acoustics, Speech and Signal Processing}, + pages = {3617--3621}, + title = {Waveglow: A flow-based generative network for speech synthesis}, + year = 2019 +} +@article{madhawa2019graphnvp, + author = {Madhawa, Kaushalya and Ishiguro, Katushiko and Nakago, Kosuke and Abe, Motoki}, + journal = {arXiv:1905.11600}, + title = {Graph{NVP}: An invertible flow model for generating molecular graphs}, + year = 2019 +} +@article{muller2019neural, + author = {M{\"u}ller, Thomas and McWilliams, Brian and Rousselle, Fabrice and Gross, Markus and Nov{\'a}k, Jan}, + journal = {ACM Transactions on Graphics (TOG)}, + number = 5, + pages = {1--19}, + publisher = {ACM New York, NY, USA}, + title = {Neural importance sampling}, + volume = 38, + year = 2019 +} +@article{noe2019boltzmann, + author = {No{\'e}, Frank and Olsson, Simon and K{\"o}hler, Jonas and Wu, Hao}, + journal = {Science}, + number = 6457, + publisher = {American Association for the Advancement of Science}, + title = {Boltzmann generators: Sampling equilibrium states of many-body systems with deep learning}, + volume = 365, + year = 2019 +} +@article{tran2019discrete, + author = {Tran, Dustin and Vafa, Keyon and Agrawal, Kumar and Dinh, Laurent and Poole, Ben}, + journal = {Neural Information Processing Systems}, + pages = {14692--14701}, + title = {Discrete flows: Invertible generative models of discrete data}, + volume = 32, + year = 2019 +} +@inproceedings{ziegler2019latent, + author = {Ziegler, Zachary and Rush, Alexander}, + booktitle = {International Conference on Machine Learning}, + pages = {7673--7682}, + title = {Latent normalizing flows for discrete sequences}, + year = 2019 +} +@article{zhou2019density, + author = {Zhou, Chunting and Ma, Xuezhe and Wang, Di and Neubig, Graham}, + journal = {ACL Human Language Technologies}, + pages = {1588--1598}, + title = {Density matching for bilingual word embedding}, + year = 2019 +} +@inproceedings{jin2019unsupervised, + author = {Jin, Lifeng and Doshi-Velez, Finale and Miller, Timothy and Schwartz, Lane and Schuler, William}, + booktitle = {Meeting of the Association for Computational Linguistics}, + pages = {2442--2452}, + title = {Unsupervised learning of {PCFG}s with normalizing flow}, + year = 2019 +} +@article{schroecker2019generative, + author = {Schroecker, Yannick and Vecerik, Mel and Scholz, Jonathan}, + journal = {International Conference on Learning Representations}, + title = {Generative predecessor models for sample-efficient imitation learning}, + year = 2019 +} +@article{ward2019improving, + author = {Ward, Patrick Nadeem and Smofsky, Ariella and Bose, Avishek Joey}, + journal = {ICML Workshop on Invertible Neural Networks and Normalizing Flows}, + title = {Improving exploration in soft-actor-critic with normalizing flows policies}, + year = 2019 +} +@inproceedings{hoogeboom2019emerging, + author = {Hoogeboom, Emiel and Van Den Berg, Rianne and Welling, Max}, + booktitle = {International Conference on Machine Learning}, + pages = {2771--2780}, + title = {Emerging convolutions for generative normalizing flows}, + year = 2019 +} +@article{durkan2019cubic, + author = {Durkan, Conor and Bekasov, Artur and Murray, Iain and Papamakarios, George}, + journal = {ICML Invertible Neural Networks and Normalizing Flows}, + title = {Cubic-spline flows}, + year = 2019 +} +@article{durkan2019neural, + author = {Durkan, Conor and Bekasov, Artur and Murray, Iain and Papamakarios, George}, + journal = {Neural Information Processing Systems}, + pages = {7509--7520}, + title = {Neural spline flows}, + volume = 32, + year = 2019 +} +@article{wehenkel2019unconstrained, + author = {Wehenkel, Antoine and Louppe, Gilles}, + journal = {Neural Information Processing Systems}, + pages = {1543--1553}, + title = {Unconstrained monotonic neural networks}, + volume = 32, + year = 2019 +} +@inproceedings{jaini2019sum, + author = {Jaini, Priyank and Selby, Kira A and Yu, Yaoliang}, + booktitle = {International Conference on Machine Learning}, + pages = {3009--3018}, + title = {Sum-of-squares polynomial flow}, + year = 2019 +} +@article{dinh2019rad, + author = {Dinh, Laurent and Sohl-Dickstein, Jascha and Larochelle, Hugo and Pascanu, Razvan}, + journal = {ICLR Workshop on Deep Generative Models for Highly Structured Data}, + title = {A {RAD} approach to deep mixture models}, + year = 2019 +} +@article{Das2019DimensionalityRF, + author = {Das, Hari Prasanna and Abbeel, Pieter and Spanos, Costas J}, + journal = {arXiv:1908.01686}, + title = {Likelihood Contribution based Multi-scale Architecture for Generative Flows}, + year = 2019 +} +@article{chang2019antisymmetricrnn, + author = {Chang, Bo and Chen, Minmin and Haber, Eldad and Chi, Ed H}, + journal = {International Conference on Learning Representations}, + title = {Antisymmetric{RNN}: A dynamical system view on recurrent neural networks}, + year = 2019 +} +@article{chen2019residual, + author = {Chen, Ricky TQ and Behrmann, Jens and Duvenaud, David K and Jacobsen, J{\"o}rn-Henrik}, + journal = {Neural Information Processing Systems}, + pages = {9913--9923}, + title = {Residual flows for invertible generative modeling}, + volume = 32, + year = 2019 +} +@article{grathwohl2018ffjord, + author = {Grathwohl, Will and Chen, Ricky TQ and Bettencourt, Jesse and Sutskever, Ilya and Duvenaud, David}, + journal = {International Conference on Learning Representations}, + title = {Ffjord: Free-form continuous dynamics for scalable reversible generative models}, + year = 2019 +} +@article{dupont2019augmented, + author = {Dupont, Emilien and Doucet, Arnaud and Teh, Yee Whye}, + journal = {Neural Information Processing Systems}, + pages = {3134--3144}, + title = {Augmented neural {ODE}s}, + volume = 32, + year = 2019 +} +@article{tzen2019neural, + author = {Tzen, Belinda and Raginsky, Maxim}, + journal = {arXiv:1905.09883}, + title = {Neural stochastic differential equations: Deep latent {G}aussian models in the diffusion limit}, + year = 2019 +} +@article{hoogeboom2019integer, + author = {Hoogeboom, Emiel and Peters, Jorn and Van Den Berg, Rianne and Welling, Max}, + journal = {Neural Information Processing Systems}, + pages = {12134--12144}, + title = {Integer discrete flows and lossless compression}, + volume = 32, + year = 2019 +} +@article{wang2019riemannian, + author = {Wang, Prince Zizhuang and Wang, William Yang}, + journal = {ACL Human Language Technologies}, + pages = {284--294}, + title = {Riemannian normalizing flow on variational {W}asserstein autoencoder for text modeling}, + year = 2019 +} +@article{rezende2019equivariant, + author = {Rezende, Danilo Jimenez and Racani{\`e}re, S{\'e}bastien and Higgins, Irina and Toth, Peter}, + journal = {arXiv:1909.13739}, + title = {Equivariant {H}amiltonian flows}, + year = 2019 +} +@article{kingma2019introduction, + author = {Kingma, Diederik P and Welling, Max and others}, + journal = {Foundations and Trends in Machine Learning}, + number = 4, + pages = {307--392}, + publisher = {Now Publishers, Inc.}, + title = {An introduction to variational autoencoders}, + volume = 12, + year = 2019 +} +@article{gregor2018temporal, + author = {Gregor, Karol and Papamakarios, George and Besse, Frederic and Buesing, Lars and Weber, Theophane}, + journal = {International Conference on Learning Representations}, + title = {Temporal difference variational auto-encoder}, + year = 2019 +} +@article{razavi2019generating, + author = {Razavi, Ali and Van den Oord, Aaron and Vinyals, Oriol}, + journal = {Neural Information Processing Systems}, + pages = {14837--14847}, + title = {Generating diverse high-fidelity images with {VQ-VAE}-2}, + volume = 32, + year = 2019 +} +@article{razavi2019preventing, + 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Conference on Machine Learning}, + pages = {4402--4412}, + title = {Disentangling disentanglement in variational autoencoders}, + year = 2019 +} +@article{masrani2019thermodynamic, + author = {Masrani, Vaden and Le, Tuan Anh and Wood, Frank}, + journal = {Neural Information Processing Systems}, + pages = {11521--11530}, + title = {The thermodynamic variational objective}, + volume = 32, + year = 2019 +} +@article{song2019generative, + author = {Song, Yang and Ermon, Stefano}, + journal = {Neural Information Processing Systems}, + pages = {11895--11907}, + title = {Generative modeling by estimating gradients of the data distribution}, + volume = 32, + year = 2019 +} +@article{berner2019dota, + author = {Berner, Christopher and Brockman, Greg and Chan, Brooke and Cheung, Vicki and D{\k{e}}biak, Przemys{\l}aw and Dennison, Christy and Farhi, David and Fischer, Quirin and Hashme, Shariq and Hesse, Chris and others}, + journal = {arXiv:1912.06680}, + title = {{DOTA} 2 with large scale deep reinforcement learning}, + year = 2019 +} +@article{kool2018attention, + author = {Kool, Wouter and van Hoof, Herke and Welling, Max}, + journal = {International Conference on Learning Representations}, + title = {Attention, Learn to Solve Routing Problems!}, + year = 2019 +} +@inproceedings{fujimoto2019off, + author = {Fujimoto, Scott and Meger, David and Precup, Doina}, + booktitle = {International Conference on Machine Learning}, + pages = {2052--2062}, + title = {Off-policy deep reinforcement learning without exploration}, + year = 2019 +} +@article{kumar2019stabilizing, + author = {Kumar, Aviral and Fu, Justin and Soh, Matthew and Tucker, George and Levine, Sergey}, + journal = {Neural Information Processing Systems}, + pages = {11761--11771}, + title = {Stabilizing off-policy {Q}-learning via bootstrapping error reduction}, + volume = 32, + year = 2019 +} +@article{aubret2019survey, + author = {Aubret, Arthur and Matignon, Laetitia and Hassas, Salima}, + journal = 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neural networks}, + year = 2019 +} +@inproceedings{du2019gradient, + author = {Simon S. 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Conference on Learning Representations}, + title = {The Lottery Ticket Hypothesis: Finding Sparse, Trainable Neural Networks}, + year = 2019 +} +@inproceedings{fort2019large, + author = {Stanislav Fort and Jastrz{\k{e}}bski, Stanis{\l}aw}, + booktitle = {Neural Information Processing Systems}, + pages = {6706--6714}, + title = {Large Scale Structure of Neural Network Loss Landscapes}, + volume = 32, + year = 2019 +} +@inproceedings{fort2019goldilocks, + author = {Fort, Stanislav and Scherlis, Adam}, + booktitle = {AAAI Conference on Artificial Intelligence}, + pages = {3574--3581}, + title = {The {G}oldilocks zone: Towards better understanding of neural network loss landscapes}, + year = 2019 +} +@article{ilyas2019adversarial, + author = {Ilyas, Andrew and Santurkar, Shibani and Tsipras, Dimitris and Engstrom, Logan and Tran, Brandon and Madry, Aleksander}, + journal = {Neural Information Processing Systems}, + pages = {125--136}, + title = {Adversarial examples are not bugs, they are 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author = {Inioluwa Deborah Raji and Joy Buolamwini}, + booktitle = {AAAI/ACM Conference on AI, Ethics, and Society}, + pages = {429--435}, + title = {Actionable Auditing: Investigating the Impact of Publicly Naming Biased Performance Results of Commercial {AI} Products}, + year = 2019 +} +@article{Van-Wynsberghe-Robbins-2019, + author = {Aimee van Wynsberghe and Scott Robbins}, + journal = {Science and Engineering Ethics}, + pages = {719--735}, + title = {Critiquing the Reasons for Making Artificial Moral Agents}, + volume = 25, + year = 2019 +} +@article{Cervantes-et-al-2019, + author = {Jos{\'e}-Antonio Cervantes and Sonia L{\'o}pez and Luis-Felipe Rodr{\'i}guez and Salvador Cervantes and Francisco Cervantes and F{\'e}lix Ramos}, + journal = {Science and Engineering Ethics}, + pages = {501--532}, + title = {Artificial Moral Agents: A Survey of the Current Status}, + volume = 26, + year = 2019 +} +@misc{Berger-2019, + author = {Paul Berger}, + howpublished = {April 07, 2019. 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Xiaohui and Zhang, Frank and others}, + booktitle = {IEEE International Conference on Acoustics, Speech and Signal Processing}, + pages = {6874--6878}, + title = {Transformer-based acoustic modeling for hybrid speech recognition}, + year = 2020 +} +@inproceedings{Lample2020Deep, + author = {Guillaume Lample and François Charton}, + booktitle = {International Conference on Learning Representations}, + title = {Deep Learning For Symbolic Mathematics}, + year = 2020 +} +@article{wu2020deep, + author = {Wu, Neo and Green, Bradley and Ben, Xue and O'Banion, Shawn}, + journal = {arXiv:2001.08317}, + title = {Deep transformer models for time series forecasting: The influenza prevalence case}, + year = 2020 +} +@inproceedings{bender2020climbing, + author = {Bender, Emily M and Koller, Alexander}, + booktitle = {Meeting of the Association for Computational Linguistics}, + pages = {5185--5198}, + title = {Climbing towards {NLU}: On meaning, form, and understanding in the age of data}, + year = 2020 +} +@article{provilkov2019bpe, + author = {Provilkov, Ivan and Emelianenko, Dmitrii and Voita, Elena}, + journal = {Meeting of the Association for Computational Linguistics}, + pages = {1882--1892}, + title = {{BPE}-{D}ropout: Simple and effective subword regularization}, + year = 2020 +} +@article{he2020dynamic, + author = {He, Xuanli and Haffari, Gholamreza and Norouzi, Mohammad}, + journal = {Meeting of the Association for Computational Linguistics}, + pages = {3042--3051}, + title = {Dynamic programming encoding for subword segmentation in neural machine translation}, + year = 2020 +} +@article{holtzman2019curious, + author = {Holtzman, Ari and Buys, Jan and Du, Li and Forbes, Maxwell and Choi, Yejin}, + journal = {International Conference on Learning Representations}, + title = {The curious case of neural text degeneration}, + year = 2020 +} +@misc{elasri2020neural1, + author = {El Asri, Layla and Prince, Simon, Jeremy Damion}, + howpublished = {\url{https://www.borealisai.com/research-blogs/tutorial-6-neural-natural-language-generation-decoding-algorithms/}}, + title = {Tutorial \#6: Neural natural language generation – decoding algorithms}, + year = 2020 +} +@article{cordonnier2019relationship, + author = {Cordonnier, Jean-Baptiste and Loukas, Andreas and Jaggi, Martin}, + journal = {International Conference on Learning Representations}, + title = {On the relationship between self-attention and convolutional layers}, + year = 2020 +} +@article{choromanski2020rethinking, + author = {Choromanski, Krzysztof and Likhosherstov, Valerii and Dohan, David and Song, Xingyou and Gane, Andreea and Sarlos, Tamas and Hawkins, Peter and Davis, Jared and Mohiuddin, Afroz and Kaiser, Lukasz and others}, + journal = {International Conference on Learning Representations}, + title = {Rethinking attention with {P}erformers}, + year = 2020 +} +@inproceedings{wang2020position, + author = {Wang, Benyou and Shang, Lifeng and Lioma, Christina and Jiang, Xin and Yang, Hao and Liu, Qun and Simonsen, Jakob Grue}, + booktitle = {International Conference on Learning Representations}, + title = {On position embeddings in {BERT}}, + year = 2020 +} +@article{raffel2020exploring, + author = {Raffel, Colin and Shazeer, Noam and Roberts, Adam and Lee, Katherine and Narang, Sharan and Matena, Michael and Zhou, Yanqi and Li, Wei and Liu, Peter J and others}, + journal = {Journal of Machine Learning Research}, + number = 140, + pages = {1--67}, + title = {Exploring the limits of transfer learning with a unified text-to-text transformer}, + volume = 21, + year = 2020 +} +@article{huang2020improve, + author = {Huang, Zhiheng and Liang, Davis and Xu, Peng and Xiang, Bing}, + journal = {Empirical Methods in Natural Language Processing}, + title = {Improve transformer models with better relative position embeddings}, + year = 2020 +} +@article{wang2020linformer, + author = {Wang, Sinong and Li, Belinda Z and Khabsa, Madian and Fang, Han and Ma, Hao}, + journal = {arXiv:2006.04768}, + title = {Linformer: Self-attention with linear complexity}, + year = 2020 +} +@article{beltagy2020longformer, + author = {Beltagy, Iz and Peters, Matthew E and Cohan, Arman}, + journal = {arXiv:2004.05150}, + title = {Longformer: The long-document transformer}, + year = 2020 +} +@inproceedings{ainslie2020etc, + author = {Joshua Ainslie and Santiago Onta{\~{n}}{\'{o}}n and Chris Alberti and Vaclav Cvicek and Zachary Fisher and Philip Pham and Anirudh Ravula and Sumit Sanghai and Qifan Wang and Li Yang}, + booktitle = {ACL Empirical Methods in Natural Language Processing}, + pages = {268--284}, + title = {{ETC:} {E}ncoding Long and Structured Inputs in Transformers}, + year = 2020 +} +@article{kitaev2020reformer, + author = {Kitaev, Nikita and Kaiser, {\L}ukasz and Levskaya, Anselm}, + journal = {International Conference on Learning Representations}, + title = {Reformer: The efficient transformer}, + year = 2020 +} +@inproceedings{tay2020sparse, + author = {Tay, Yi and Bahri, Dara and Yang, Liu and Metzler, Donald and Juan, Da-Cheng}, + booktitle = {International Conference on Machine Learning}, + pages = {9438--9447}, + title = {Sparse {S}inkhorn attention}, + year = 2020 +} +@inproceedings{katharopoulos2020transformers, + author = {Katharopoulos, Angelos and Vyas, Apoorv and Pappas, Nikolaos and Fleuret, Fran{\c{c}}ois}, + booktitle = {International Conference on Machine Learning}, + pages = {5156--5165}, + title = {Transformers are {RNN}s: Fast autoregressive transformers with linear attention}, + year = 2020 +} +@inproceedings{xiong2020layer, + author = {Xiong, Ruibin and Yang, Yunchang and He, Di and Zheng, Kai and Zheng, Shuxin and Xing, Chen and Zhang, Huishuai and Lan, Yanyan and Wang, Liwei and Liu, Tieyan}, + booktitle = {International Conference on Machine Learning}, + pages = {10524--10533}, + title = {On layer normalization in the transformer architecture}, + year = 2020 +} +@inproceedings{shen2020powernorm, + author = {Shen, Sheng and Yao, Zhewei and Gholami, Amir and Mahoney, Michael and Keutzer, Kurt}, + booktitle = {International Conference on Machine Learning}, + pages = {8741--8751}, + title = {Power{N}orm: Rethinking batch normalization in transformers}, + year = 2020 +} +@article{liu2020understanding, + author = {Liu, Liyuan and Liu, Xiaodong and Gao, Jianfeng and Chen, Weizhu and Han, Jiawei}, + journal = {Empirical Methods in Natural Language Processing}, + pages = {5747--5763}, + title = {Understanding the difficulty of training transformers}, + year = 2020 +} +@inproceedings{chen2020generative, + author = {Chen, Mark and Radford, Alec and Child, Rewon and Wu, Jeffrey and Jun, Heewoo and Luan, David and Sutskever, Ilya}, + booktitle = {International Conference on Machine Learning}, + pages = {1691--1703}, + title = {Generative pretraining from pixels}, + year = 2020 +} +@inproceedings{carion2020end, + author = {Carion, Nicolas and Massa, Francisco and Synnaeve, Gabriel and Usunier, Nicolas and Kirillov, Alexander and Zagoruyko, Sergey}, + booktitle = {European Conference on Computer Vision}, + pages = {213--229}, + title = {End-to-end object detection with transformers}, + year = 2020 +} +@article{zhu2020deformable, + author = {Zhu, Xizhou and Su, Weijie and Lu, Lewei and Li, Bin and Wang, Xiaogang and Dai, Jifeng}, + journal = {International Conference on Learning Representations}, + title = {Deformable {DETR}: Deformable transformers for end-to-end object detection}, + year = 2020 +} +@inproceedings{yang2020learning, + author = {Yang, Fuzhi and Yang, Huan and Fu, Jianlong and Lu, Hongtao and Guo, Baining}, + booktitle = {IEEE/CVF Computer Vision \& Pattern Recognition}, + pages = {5791--5800}, + title = {Learning texture transformer network for image super-resolution}, + year = 2020 +} +@inproceedings{zhao2020uctgan, + author = {Zhao, Lei and Mo, Qihang and Lin, Sihuan and Wang, Zhizhong and Zuo, Zhiwen and Chen, Haibo and Xing, Wei and Lu, Dongming}, + booktitle = {IEEE/CVF Computer Vision \& Pattern Recognition}, + pages = {5741--5750}, + title = {{UCTGAN}: Diverse image inpainting based on unsupervised cross-space translation}, + year = 2020 +} +@article{wu2020visual, + author = {Wu, Bichen and Xu, Chenfeng and Dai, Xiaoliang and Wan, Alvin and Zhang, Peizhao and Yan, Zhicheng and Tomizuka, Masayoshi and Gonzalez, Joseph and Keutzer, Kurt and Vajda, Peter}, + journal = {arXiv:2006.03677}, + title = {Visual transformers: Token-based image representation and processing for computer vision}, + year = 2020 +} +@article{locatello2020object, + author = {Locatello, Francesco and Weissenborn, Dirk and Unterthiner, Thomas and Mahendran, Aravindh and Heigold, Georg and Uszkoreit, Jakob and Dosovitskiy, Alexey and Kipf, Thomas}, + journal = {Neural Information Processing Systems}, + pages = {11525--11538}, + title = {Object-centric learning with slot attention}, + volume = 33, + year = 2020 +} +@inproceedings{chen2020uniter, + author = {Chen, Yen-Chun and Li, Linjie and Yu, Licheng and El Kholy, Ahmed and Ahmed, Faisal and Gan, Zhe and Cheng, Yu and Liu, Jingjing}, + booktitle = {European Conference on Computer Vision}, + pages = {104--120}, + title = {{UNITER}: Universal image-text representation learning}, + year = 2020 +} +@inproceedings{zhao2020exploring, + author = {Zhao, Hengshuang and Jia, Jiaya and Koltun, Vladlen}, + booktitle = {IEEE/CVF Computer Vision \& Pattern Recognition}, + pages = {10076--10085}, + title = {Exploring self-attention for image recognition}, + year = 2020 +} +@article{zhou2020graph, + author = {Zhou, Jie and Cui, Ganqu and Hu, Shengding and Zhang, Zhengyan and Yang, Cheng and Liu, Zhiyuan and Wang, Lifeng and Li, Changcheng and Sun, Maosong}, + journal = {AI Open}, + pages = {57--81}, + publisher = {Elsevier}, + title = {Graph neural networks: A review of methods and applications}, + volume = 1, + year = 2020 +} +@article{wu2020comprehensive, + author = {Wu, Zonghan and Pan, Shirui and Chen, Fengwen and Long, Guodong and Zhang, Chengqi and Philip, S Yu}, + journal = {IEEE Transactions on Neural Networks and Learning Systems}, + number = 1, + pages = {4--24}, + title = {A comprehensive survey on graph neural networks}, + volume = 32, + year = 2020 +} +@article{chami2020machine, + author = {Chami, Ines and Abu-El-Haija, Sami and Perozzi, Bryan and R{\'e}, Christopher and Murphy, Kevin}, + journal = {arXiv:2005.03675}, + title = {Machine learning on graphs: A model and comprehensive taxonomy}, + year = 2020 +} +@inproceedings{loukas2020what, + author = {Andreas Loukas}, + booktitle = {International Conference on Learning Representations}, + title = {What graph neural networks cannot learn: {D}epth vs width}, + year = 2020 +} +@article{tsitsulin2020graph, + author = {Tsitsulin, Anton and Palowitch, John and Perozzi, Bryan and M{\"u}ller, Emmanuel}, + journal = {arXiv:2006.16904}, + title = {Graph clustering with graph neural networks}, + year = 2020 +} +@article{mutlu2020review, + author = {Mutlu, Ece C and Oghaz, Toktam and Rajabi, Amirarsalan and Garibay, Ivan}, + journal = {Machine Learning and Knowledge Extraction}, + number = 4, + pages = {672--704}, + publisher = {MDPI}, + title = {Review on learning and extracting graph features for link prediction}, + volume = 2, + year = 2020 +} +@article{kumar2020link, + author = {Kumar, Ajay and Singh, Shashank Sheshar and Singh, Kuldeep and Biswas, Bhaskar}, + journal = {Physica A: Statistical Mechanics and its Applications}, + pages = 124289, + publisher = {Elsevier}, + title = {Link prediction techniques, applications, and performance: A survey}, + volume = 553, + year = 2020 +} +@article{rossi2020sign, + author = {Rossi, Emanuele and Frasca, Fabrizio and Chamberlain, Ben and Eynard, Davide and Bronstein, Michael and Monti, Federico}, + journal = {ICML Graph Representation Learning and Beyond Workshop}, + pages = 15, + publisher = {2020b}, + title = {{SIGN}: Scalable inception graph neural networks}, + volume = 7, + year = 2020 +} +@article{zhang2020graph, + author = {Zhang, Jiawei and Zhang, Haopeng and Xia, Congying and Sun, Li}, + journal = {arXiv:2001.05140}, + title = {Graph-{B}ert: Only attention is needed for learning graph representations}, + year = 2020 +} +@article{zeng2019graphsaint, + author = {Zeng, Hanqing and Zhou, Hongkuan and Srivastava, Ajitesh and Kannan, Rajgopal and Prasanna, Viktor}, + journal = {International Conference on Learning Representations}, + title = {Graph{SAINT}: Graph sampling based inductive learning method}, + year = 2020 +} +@inproceedings{rozemberczki2020little, + author = {Rozemberczki, Benedek and Kiss, Oliver and Sarkar, Rik}, + booktitle = {ACM International Conference on Information \& Knowledge Management}, + pages = {3133--3140}, + title = {Little ball of fur: A {P}ython library for graph sampling}, + year = 2020 +} +@inproceedings{Rong2020DropEdge, + author = {Yu Rong and Wenbing Huang and Tingyang Xu and Junzhou Huang}, + booktitle = {International Conference on Learning Representations}, + title = {Drop{E}dge: Towards Deep Graph Convolutional Networks on Node Classification}, + year = 2020 +} +@inproceedings{teru2020inductive, + author = {Teru, Komal and Denis, Etienne and Hamilton, Will}, + booktitle = {International Conference on Machine Learning}, + pages = {9448--9457}, + title = {Inductive relation prediction by subgraph reasoning}, + year = 2020 +} +@inproceedings{hasanzadeh2020bayesian, + author = {Hasanzadeh, Arman and Hajiramezanali, Ehsan and Boluki, Shahin and Zhou, Mingyuan and Duffield, Nick and Narayanan, Krishna and Qian, Xiaoning}, + booktitle = {International Conference on Machine Learning}, + pages = {4094--4104}, + title = {Bayesian graph neural networks with adaptive connection sampling}, + year = 2020 +} +@article{zhao2019pairnorm, + author = {Zhao, Lingxiao and Akoglu, Leman}, + journal = {International Conference on Learning Representations}, + title = {Pair{N}orm: Tackling oversmoothing in {GNN}s}, + year = 2020 +} +@article{zhou2020towards, + author = {Zhou, Kaixiong and Huang, Xiao and Li, Yuening and Zha, Daochen and Chen, Rui and Hu, Xia}, + journal = {Neural Information Processing Systems}, + pages = {4917--4928}, + title = {Towards deeper graph neural networks with differentiable group normalization}, + volume = 33, + year = 2020 +} +@article{li2020deepergcn, + author = {Li, Guohao and Xiong, Chenxin and Thabet, Ali and Ghanem, Bernard}, + journal = {arXiv:2006.07739}, + title = {Deeper{GCN}: All you need to train deeper {GCN}s}, + year = 2020 +} +@inproceedings{gong2020geometrically, + author = {Gong, Shunwang and Bahri, Mehdi and Bronstein, Michael M and Zafeiriou, Stefanos}, + booktitle = {IEEE/CVF Computer Vision \& Pattern Recognition}, + pages = {11415--11424}, + title = {Geometrically principled connections in graph neural networks}, + year = 2020 +} +@inproceedings{chen2020simple2, + author = {Chen, Ming and Wei, Zhewei and Huang, Zengfeng and Ding, Bolin and Li, Yaliang}, + booktitle = {International Conference on Machine Learning}, + pages = {1725--1735}, + title = {Simple and deep graph convolutional networks}, + year = 2020 +} +@article{ho2020denoising, + author = {Ho, Jonathan and Jain, Ajay and Abbeel, Pieter}, + journal = {Neural Information Processing Systems}, + pages = {6840--6851}, + title = {Denoising diffusion probabilistic models}, + volume = 33, + year = 2020 +} +@inproceedings{farnia2020gans, + author = {Farnia, Farzan and Ozdaglar, Asuman}, + booktitle = {International Conference on Machine Learning}, + pages = {3029--3039}, + title = {Do {GAN}s always have {N}ash equilibria?}, + year = 2020 +} +@inproceedings{jin2020local, + author = {Jin, Chi and Netrapalli, Praneeth and Jordan, Michael}, + booktitle = {International Conference on Machine Learning}, + pages = {4880--4889}, + title = {What is local optimality in nonconvex-nonconcave minimax optimization?}, + year = 2020 +} +@article{qi2020loss, + author = {Qi, Guo-Jun}, + journal = {International Journal of Computer Vision}, + number = 5, + pages = {1118--1140}, + publisher = {Springer}, + title = {Loss-sensitive generative adversarial networks on {L}ipschitz densities}, + volume = 128, + year = 2020 +} +@article{sinha2020top, + author = {Samarth Sinha and Zhengli Zhao and Anirudh Goyal and Colin Raffel and Augustus Odena}, + journal = {Neural Information Processing Systems}, + pages = {14638--14649}, + title = {Top-k training of {GAN}s: Improving {GAN} performance by throwing away bad samples}, + volume = 33, + year = 2020 +} +@misc{chintala2020how, + author = {Chintala, Soumith and Denton, Emily and Arjovsky, Martin and Matheiu, Michael}, + howpublished = {\url{https://github.com/soumith/ganhacks}}, + title = {How to Train a {GAN}? {T}ips and tricks to make {GAN}s work}, + year = 2020 +} +@article{karras2020training, + author = {Karras, Tero and Aittala, Miika and Hellsten, Janne and Laine, Samuli and Lehtinen, Jaakko and Aila, Timo}, + journal = {Neural Information Processing Systems}, + pages = {12104--12114}, + title = {Training generative adversarial networks with limited data}, + volume = 33, + year = 2020 +} +@inproceedings{collins2020editing, + author = {Collins, Edo and Bala, Raja and Price, Bob and Susstrunk, Sabine}, + booktitle = {IEEE/CVF Computer Vision \& Pattern Recognition}, + pages = {5771--5780}, + title = {Editing in style: Uncovering the local semantics of {GAN}s}, + year = 2020 +} +@article{harkonen2020ganspace, + author = {H{\"a}rk{\"o}nen, Erik and Hertzmann, Aaron and Lehtinen, Jaakko and Paris, Sylvain}, + journal = {Neural Information Processing Systems}, + pages = {9841--9850}, + title = {{GANS}pace: Discovering interpretable {GAN} controls}, + volume = 33, + year = 2020 +} +@inproceedings{shen2020interpreting, + author = {Shen, Yujun and Gu, Jinjin and Tang, Xiaoou and Zhou, Bolei}, + booktitle = {IEEE/CVF Computer Vision \& Pattern Recognition}, + pages = {9243--9252}, + title = {Interpreting the latent space of {GAN}s for semantic face editing}, + year = 2020 +} +@inproceedings{tewari2020stylerig, + author = {Tewari, Ayush and Elgharib, Mohamed and Bharaj, Gaurav and Bernard, Florian and Seidel, Hans-Peter and P{\'e}rez, Patrick and Zollhofer, Michael and Theobalt, Christian}, + booktitle = {IEEE/CVF Computer Vision \& Pattern Recognition}, + pages = {6142--6151}, + title = {Style{R}ig: Rigging {S}tyle{GAN} for 3{D} control over portrait images}, + year = 2020 +} +@article{guan2020collaborative, + author = {Guan, Shanyan and Tai, Ying and Ni, Bingbing and Zhu, Feida and Huang, Feiyue and Yang, Xiaokang}, + journal = {arXiv:2007.01758}, + title = {Collaborative learning for faster {S}tyle{GAN} embedding}, + year = 2020 +} +@inproceedings{zhu2020domain, + author = {Zhu, Jiapeng and Shen, Yujun and Zhao, Deli and Zhou, Bolei}, + booktitle = {European Conference on Computer Vision}, + pages = {592--608}, + title = {In-domain {GAN} inversion for real image editing}, + year = 2020 +} +@inproceedings{abdal2020image2stylegan++, + author = {Abdal, Rameen and Qin, Yipeng and Wonka, Peter}, + booktitle = {IEEE/CVF Computer Vision \& Pattern Recognition}, + pages = {8296--8305}, + title = {Image2{S}tyle{GAN}++: How to edit the embedded images?}, + year = 2020 +} +@article{etmann2020iunets, + author = {Etmann, Christian and Ke, Rihuan and Sch{\"o}nlieb, Carola-Bibiane}, + journal = {IEEE International Workshop on Machine Learning for Signal Processing}, + title = {iUNets: Fully invertible {U}-{N}ets with learnable up-and downsampling}, + year = 2020 +} +@article{yu2020wavelet, + author = {Yu, Jason J and Derpanis, Konstantinos G and Brubaker, Marcus A}, + journal = {Neural Information Processing Systems}, + pages = {6184--6196}, + title = {Wavelet flow: Fast training of high resolution normalizing flows}, + volume = 33, + year = 2020 +} +@inproceedings{ardizzone2020conditional, + author = {Ardizzone, Lynton and Kruse, Jakob and L{\"u}th, Carsten and Bracher, Niels and Rother, Carsten and K{\"o}the, Ullrich}, + booktitle = {DAGM German Conference on Pattern Recognition}, + pages = {373--387}, + title = {Conditional invertible neural networks for diverse image-to-image translation}, + year = 2020 +} +@article{kanwar2020equivariant, + author = {Kanwar, Gurtej and Albergo, Michael S and Boyda, Denis and Cranmer, Kyle and Hackett, Daniel C and Racaniere, S{\'e}bastien and Rezende, Danilo Jimenez and Shanahan, Phiala E}, + journal = {Physical Review Letters}, + number = 12, + pages = 121601, + publisher = {APS}, + title = {Equivariant flow-based sampling for lattice gauge theory}, + volume = 125, + year = 2020 +} +@inproceedings{kohler2020equivariant, + author = {K{\"o}hler, Jonas and Klein, Leon and No{\'e}, Frank}, + booktitle = {International Conference on Machine Learning}, + pages = {5361--5370}, + title = {Equivariant flows: Exact likelihood generative learning for symmetric densities}, + year = 2020 +} +@article{wirnsberger2020targeted, + author = {Wirnsberger, Peter and Ballard, Andrew J and Papamakarios, George and Abercrombie, Stuart and Racani{\`e}re, S{\'e}bastien and Pritzel, Alexander and Jimenez Rezende, Danilo and Blundell, Charles}, + journal = {The Journal of Chemical Physics}, + number = 14, + pages = 144112, + publisher = {AIP Publishing LLC}, + title = {Targeted free energy estimation via learned mappings}, + volume = 153, + year = 2020 +} +@article{wong2020gravitational, + author = {Wong, Kaze WK and Contardo, Gabriella and Ho, Shirley}, + journal = {Physical Review D}, + number = 12, + pages = 123005, + publisher = {APS}, + title = {Gravitational-wave population inference with deep flow-based generative network}, + volume = 101, + year = 2020 +} +@inproceedings{mazoure2020leveraging, + author = {Mazoure, Bogdan and Doan, Thang and Durand, Audrey and Pineau, Joelle and Hjelm, R Devon}, + booktitle = {Conference on Robot Learning}, + pages = {430--444}, + title = {Leveraging exploration in off-policy algorithms via normalizing flows}, + year = 2020 +} +@inproceedings{touati2020randomized, + author = {Touati, Ahmed and Satija, Harsh and Romoff, Joshua and Pineau, Joelle and Vincent, Pascal}, + booktitle = {Uncertainty in Artificial Intelligence}, + pages = {422--432}, + title = {Randomized value functions via multiplicative normalizing flows}, + year = 2020 +} +@inproceedings{finlay2020train, + author = {Chris Finlay and J{\"{o}}rn{-}Henrik Jacobsen and Levon Nurbekyan and Adam M. Oberman}, + booktitle = {International Conference on Machine Learning}, + pages = {3154--3164}, + title = {How to Train Your Neural {ODE:} {T}he World of {J}acobian and Kinetic Regularization}, + year = 2020 +} +@inproceedings{peluchetti2020infinitely, + author = {Peluchetti, Stefano and Favaro, Stefano}, + booktitle = {International Conference on Artificial Intelligence and Statistics}, + pages = {1126--1136}, + title = {Infinitely deep neural networks as diffusion processes}, + year = 2020 +} +@inproceedings{jaini2019tails, + author = {Priyank Jaini and Ivan Kobyzev and Yaoliang Yu and Marcus A. Brubaker}, + booktitle = {International Conference on Machine Learning}, + pages = {4673--4681}, + title = {Tails of {L}ipschitz Triangular Flows}, + year = 2020 +} +@article{vahdat2020nvae, + author = {Vahdat, Arash and Kautz, Jan}, + journal = {Neural Information Processing Systems}, + pages = {19667--19679}, + title = {N{VAE}: A deep hierarchical variational autoencoder}, + volume = 33, + year = 2020 +} +@inproceedings{xu2020variational, + author = {Xu, Peng and Cheung, Jackie Chi Kit and Cao, Yanshuai}, + booktitle = {International Conference on Machine Learning}, + pages = {10534--10543}, + title = {On variational learning of controllable representations for text without supervision}, + year = 2020 +} +@inproceedings{vahdat2020undirected, + author = {Vahdat, Arash and Andriyash, Evgeny and Macready, William}, + booktitle = {International Conference on Machine Learning}, + pages = {9680--9689}, + title = {Undirected graphical models as approximate posteriors}, + year = 2020 +} +@article{lucke2020evidence, + author = {L{\"u}cke, J{\"o}rg and Forster, Dennis and Dai, Zhenwen}, + journal = {arXiv:2010.14860}, + title = {The Evidence Lower Bound of Variational Autoencoders Converges to a Sum of Three Entropies}, + year = 2020 +} +@article{song2020improved, + author = {Song, Yang and Ermon, Stefano}, + journal = {Neural Information Processing Systems}, + pages = {12438--12448}, + title = {Improved techniques for training score-based generative models}, + volume = 33, + year = 2020 +} +@inproceedings{agarwal2020optimistic, + author = {Agarwal, Rishabh and Schuurmans, Dale and Norouzi, Mohammad}, + booktitle = {International Conference on Machine Learning}, + pages = {104--114}, + title = {An optimistic perspective on offline reinforcement learning}, + year = 2020 +} +@article{kumar2020conservative, + author = {Kumar, Aviral and Zhou, Aurick and Tucker, George and Levine, Sergey}, + journal = {Neural Information Processing Systems}, + pages = {1179--1191}, + title = {Conservative {Q}-learning for offline reinforcement learning}, + volume = 33, + year = 2020 +} +@article{stiennon2020learning, + author = {Stiennon, Nisan and Ouyang, Long and Wu, Jeffrey and Ziegler, Daniel and Lowe, Ryan and Voss, Chelsea and Radford, Alec and Amodei, Dario and Christiano, Paul F}, + journal = {Neural Information Processing Systems}, + pages = {3008--3021}, + title = {Learning to summarize with human feedback}, + volume = 33, + year = 2020 +} +@article{schrittwieser2020mastering, + author = {Schrittwieser, Julian and Antonoglou, Ioannis and Hubert, Thomas and Simonyan, Karen and Sifre, Laurent and Schmitt, Simon and Guez, Arthur and Lockhart, Edward and Hassabis, Demis and Graepel, Thore and others}, + journal = {Nature}, + number = 7839, + pages = {604--609}, + publisher = {Nature Publishing Group}, + title = {Mastering {A}tari, {G}o, chess and shogi by planning with a learned model}, + volume = 588, + year = 2020 +} +@article{sejnowski2020unreasonable, + abstract = {Deep learning networks have been trained to recognize speech, caption photographs, and translate text between languages at high levels of performance. Although applications of deep learning networks to real-world problems have become ubiquitous, our understanding of why they are so effective is lacking. These empirical results should not be possible according to sample complexity in statistics and nonconvex optimization theory. However, paradoxes in the training and effectiveness of deep learning networks are being investigated and insights are being found in the geometry of high-dimensional spaces. A mathematical theory of deep learning would illuminate how they function, allow us to assess the strengths and weaknesses of different network architectures, and lead to major improvements. Deep learning has provided natural ways for humans to communicate with digital devices and is foundational for building artificial general intelligence. Deep learning was inspired by the architecture of the cerebral cortex and insights into autonomy and general intelligence may be found in other brain regions that are essential for planning and survival, but major breakthroughs will be needed to achieve these goals.There are no data associated with this paper.}, + author = {Sejnowski, Terrence J.}, + doi = {10.1073/pnas.1907373117}, + issn = {0027-8424}, + journal = {Proceedings of the National Academy of Sciences}, + number = 48, + pages = {30033--30038}, + publisher = {National Academy of Sciences}, + title = {The unreasonable effectiveness of deep learning in artificial intelligence}, + volume = 117, + year = 2020 +} +@article{zou2018stochastic, + author = {Difan Zou and Yuan Cao and Dongruo Zhou and Quanquan Gu}, + journal = {Machine Learning}, + pages = {467--492}, + title = {Gradient Descent Optimizes Over-parameterized Deep {R}e{LU} Networks}, + volume = 109, + year = 2020 +} +@article{bahri2020statistical, + author = {Bahri, Yasaman and Kadmon, Jonathan and Pennington, Jeffrey and Schoenholz, Sam S and Sohl-Dickstein, Jascha and Ganguli, Surya}, + journal = {Annual Review of Condensed Matter Physics}, + pages = {501--528}, + publisher = {Annual Reviews}, + title = {Statistical mechanics of deep learning}, + volume = 11, + year = 2020 +} +@article{d2020underspecification, + author = {D’Amour, Alexander and Heller, Katherine and Moldovan, Dan and Adlam, Ben and Alipanahi, Babak and Beutel, Alex and Chen, Christina and Deaton, Jonathan and Eisenstein, Jacob and Hoffman, Matthew D and others}, + journal = {Journal of Machine Learning Research}, + pages = {1--61}, + title = {Underspecification presents challenges for credibility in modern machine learning}, + year = 2020 +} +@article{tomavsev2020ai, + author = {Toma{\v{s}}ev, Nenad and Cornebise, Julien and Hutter, Frank and Mohamed, Shakir and Picciariello, Angela and Connelly, Bec and Belgrave, Danielle CM and Ezer, Daphne and Haert, Fanny Cachat van der and Mugisha, Frank and others}, + journal = {Nature Communications}, + number = 1, + pages = 2468, + title = {{AI} for social good: {U}nlocking the opportunity for positive impact}, + volume = 11, + year = 2020 +} +@article{Hagendorff-2019, + author = {Thilo Hagendorff}, + journal = {Minds and Machines}, + number = 1, + pages = {99--120}, + title = {The ethics of {AI} ethics: {A}n evaluation of guidelines}, + volume = 30, + year = 2020 +} +@book{Christian-2020, + author = {Brian Christian}, + publisher = {W. W. Norton}, + title = {The Alignment Problem: Machine Learning and Human Values}, + year = 2020 +} +@article{Gabriel-2020, + author = {Iason Gabriel}, + journal = {Minds and Machines}, + pages = {411--437}, + title = {Artificial Intelligence, Values, and Alignment}, + volume = 30, + year = 2020 +} +@inproceedings{menon2020pulse, + author = {Sachit Menon and Alexandru Damian and Shijia Hu and Nikhil Ravi and Cynthia Rudin}, + booktitle = {IEEE/CVF Computer Vision \& Pattern Recognition}, + pages = {2434--2442}, + title = {{PULSE:} Self-Supervised Photo Upsampling via Latent Space Exploration of Generative Models}, + year = 2020 +} +@misc{vincent2020what, + author = {Vincent, James}, + howpublished = {The Verge, June 23, 2020. \url{https://www.theverge.com/21298762/face-depixelizer-ai-machine-learning-tool-pulse-stylegan-obama-bias}}, + title = {What a machine learning tool that turns {O}bama white can (and can’t) tell us about {AI} bias / A striking image that only hints at a much bigger problem}, + year = 2020 +} +@article{Evans-et-al-2022, + author = {Evans, K. and de Moura, N. and Chauvier, S. and Chatila, R. and Dogan, E.}, + journal = {Science and Engineering Ethics}, + number = 6, + pages = {3285--3312}, + title = {Ethical decision making in autonomous vehicles: the {AV} ethics project}, + volume = 26, + year = 2020 +} +@article{Tolmeijer-et-al-2020, + author = {Suzanne Tolmeijer and Markus Kneer and Cristina Sarasua and Markus Christen and Abraham Bernstein}, + journal = {ACM Computing Surveys}, + number = 6, + pages = {1--38}, + title = {Implementations in Machine Ethics: A Survey}, + volume = 53, + year = 2020 +} +@article{Creel-2020, + author = {Kathleen A. Creel}, + journal = {Philosophy of Science}, + number = 4, + pages = {568--589}, + title = {Transparency in Complex Computational Systems}, + volume = 87, + year = 2020 +} +@article{Barrett-2020, + author = {Lindsey Barrett}, + journal = {Boston University Journal of Science and Technology Law}, + number = 2, + pages = {223--285}, + title = {Ban Facial Recognition Technologies for Children --- And for Everyone Else}, + volume = 26, + year = 2020 +} +@inproceedings{Raji-Fried-2021, + author = {Inioluwa Deborah Raji and Genevieve Fried}, + booktitle = {AAAI Workshop on AI Evaluation}, + title = {About Face: A Survey of Facial Recognition Evaluation}, + year = 2020 +} +@article{Fetscherin-et-al-2020, + author = {Marc Fetscherin and Stacey Tantleff-Dunn and Arne Klumb}, + journal = {The Journal of Social Psychology}, + number = 3, + pages = {332--345}, + title = {Effects of facial features and styling elements on perceptions of competence, warmth, and hireability of male professionals}, + volume = 160, + year = 2020 +} +@article{Sisson-et-al-2020, + author = {Melanie Sisson and Jennifer Spindel and Paul Scharre and Vadim Kozyulin}, + journal = {United Nations Office for Disarmament Affairs}, + title = {The Militarization of Artificial Intelligence}, + year = 2020 +} +@article{Mireshghallah-et-al-2020, + author = {Fatemehsadat Mireshghallah and Mohammadkazem Taram and Praneeth Vepakomma and Abhishek Singh and Ramesh Raskar and Hadi Esmaeilzadeh}, + journal = {arXiv:2004.12254}, + title = {Privacy in Deep Learning: A Survey}, + year = 2020 +} +@article{Boulemtafes-et-al-2020, + author = {Amine Boulemtafes and Abdelouahid Derhab and Yacine Challal}, + journal = {Neurocomputing}, + pages = {21--45}, + title = {A Review of Privacy-preserving Techniques for Deep Learning}, + volume = 384, + year = 2020 +} +@article{Strubell-et-al-2020, + author = {Emma Strubell and Ananya Ganesh and Andrew McCallum}, + journal = {Meeting of the Association for Computational Linguistics}, + pages = {13693--13696}, + title = {Energy and Policy Considerations for Modern Deep Learning Research}, + year = 2020 +} +@misc{Riedl-2020, + author = {Mark Riedl}, + howpublished = {The Gradient, Sept 25, 2020. \url{https://thegradient.pub/ai-democratization-in-the-era-of-gpt-3/}}, + title = {{AI} Democratization in the Era of {GPT-3}}, + year = 2020 +} +@article{Ahmed-et-al-2020, + author = {Shakeel Ahmed and Ravi S. Mula and Soma S. Dhavala}, + journal = {arXiv:2001.00818}, + title = {A Framework for Democratizing {AI}}, + year = 2020 +} +@book{Costanza-Chock-2020, + address = {Cambridge, MA}, + author = {Sasha Constanza-Chock}, + publisher = {The MIT Press}, + title = {Design Justice: Community-Led Practices to Build the Worlds We Need}, + year = 2020 +} +@book{Veliz-2020, + author = {Carissa V{\'e}liz}, + publisher = {Bantam Press}, + title = {Privacy is Power: Why and How You Should Take Back Control of Your Data}, + year = 2020 +} +@misc{dao2021awful, + author = {Dao, David}, + howpublished = {Github. {R}etrieved {J}anuary 17, 2023. \url{https://github.com/daviddao/awful-ai}}, + title = {{\em {A}wful {AI}}}, + year = 2021 +} +@article{liu2021variance, + author = {Liyuan Liu and Haoming Jiang and Pengcheng He and Weizhu Chen and Xiaodong Liu and Jianfeng Gao and Jiawei Han}, + journal = {International Conference on Learning Representations}, + title = {On the Variance of the Adaptive Learning Rate and Beyond}, + year = 2021 +} +@article{zhu2021gradinit, + author = {Zhu, Chen and Ni, Renkun and Xu, Zheng and Kong, Kezhi and Huang, W Ronny and Goldstein, Tom}, + journal = {Neural Information Processing Systems}, + pages = {16410--16422}, + title = {Grad{I}nit: Learning to initialize neural networks for stable and efficient training}, + volume = 34, + year = 2021 +} +@article{xu2021optimizing, + author = {Xu, Peng and Kumar, Dhruv and Yang, Wei and Zi, Wenjie and Tang, Keyi and Huang, Chenyang and Cheung, Jackie Chi Kit and Prince, Simon Jeremy Damion and Cao, Yanshuai}, + journal = {Meeting of the Association for Computational Linguistics}, + title = {Optimizing deeper transformers on small datasets}, + year = 2021 +} +@inproceedings{narayanan2021memoryefficient, + author = {Narayanan, Deepak and Phanishayee, Amar and Shi, Kaiyu and Chen, Xie and Zaharia, Matei}, + booktitle = {International Conference on Machine Learning}, + pages = {7937--7947}, + title = {Memory-efficient pipeline-parallel {DNN} training}, + year = 2021 +} +@inproceedings{narayanan2021efficient, + author = {Narayanan, Deepak and Shoeybi, Mohammad and Casper, Jared and LeGresley, Patrick and Patwary, Mostofa and Korthikanti, Vijay and Vainbrand, Dmitri and Kashinkunti, Prethvi and Bernauer, Julie and Catanzaro, Bryan and others}, + booktitle = {International Conference for High Performance Computing, Networking, Storage and Analysis}, + pages = {1--15}, + title = {Efficient large-scale language model training on {GPU} clusters using {M}egatron-{LM}}, + year = 2021 +} +@article{nakkiran2019deep, + author = {Nakkiran, Preetum and Kaplun, Gal and Bansal, Yamini and Yang, Tristan and Barak, Boaz and Sutskever, Ilya}, + journal = {Journal of Statistical Mechanics: Theory and Experiment}, + number = 12, + pages = 124003, + publisher = {IOP Publishing}, + title = {Deep double descent: {W}here bigger models and more data hurt}, + volume = 2021, + year = 2021 +} +@article{buschjager2021doubledescent, + author = {Buschj{\"a}ger, Sebastian and Morik, Katharina}, + journal = {arXiv:2111.04409}, + title = {There is no Double-Descent in Random Forests}, + year = 2021 +} +@article{bubeck2021universal, + author = {Bubeck, S{\'e}bastien and Sellke, Mark}, + journal = {Neural Information Processing Systems}, + pages = {28811--28822}, + title = {A universal law of robustness via isoperimetry}, + volume = 34, + year = 2021 +} +@article{dar2021farewell, + author = {Dar, Yehuda and Muthukumar, Vidya and Baraniuk, Richard G}, + journal = {arXiv:2109.02355}, + title = {A farewell to the bias-variance tradeoff? {A}n overview of the theory of overparameterized machine learning}, + year = 2021 +} +@article{schwarz2021powerpropagation, + author = {Schwarz, Jonathan and Jayakumar, Siddhant and Pascanu, Razvan and Latham, Peter and Teh, Yee}, + journal = {Neural Information Processing Systems}, + pages = {28889--28903}, + title = {Powerpropagation: A sparsity inducing weight reparameterisation}, + volume = 34, + year = 2021 +} +@article{gouk2018regularisation, + author = {Gouk, Henry and Frank, Eibe and Pfahringer, Bernhard and Cree, Michael J.}, + journal = {Machine Learning}, + number = 2, + pages = {393-–416}, + title = {Regularisation of Neural Networks by Enforcing {L}ipschitz Continuity}, + volume = 110, + year = 2021 +} +@article{barrett2021implicit, + author = {David G. T. Barrett and Benoit Dherin}, + journal = {International Conference on Learning Representations}, + title = {Implicit Gradient Regularization}, + year = 2021 +} +@inproceedings{smith2021origin, + author = {Samuel L. Smith and Benoit Dherin and David G. T. Barrett and Soham De}, + booktitle = {International Conference on Learning Representations}, + title = {On the Origin of Implicit Regularization in Stochastic Gradient Descent}, + year = 2021 +} +@inproceedings{jastrzebski2021catastrophic, + author = {Jastrz{\k{e}}bski, Stanis{\l}aw and Arpit, Devansh and Astrand, Oliver and Kerg, Giancarlo B and Wang, Huan and Xiong, Caiming and Socher, Richard and Cho, Kyunghyun and Geras, Krzysztof J}, + booktitle = {International Conference on Machine Learning}, + pages = {4772--4784}, + title = {Catastrophic fisher explosion: Early phase fisher matrix impacts generalization}, + year = 2021 +} +@article{cohen2021gradient, + author = {Cohen, Jeremy M and Kaur, Simran and Li, Yuanzhi and Kolter, J Zico and Talwalkar, Ameet}, + journal = {International Conference on Learning Representations}, + title = {Gradient descent on neural networks typically occurs at the edge of stability}, + year = 2021 +} +@article{noci2021disentangling, + author = {Noci, Lorenzo and Roth, Kevin and Bachmann, Gregor and Nowozin, Sebastian and Hofmann, Thomas}, + journal = {Neural Information Processing Systems}, + pages = {12738--12748}, + title = {Disentangling the Roles of Curation, Data-Augmentation and the Prior in the Cold Posterior Effect}, + volume = 34, + year = 2021 +} +@article{bousselham2021efficient, + author = {Bousselham, Walid and Thibault, Guillaume and Pagano, Lucas and Machireddy, Archana and Gray, Joe and Chang, Young Hwan and Song, Xubo}, + journal = {arXiv:2111.13280}, + title = {Efficient Self-Ensemble Framework for Semantic Segmentation}, + year = 2021 +} +@article{kiranyaz20211d, + author = {Kiranyaz, Serkan and Avci, Onur and Abdeljaber, Osama and Ince, Turker and Gabbouj, Moncef and Inman, Daniel J}, + journal = {Mechanical Systems and Signal Processing}, + pages = 107398, + title = {{1D} convolutional neural networks and applications: A survey}, + volume = 151, + year = 2021 +} +@article{pintea2021resolution, + author = {Pintea, Silvia L and T{\"o}men, Nergis and Goes, Stanley F and Loog, Marco and van Gemert, Jan C}, + journal = {IEEE Transactions on Image Processing}, + pages = {8342--8353}, + publisher = {IEEE}, + title = {Resolution learning in deep convolutional networks using scale-space theory}, + volume = 30, + year = 2021 +} +@article{romero2021flexconv, + author = {Romero, David W and Bruintjes, Robert-Jan and Tomczak, Jakub M and Bekkers, Erik J and Hoogendoorn, Mark and van Gemert, Jan C}, + journal = {International Conference on Learning Representations}, + title = {Flex{C}onv: Continuous kernel convolutions with differentiable kernel sizes}, + year = 2021 +} +@article{minaee2021image, + author = {Minaee, Shervin and Boykov, Yuri Y and Porikli, Fatih and Plaza, Antonio J and Kehtarnavaz, Nasser and Terzopoulos, Demetri}, + journal = {IEEE Transactions on Pattern Analysis \& Machine Intelligence}, + number = 7, + pages = {3523--3542}, + publisher = {IEEE}, + title = {Image segmentation using deep learning: A survey}, + volume = 44, + year = 2021 +} +@article{wightman2021resnet, + author = {Wightman, Ross and Touvron, Hugo and J{\'e}gou, Herv{\'e}}, + journal = {Neural Information Processing Systems Workshop}, + title = {Res{N}et strikes back: An improved training procedure in timm}, + year = 2021 +} +@article{lubana2021beyond, + author = {Lubana, Ekdeep S and Dick, Robert and Tanaka, Hidenori}, + journal = {Neural Information Processing Systems}, + pages = {4778--4791}, + title = {Beyond {B}atch{N}orm: {T}owards a unified understanding of normalization in deep learning}, + volume = 34, + year = 2021 +} +@inproceedings{hayou2021stable, + author = {Hayou, Soufiane and Clerico, Eugenio and He, Bobby and Deligiannidis, George and Doucet, Arnaud and Rousseau, Judith}, + booktitle = {International Conference on Artificial Intelligence and Statistics}, + pages = {1324--1332}, + title = {Stable {R}es{N}et}, + year = 2021 +} +@article{siddique2021u, + author = {Siddique, Nahian and Paheding, Sidike and Elkin, Colin P and Devabhaktuni, Vijay}, + journal = {IEEE Access}, + pages = {82031--82057}, + title = {U-{N}et and its variants for medical image segmentation: A review of theory and applications}, + year = 2021 +} +@inproceedings{liu2021swin, + author = {Liu, Ze and Lin, Yutong and Cao, Yue and Hu, Han and Wei, Yixuan and Zhang, Zheng and Lin, Stephen and Guo, Baining}, + booktitle = {IEEE/CVF International Conference on Computer Vision}, + pages = {10012--10022}, + title = {Swin transformer: Hierarchical vision transformer using shifted windows}, + year = 2021 +} +@article{narang2021transformer, + author = {Narang, Sharan and Chung, Hyung Won and Tay, Yi and Fedus, William and Fevry, Thibault and Matena, Michael and Malkan, Karishma and Fiedel, Noah and Shazeer, Noam and Lan, Zhenzhong and others}, + journal = {Empirical Methods in Natural Language Processing}, + pages = {5758--5773}, + title = {Do transformer modifications transfer across implementations and applications?}, + year = 2021 +} +@article{dosovitskiy2020image, + author = {Dosovitskiy, Alexey and Beyer, Lucas and Kolesnikov, Alexander and Weissenborn, Dirk and Zhai, Xiaohua and Unterthiner, Thomas and Dehghani, Mostafa and Minderer, Matthias and Heigold, Georg and Gelly, Sylvain and others}, + journal = {International Conference on Learning Representations}, + title = {An image is worth 16x16 words: Transformers for image recognition at scale}, + year = 2021 +} +@article{rives2021biological, + author = {Rives, Alexander and Meier, Joshua and Sercu, Tom and Goyal, Siddharth and Lin, Zeming and Liu, Jason and Guo, Demi and Ott, Myle and Zitnick, C Lawrence and Ma, Jerry and others}, + journal = {Proceedings of the National Academy of Sciences}, + number = 15, + publisher = {National Acad Sciences}, + title = {Biological structure and function emerge from scaling unsupervised learning to 250 million protein sequences}, + volume = 118, + year = 2021 +} +@article{rae2021scaling, + author = {Rae, Jack W and Borgeaud, Sebastian and Cai, Trevor and Millican, Katie and Hoffmann, Jordan and Song, Francis and Aslanides, John and Henderson, Sarah and Ring, Roman and Young, Susannah and others}, + journal = {arXiv:2112.11446}, + title = {Scaling language models: Methods, analysis \& insights from training {G}opher}, + year = 2021 +} +@article{ribeiro2020beyond, + author = {Ribeiro, Marco Tulio and Wu, Tongshuang and Guestrin, Carlos and Singh, Sameer}, + journal = {}, + pages = {4824--4828}, + title = {Beyond accuracy: Behavioral testing of {NLP} models with {C}heck{L}ist}, + year = 2021 +} +@inproceedings{bowman2021will, + author = {Samuel R. Bowman and George E. Dahl}, + booktitle = {ACL Human Language Technologies}, + pages = {4843--4855}, + title = {What Will it Take to Fix Benchmarking in Natural Language Understanding?}, + year = 2021 +} +@article{dehghani2021benchmark, + author = {Dehghani, Mostafa and Tay, Yi and Gritsenko, Alexey A and Zhao, Zhe and Houlsby, Neil and Diaz, Fernando and Metzler, Donald and Vinyals, Oriol}, + journal = {arXiv:2107.07002}, + title = {The benchmark lottery}, + year = 2021 +} +@misc{zhai2021attention, + author = {Shuangfei Zhai and Walter Talbott and Nitish Srivastava and Chen Huang and Hanlin Goh and Ruixiang Zhang and Josh Susskind}, + journal = {arXiv:2105.14103}, + title = {An Attention Free Transformer}, + year = 2021 +} +@inproceedings{tay2021synthesizer, + author = {Tay, Yi and Bahri, Dara and Metzler, Donald and Juan, Da-Cheng and Zhao, Zhe and Zheng, Che}, + booktitle = {International Conference on Machine Learning}, + pages = {10183--10192}, + title = {Synthesizer: Rethinking self-attention for transformer models}, + year = 2021 +} +@inproceedings{schlag2021linear, + author = {Schlag, Imanol and Irie, Kazuki and Schmidhuber, J{\"u}rgen}, + booktitle = {International Conference on Machine Learning}, + pages = {9355--9366}, + title = {Linear transformers are secretly fast weight programmers}, + year = 2021 +} +@article{ramsauer2020hopfield, + author = {Ramsauer, Hubert and Sch{\"a}fl, Bernhard and Lehner, Johannes and Seidl, Philipp and Widrich, Michael and Adler, Thomas and Gruber, Lukas and Holzleitner, Markus and Pavlovi{\'c}, Milena and Sandve, Geir Kjetil and others}, + journal = {International Conference on Learning Representations}, + title = {Hopfield networks is all you need}, + year = 2021 +} +@misc{prince2021transext, + author = {Prince, Simon Jeremy Damion}, + howpublished = {\url{https://www.borealisai.com/en/blog/tutorial-16-transformers-ii-extensions/}}, + title = {Transformers {II}: {E}xtensions}, + year = 2021 +} +@article{ke2020rethinking, + author = {Ke, Guolin and He, Di and Liu, Tie-Yan}, + journal = {International Conference on Learning Representations}, + title = {Rethinking positional encoding in language pre-training}, + year = 2021 +} +@article{he2020deberta, + author = {He, Pengcheng and Liu, Xiaodong and Gao, Jianfeng and Chen, Weizhu}, + journal = {International Conference on Learning Representations}, + title = {De{BERT}a: Decoding-enhanced {BERT} with disentangled attention}, + year = 2021 +} +@article{su2021roformer, + author = {Su, Jianlin and Lu, Yu and Pan, Shengfeng and Wen, Bo and Liu, Yunfeng}, + journal = {arXiv:2104.09864}, + title = {Roformer: Enhanced transformer with rotary position embedding}, + year = 2021 +} +@article{dufter2021position, + author = {Dufter, Philipp and Schmitt, Martin and Sch{\"u}tze, Hinrich}, + journal = {Computational Linguistics}, + pages = {1--31}, + title = {Position information in transformers: An overview}, + year = 2021 +} +@article{roy2021efficient, + author = {Roy, Aurko and Saffar, Mohammad and Vaswani, Ashish and Grangier, David}, + journal = {Transactions of the Association for Computational Linguistics}, + pages = {53--68}, + publisher = {MIT Press}, + title = {Efficient content-based sparse attention with routing transformers}, + volume = 9, + year = 2021 +} +@inproceedings{bachlechner2021rezero, + author = {Bachlechner, Thomas and Majumder, Bodhisattwa Prasad and Mao, Henry and Cottrell, Gary and McAuley, Julian}, + booktitle = {Uncertainty in Artificial Intelligence}, + pages = {1352--1361}, + title = {Re{Z}ero is all you need: Fast convergence at large depth}, + year = 2021 +} +@misc{prince2021transtrain, + author = {Prince, Simon Jeremy Damion}, + howpublished = {\url{https://www.borealisai.com/en/blog/tutorial-17-transformers-iii-training/}}, + title = {Transformers {III}: {T}raining}, + year = 2021 +} +@inproceedings{touvron2021training, + author = {Touvron, Hugo and Cord, Matthieu and Douze, Matthijs and Massa, Francisco and Sablayrolles, Alexandre and J{\'e}gou, Herv{\'e}}, + booktitle = {International Conference on Machine Learning}, + pages = {10347--10357}, + title = {Training data-efficient image transformers \& distillation through attention}, + year = 2021 +} +@article{fang2021you, + author = {Fang, Yuxin and Liao, Bencheng and Wang, Xinggang and Fang, Jiemin and Qi, Jiyang and Wu, Rui and Niu, Jianwei and Liu, Wenyu}, + journal = {Neural Information Processing Systems}, + pages = {26183--26197}, + title = {You only look at one sequence: Rethinking transformer in vision through object detection}, + volume = 34, + year = 2021 +} +@article{xie2021segformer, + author = {Xie, Enze and Wang, Wenhai and Yu, Zhiding and Anandkumar, Anima and Alvarez, Jose M and Luo, Ping}, + journal = {Neural Information Processing Systems}, + pages = {12077--12090}, + title = {Seg{F}ormer: Simple and efficient design for semantic segmentation with transformers}, + volume = 34, + year = 2021 +} +@inproceedings{chen2021pre, + author = {Chen, Hanting and Wang, Yunhe and Guo, Tianyu and Xu, Chang and Deng, Yiping and Liu, Zhenhua and Ma, Siwei and Xu, Chunjing and Xu, Chao and Gao, Wen}, + booktitle = {IEEE/CVF Computer Vision \& Pattern Recognition}, + pages = {12299--12310}, + title = {Pre-trained image processing transformer}, + year = 2021 +} +@article{nash2021generating, + author = {Nash, Charlie and Menick, Jacob and Dieleman, Sander and Battaglia, Peter W}, + journal = {International Conference on Machine Learning}, + pages = {7958--7968}, + title = {Generating images with sparse representations}, + year = 2021 +} +@inproceedings{wan2021high, + author = {Wan, Ziyu and Zhang, Jingbo and Chen, Dongdong and Liao, Jing}, + booktitle = {IEEE/CVF International Conference on Computer Vision}, + pages = {4692--4701}, + title = {High-fidelity pluralistic image completion with transformers}, + year = 2021 +} +@article{zheng2021tfill, + author = {Zheng, Chuanxia and Cham, Tat-Jen and Cai, Jianfei}, + journal = {arXiv:2104.00845}, + title = {T{F}ill: Image completion via a transformer-based architecture}, + year = 2021 +} +@article{kumar2021colorization, + author = {Kumar, Manoj and Weissenborn, Dirk and Kalchbrenner, Nal}, + journal = {International Conference on Learning Representations}, + title = {Colorization transformer}, + year = 2021 +} +@article{schuhmann2021laion, + author = {Schuhmann, Christoph and Vencu, Richard and Beaumont, Romain and Kaczmarczyk, Robert and Mullis, Clayton and Katta, Aarush and Coombes, Theo and Jitsev, Jenia and Komatsuzaki, Aran}, + journal = {NeurIPS Workshop on Data-centric AI}, + title = {Laion-400m: Open dataset of clip-filtered 400 million image-text pairs}, + year = 2021 +} +@inproceedings{fan2021multiscale, + author = {Fan, Haoqi and Xiong, Bo and Mangalam, Karttikeya and Li, Yanghao and Yan, Zhicheng and Malik, Jitendra and Feichtenhofer, Christoph}, + booktitle = {IEEE/CVF International Conference on Computer Vision}, + pages = {6824--6835}, + title = {Multiscale vision transformers}, + year = 2021 +} +@inproceedings{wang2021pyramid, + author = {Wang, Wenhai and Xie, Enze and Li, Xiang and Fan, Deng-Ping and Song, Kaitao and Liang, Ding and Lu, Tong and Luo, Ping and Shao, Ling}, + booktitle = {IEEE/CVF International Conference on Computer Vision}, + pages = {568--578}, + title = {Pyramid vision transformer: A versatile backbone for dense prediction without convolutions}, + year = 2021 +} +@article{ali2021xcit, + author = {Ali, Alaaeldin and Touvron, Hugo and Caron, Mathilde and Bojanowski, Piotr and Douze, Matthijs and Joulin, Armand and Laptev, Ivan and Neverova, Natalia and Synnaeve, Gabriel and Verbeek, Jakob and others}, + journal = {Neural Information Processing Systems}, + pages = {20014--20027}, + title = {X{C}i{T}: Cross-covariance image transformers}, + volume = 34, + year = 2021 +} +@article{chu2021twins, + author = {Chu, Xiangxiang and Tian, Zhi and Wang, Yuqing and Zhang, Bo and Ren, Haibing and Wei, Xiaolin and Xia, Huaxia and Shen, Chunhua}, + journal = {Neural Information Processing Systems}, + pages = {9355--9366}, + title = {Twins: Revisiting the design of spatial attention in vision transformers}, + volume = 34, + year = 2021 +} +@inproceedings{arnab2021vivit, + author = {Arnab, Anurag and Dehghani, Mostafa and Heigold, Georg and Sun, Chen and Lu{\v{c}}i{\'c}, Mario and Schmid, Cordelia}, + booktitle = {IEEE/CVF International Conference on Computer Vision}, + pages = {6836--6846}, + title = {Vi{V}it: A video vision transformer}, + year = 2021 +} +@inproceedings{bertasius2021space, + author = {Bertasius, Gedas and Wang, Heng and Torresani, Lorenzo}, + booktitle = {International Conference on Machine Learning}, + number = 3, + pages = {813--824}, + title = {Is space-time attention all you need for video understanding?}, + year = 2021 +} +@inproceedings{neimark2021video, + author = {Neimark, Daniel and Bar, Omri and Zohar, Maya and Asselmann, Dotan}, + booktitle = {IEEE/CVF International Conference on Computer Vision}, + pages = {3163--3172}, + title = {Video transformer network}, + year = 2021 +} +@article{patrick2021keeping, + author = {Patrick, Mandela and Campbell, Dylan and Asano, Yuki and Misra, Ishan and Metze, Florian and Feichtenhofer, Christoph and Vedaldi, Andrea and Henriques, Jo{\~a}o F}, + journal = {Neural Information Processing Systems}, + pages = {12493--12506}, + title = {Keeping your eye on the ball: Trajectory attention in video transformers}, + volume = 34, + year = 2021 +} +@inproceedings{radford2021learning, + author = {Radford, Alec and Kim, Jong Wook and Hallacy, Chris and Ramesh, Aditya and Goh, Gabriel and Agarwal, Sandhini and Sastry, Girish and Askell, Amanda and Mishkin, Pamela and Clark, Jack and others}, + booktitle = {International Conference on Machine Learning}, + pages = {8748--8763}, + title = {Learning transferable visual models from natural language supervision}, + year = 2021 +} +@inproceedings{ramesh2021zero, + author = {Ramesh, Aditya and Pavlov, Mikhail and Goh, Gabriel and Gray, Scott and Voss, Chelsea and Radford, Alec and Chen, Mark and Sutskever, Ilya}, + booktitle = {International Conference on Machine Learning}, + pages = {8821--8831}, + title = {Zero-shot text-to-image generation}, + year = 2021 +} +@misc{sanchez2021gentle, + author = {Sanchez-Lengeling, Benjamin and Reif, Emily and Pearce, Adam and Wiltschko, Alexander B.}, + howpublished = {Distill, \url{https://distill.pub/2021/gnn-intro/}}, + journal = {Distill}, + title = {A Gentle Introduction to Graph Neural Networks}, + year = 2021 +} +@misc{daigavane2021understanding, + author = {Daigavane, Ameya and Balaraman, Ravindran and Aggarwal, Gaurav}, + howpublished = {Distill, \url{https://distill.pub/2021/understanding-gnns/}}, + journal = {Distill}, + title = {Understanding Convolutions on Graphs}, + year = 2021 +} +@book{ma2021deep, + author = {Yao Ma and Jiliang Tang}, + publisher = {Cambridge University Press}, + title = {Deep learning on graphs}, + year = 2021 +} +@article{bronstein2021geometric, + author = {Bronstein, Michael M and Bruna, Joan and Cohen, Taco and Veli{\v{c}}kovi{\'c}, Petar}, + journal = {arXiv:2104.13478}, + title = {Geometric deep learning: Grids, groups, graphs, geodesics, and gauges}, + year = 2021 +} +@article{wolfe2021gist, + author = {Wolfe, Cameron R and Yang, Jingkang and Chowdhury, Arindam and Dun, Chen and Bayer, Artun and Segarra, Santiago and Kyrillidis, Anastasios}, + journal = {NeurIPS Workshop on New Frontiers in Graph Learning}, + title = {{GIST}: Distributed training for large-scale graph convolutional networks}, + year = 2021 +} +@inproceedings{cai2021graphnorm, + author = {Cai, Tianle and Luo, Shengjie and Xu, Keyulu and He, Di and Liu, Tie-yan and Wang, Liwei}, + booktitle = {International Conference on Machine Learning}, + pages = {1204--1215}, + title = {Graph{N}orm: A principled approach to accelerating graph neural network training}, + year = 2021 +} 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Cohen-Or, Daniel and Lischinski, Dani}, + booktitle = {IEEE/CVF International Conference on Computer Vision}, + pages = {2085--2094}, + title = {Style{CLIP}: Text-driven manipulation of {S}tyle{GAN} imagery}, + year = 2021 +} +@inproceedings{wu2021stylespace, + author = {Wu, Zongze and Lischinski, Dani and Shechtman, Eli}, + booktitle = {IEEE/CVF Computer Vision \& Pattern Recognition}, + pages = {12863--12872}, + title = {Stylespace analysis: Disentangled controls for {S}tyle{GAN} image generation}, + year = 2021 +} +@inproceedings{esser2021taming, + author = {Patrick Esser and Robin Rombach and Bj{\"{o}}rn Ommer}, + booktitle = {IEEE/CVF Computer Vision \& Pattern Recognition}, + pages = {12873--12883}, + title = {Taming Transformers for High-Resolution Image Synthesis}, + year = 2021 +} +@inproceedings{richardson2021encoding, + author = {Richardson, Elad and Alaluf, Yuval and Patashnik, Or and Nitzan, Yotam and Azar, Yaniv and Shapiro, Stav and Cohen-Or, Daniel}, + booktitle = 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Abbeel, Pieter and Srinivas, Aravind and Mordatch, Igor}, + journal = {Neural Information Processing Systems}, + pages = {15084--15097}, + title = {Decision transformer: Reinforcement learning via sequence modeling}, + volume = 34, + year = 2021 +} +@article{pateria2021hierarchical, + author = {Pateria, Shubham and Subagdja, Budhitama and Tan, Ah-hwee and Quek, Chai}, + journal = {ACM Computing Surveys}, + number = 5, + pages = {1--35}, + publisher = {ACM New York, NY, USA}, + title = {Hierarchical reinforcement learning: A comprehensive survey}, + volume = 54, + year = 2021 +} +@article{zhang2021multi, + author = {Zhang, Kaiqing and Yang, Zhuoran and Ba{\c{s}}ar, Tamer}, + journal = {Handbook of Reinforcement Learning and Control}, + pages = {321--384}, + publisher = {Springer}, + title = {Multi-agent reinforcement learning: A selective overview of theories and algorithms}, + year = 2021 +} +@inproceedings{chen2021cross, + author = {Chen, Defang and Mei, Jian-Ping and Zhang, Yuan and Wang, Can and Wang, Zhe and Feng, Yan and Chen, Chun}, + booktitle = {AAAI Conference on Artificial Intelligence}, + pages = {7028--7036}, + title = {Cross-layer distillation with semantic calibration}, + year = 2021 +} +@article{goyal2021non, + author = {Goyal, Ankit and Bochkovskiy, Alexey and Deng, Jia and Koltun, Vladlen}, + journal = {arXiv:2110.07641}, + title = {Non-deep networks}, + year = 2021 +} +@article{frankle2020training, + author = {Frankle, Jonathan and Schwab, David J and Morcos, Ari S}, + journal = {International Conference on Learning Representations}, + title = {Training {B}atch{N}orm and only {B}atch{N}orm: On the expressive power of random features in {CNN}s}, + year = 2021 +} +@article{Fazelpour-Danks-2021, + author = {Sina Fazelpour and David Danks}, + journal = {Philosophy Compass}, + title = {Algorithmic Bias: Senses, Sources, Solutions}, + volume = 16, + year = 2021 +} +@article{Mitchell-et-al-2021, + author = {Shira Mitchell and Eric Potash and Solon Barocas 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Taylor and Baker, Leslie and Du, Yu and others}, + journal = {arXiv:2201.08239}, + title = {La{MDA}: Language models for dialog applications}, + year = 2022 +} +@article{chowdhery2022palm, + author = {Chowdhery, Aakanksha and Narang, Sharan and Devlin, Jacob and Bosma, Maarten and Mishra, Gaurav and Roberts, Adam and Barham, Paul and Chung, Hyung Won and Sutton, Charles and Gehrmann, Sebastian and others}, + journal = {arXiv:2204.02311}, + title = {Pa{LM}: Scaling language modeling with pathways}, + year = 2022 +} +@misc{clark2022engineer, + author = {Clark, Mitchell}, + howpublished = {The Verge, July 22, 2022. \url{https://www.theverge.com/2022/7/22/23274958/google-ai-engineer-blake-lemoine-chatbot-lamda-2-sentience}}, + title = {The engineer who claimed a {G}oogle {AI} is sentient has been fired}, + year = 2022 +} +@inproceedings{gu2022multi, + author = {Gu, Jiaqi and Kwon, Hyoukjun and Wang, Dilin and Ye, Wei and Li, Meng and Chen, Yu-Hsin and Lai, Liangzhen and Chandra, Vikas and 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Alex and others}, + journal = {Neural Information Processing Systems}, + pages = {27730--27744}, + title = {Training language models to follow instructions with human feedback}, + volume = 35, + year = 2022 +} +@article{power2022grokking, + author = {Power, Alethea and Burda, Yuri and Edwards, Harri and Babuschkin, Igor and Misra, Vedant}, + journal = {arXiv:2201.02177}, + title = {Grokking: Generalization beyond overfitting on small algorithmic datasets}, + year = 2022 +} +@article{rajpurkar2022ai, + author = {Rajpurkar, Pranav and Chen, Emma and Banerjee, Oishi and Topol, Eric J}, + journal = {Nature Medicine}, + number = 1, + pages = {31--38}, + title = {{AI} in health and medicine}, + volume = 28, + year = 2022 +} +@article{LaCroix-Mohseni-2022, + author = {Travis LaCroix and Aydin Mohseni}, + journal = {Synthese}, + number = 289, + title = {The Tragedy of the {AI} Commons}, + volume = 200, + year = 2022 +} +@article{LaCroix-2022-Linguistic, + author = {Travis LaCroix}, + journal = 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Elizabeth Kumar and Aaron Horowitz and Andrew Selbst}, + booktitle = {ACM Conference on Fairness, Accountability, and Transparency}, + pages = {959--972}, + title = {The Fallacy of {AI} Functionality}, + year = 2022 +} +@article{Micelli-et-al-2022, + author = {Milagros Micelli and Julian Posada and Tianling Yang}, + journal = {Proceedngs of ACM on Human-Computer Interaction}, + title = {Studying Up Machine Learning Data: Why Talk About Bias When We Mean Power?}, + volume = 6, + year = 2022 +} +@article{mehrabi2022survey, + author = {Ninareh Mehrabi and Fred Morstatter and Nripsuta Saxena and Kristina Lerman and Aram Galstyan}, + journal = {ACM Computing Surveys}, + number = 6, + pages = {1--35}, + title = {A Survey on Bias and Fairness in Machine Learning}, + volume = 54, + year = 2022 +} +@misc{prince2022explain, + author = {Prince, Simon Jeremy Damion}, + howpublished = {\url{https://www.borealisai.com/research-blogs/explainability-i-local-post-hoc-explanations/}}, + title = {Explainability {I}: local post-hoc explanations}, + year = 2022 +} +@book{molnar2020interpretable, + author = {Christoph Molnar}, + publisher = {\url{https://christophm.github.io/interpretable-ml-book}}, + title = {Interpretable Machine Learning: A Guide for Making Black Box Models Explainable}, + year = 2022 +} +@article{Smith-Miller-2022, + author = {Marcus Smith and Seumas Miller}, + journal = {AI \& Society}, + pages = {167--175}, + title = {The Ethical Application of Biometric Facial Recognition Technology}, + volume = 37, + year = 2022 +} +@article{Stark-Hutson-2022, + author = {Luke Stark and Jevan Hutson}, + journal = {Fordham Intellectual Property, Media \& Entertainment Law Journal}, + number = 4, + pages = {922--978}, + title = {Physiognomic Artificial Intelligence}, + volume = {XXXII}, + year = 2022 +} +@incollection{Moore-Himma-2022, + author = {Adam Moore and Himma, Ken}, + booktitle = {The {Stanford} Encyclopedia of Philosophy}, + title = {{Intellectual Property}}, + year = 2022 +} +@article{Carlini-et-al-2022, + author = {Nicholas Carlini and Daphne Ippolito and Matthew Jagielski and Katherine Lee and Florian Tramer and and Chiyuan Zhang}, + journal = {arXiv:2202.07646}, + title = {Quantifying memorization across neural language models}, + year = 2022 +} +@inproceedings{Weidinger-et-al-2022, + author = {Laura Weidinger and Jonathan Uesato and Maribeth Rauh and Conor Griffin and Po-Sen Huang and John Mellor and Amelia Glaese and Myra Cheng and Borja Balle and Atoosa Kasirzadeh and Courtney Biles and Sasha Brown and Zac Kenton and Will Hawkins and Tom Stepleton and Abebea Birhane and Lisa Anne Hendricks and Laura Rimell and William Isaac and Julia Haas and Sean Legassick and Geoffrey Irving and Iason Gabriel}, + booktitle = {ACM Conference on Fairness, Accountability, and Transparency}, + pages = {214--229}, + title = {Taxonomy of risks posed by language models}, + year = 2022 +} +@article{Luccioni-et-al-2022, + author = {Alexandra Sasha Luccioni and Sylvain Viguier and Anne-Laure Ligozat}, + journal = {arXiv:2211.02001}, + title = {Estimating the Carbon Footprint of BLOOM, a 176B Parameter Language Model}, + year = 2022 +} +@inproceedings{Birhane-et-al-2022, + author = {Abeba Birhane and Pratyusha Kalluri and Dallas Card and William Agnew and Ravit Dotan and Michelle Bao}, + booktitle = {ACM Conference on Fairness, Accountability, and Transparency}, + pages = {173--184}, + title = {The Values Encoded in Machine Learning Research}, + year = 2022 +} +@article{Johnson-2022, + author = {Gabrielle M. Johnson}, + title = {Are Algorithms Value-Free? Feminist Theoretical Virtues in Machine Learning}, + volume = 198, + year = 2022 +} +@inproceedings{Birhane-et-al-2022-Participatory-AI, + author = {Abeba Birhane and William Isaac and Vinodkumar Prabhakaran and Mark Diaz and Madeleine Clare Elish and Iason Gabriel and Shakir Mohamed}, + booktitle = {Equity and Access in Algorithms, Mechanisms, and Optimization}, + title = {Power to the People? {O}pportunities and Challenges for Participatory {AI}}, + year = 2022 +} +@article{Himmelreich-2022, + author = {Johannes Himmelreich}, + journal = {AI \& Society}, + title = {Against `Democratizing {AI}'}, + year = 2022 +} +@book{murphy2022advanced, + author = {Murphy, Kevin P}, + publisher = {MIT Press}, + title = {Probabilistic machine learning: Advanced topics}, + year = 2023 +} +@book{zhang2021dive, + author = {Zhang, Aston and Lipton, Zachary C. and Li, Mu and Smola, Alexander J.}, + publisher = {Cambridge University Press}, + title = {Dive into deep learning}, + year = 2023 +} +@article{liu2021self, + author = {Liu, Xiao and Zhang, Fanjin and Hou, Zhenyu and Mian, Li and Wang, Zhaoyu and Zhang, Jing and Tang, Jie}, + journal = {IEEE Transactions on Knowledge and Data Engineering}, + number = 1, + pages = {857--876}, + title = {Self-supervised learning: Generative or contrastive}, + volume = 35, + year = 2023 +} +@article{zou2023object, + author = {Zou, Zhengxia and Chen, Keyan and Shi, Zhenwei and Guo, Yuhong and Ye, Jieping}, + journal = {Proceedings of the IEEE}, + title = {Object detection in 20 years: A survey}, + year = 2023 +} +@article{hoffmann2022training, + author = {Hoffmann, Jordan and Borgeaud, Sebastian and Mensch, Arthur and Buchatskaya, Elena and Cai, Trevor and Rutherford, Eliza and Casas, Diego de Las and Hendricks, Lisa Anne and Welbl, Johannes and Clark, Aidan and others}, + journal = {arXiv:2203.15556}, + title = {Training Compute-Optimal Large Language Models}, + year = 2023 +} +@article{tay2020efficient, + author = {Tay, Yi and Dehghani, Mostafa and Bahri, Dara and Metzler, Donald}, + journal = {ACM Computing Surveys}, + number = 6, + pages = {109:1--109:28}, + publisher = {ACM New York, NY}, + title = {Efficient transformers: A survey}, + volume = 55, + year = 2023 +} +@article{liu2021survey, + author = {Liu, Yang and Zhang, Yao and Wang, Yixin and Hou, Feng and Yuan, Jin and Tian, Jiang and Zhang, Yang and Shi, Zhongchao and Fan, Jianping and He, Zhiqiang}, + journal = {IEEE Transactions on Neural Networks and Learning Systems}, + title = {A survey of visual transformers}, + year = 2023 +} +@article{velivckovic2023everything, + author = {Veli{\v{c}}kovi{\'c}, Petar}, + journal = {Current Opinion in Structural Biology}, + pages = 102538, + publisher = {Elsevier}, + title = {Everything is connected: Graph neural networks}, + volume = 79, + year = 2023 +} +@article{wu2020graph, + author = {Wu, Shiwen and Sun, Fei and Zhang, Wentao and Xie, Xu and Cui, Bin}, + journal = {ACM Computing Surveys}, + number = 5, + pages = {97:1--97:37}, + publisher = {ACM New York, NY}, + title = {Graph neural networks in recommender systems: A survey}, + volume = 55, + year = 2023 +} +@article{poole2022dreamfusion, + author = {Poole, Ben and Jain, Ajay and Barron, Jonathan T and Mildenhall, Ben}, + journal = {International Conference on Learning Representations}, + title = {Dream{F}usion: Text-to-3{D} using 2{D} diffusion}, + year = 2023 +} +@article{zhang2023adding, + author = {Zhang, Lvmin and Agrawala, Maneesh}, + journal = {arXiv:2302.05543}, + title = {Adding conditional control to text-to-image diffusion models}, + year = 2023 +} +@article{moerland2020model, + author = {Moerland, Thomas M and Broekens, Joost and Plaat, Aske and Jonker, Catholijn M and others}, + journal = {Foundations and Trends in Machine Learning}, + number = 1, + pages = {1--118}, + title = {Model-based reinforcement learning: A survey}, + volume = 16, + year = 2023 +} +@article{liu2022omnigrok, + author = {Liu, Ziming and Michaud, Eric J and Tegmark, Max}, + journal = {International Conference on Learning Representations}, + title = {Omnigrok: Grokking Beyond Algorithmic Data}, + year = 2023 +} +@article{rolnick2023tackling, + author = {David Rolnick and Priya L. Donti and Lynn H. Kaack and Kelly Kochanski and Alexandre Lacoste and Kris Sankaran and Andrew Slavin Ross and Nikola Milojevic{-}Dupont and Natasha Jaques and Anna Waldman{-}Brown and Alexandra Sasha Luccioni and Tegan Maharaj and Evan D. Sherwin and S. Karthik Mukkavilli and Konrad P. Kording and Carla P. Gomes and Andrew Y. Ng and Demis Hassabis and John C. Platt and Felix Creutzig and Jennifer T. Chayes and Yoshua Bengio}, + journal = {ACM Computing Surveys}, + number = 2, + pages = {1--42}, + title = {Tackling Climate Change with Machine Learning}, + volume = 55, + year = 2023 +} +@book{LaCroix2023, + author = {Travis LaCroix}, + note = {\url{https://value-alignment.github.io}}, + title = {Artificial Intelligence and the Value-Alignment Problem: A Philosophical Introduction}, + year = 2023 +} +@book{Barocas-et-al-2019, + author = {Solon Barocas and Moritz Hardt and Arvind Narayanan}, + publisher = {MIT Press}, + title = {Fairness and Machine Learning: Limitations and Opportunities}, + year = 2023 +} +@misc{Goodin-2023, + author = {Dan Goodin}, + howpublished = {ars Technica, June 1, 2023. \url{https://arstechnica.com/information-technology/2023/01/chatgpt-is-enabling-script-kiddies-to-write-functional-malware/}}, + title = {Chat{GPT} is enabling script kiddies to write functional malware}, + year = 2023 +} +@article{Abrahams-2023, + author = {Doriel Abrahams}, + journal = {Forter Blog, March 27, 2023}, + note = {\url{https://www.forter.com/blog/lets-talk-about-generative-ai-and-fraud/}}, + title = {Let's Talk About Generative {AI} and Fraud}, + year = 2023 +} +@article{Veliz-2023, + author = {Carissa V{\'e}liz}, + journal = {Nature}, + pages = 375, + title = {Chatbots shouldn't use emojis}, + volume = 615, + year = 2023 +} +@article{Henderson-et-al-2023, + author = {Peter Henderson and Xuechen Li and Dan Jurafsky and Tatsunori Hashimoto and Mark A. Lemley and Percy Liang}, + journal = {arXiv:2303.15715}, + title = {Foundation Models and Fair Use}, + year = 2023 +} +@article{Carlini-et-al-2023, + author = {Nicholas Carlini and Jamie Hayes and Milad Nasr and Matthew Jagielski and Vikash Sehwag and Florian Tram{\`e}r and Borja Balle and Daphne Ippolito and Eric Wallace}, + journal = {arXiv:2301.13188}, + title = {Extracting training data from diffusion models}, + year = 2023 +} +@misc{Luccioni-2023, + author = {Alexandra Sasha Luccioni}, + howpublished = {ars Technica, April 12, 2023.\url{https://arstechnica.com/gadgets/2023/04/generative-ai-is-cool-but-lets-not-forget-its-human-and-environmental-costs}}, + title = {The mounting human and environmental costs of generative {AI}}, + year = 2023 +} +@article{Ceylan-et-al-2023, + author = {Gizem Ceylan and Ian A. Anderson and Wendy Wood}, + journal = {Proceedings of the National Academy of Sciences of the United States of America}, + number = 4, + title = {Sharing of misinformation is habitual, not just lazy or biased}, + volume = 120, + year = 2023 +} +@book{Broussard-2023, + author = {Meredith Broussard}, + publisher = {The MIT Press}, + title = {More than a Glitch: Confronting Race, Gender, and Ability Bias in Tech}, + year = 2023 +} +@misc{Gebru-et-al-2023, + author = {Timnit Gebru and Emily M. Bender and Angelina McMillan-Major and Margaret Mitchell}, + note = {\url{https://www.dair-institute.org/blog/letter-statement-March2023}}, + title = {Statement from the listed authors of Stochastic Parrots on the ``{AI} pause'' letter}, + year = 2023 +} +@book{torralba2023computervision, + author = {Torralba, Antonio and Freeman, William and Isola, Phillip}, + publisher = {MIT Press}, + title = {Foundations of Computer Vision}, + year = 2024 +} From 2aaaef0838fe3f74016307a34f11c6315e5e4d72 Mon Sep 17 00:00:00 2001 From: udlbook <110402648+udlbook@users.noreply.github.com> Date: Thu, 23 Jan 2025 15:47:55 -0500 Subject: [PATCH 48/49] Delete UDL_Equations.pdf --- UDL_Equations.pdf | Bin 353977 -> 0 bytes 1 file changed, 0 insertions(+), 0 deletions(-) delete mode 100644 UDL_Equations.pdf diff --git a/UDL_Equations.pdf b/UDL_Equations.pdf deleted file mode 100644 index 31e7f1d2b26033c32102693bd805ac6e6ef321f9..0000000000000000000000000000000000000000 GIT binary patch literal 0 HcmV?d00001 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zgv%mqB~eIOu%bNJ7J{;Ypd?XJP-#h9JE)|TtQ{CED}#`dfEJb8Spq`Nj$PwjLx>Ue?kz;7&0 zL^wldk$UJPW36WM%l`cSZTCVv!_b3k=?bmQxo&Y2Y}`Xlv_o`FvBA^Maf&mmpDfAY zX$I3n=Y<|0xA}S{TGPVvQ)({A6&Y>u`4?{HCA3Z97%Iz~HzrQagRy61`!K!z{|zVL he~$lX;NM9Dq;U}f?dgj~+0jTsq-g{N)pTyt{0+ok;bZ^+ From f8acbaab82e4c7de22c0a8f2563cd469d7b809d3 Mon Sep 17 00:00:00 2001 From: udlbook <110402648+udlbook@users.noreply.github.com> Date: Thu, 23 Jan 2025 15:49:08 -0500 Subject: [PATCH 49/49] Add files via upload --- UDL_Equations.tex | 2229 +++++++++++++++++++++++++++++++++++++++++++++ 1 file changed, 2229 insertions(+) create mode 100644 UDL_Equations.tex diff --git a/UDL_Equations.tex b/UDL_Equations.tex new file mode 100644 index 0000000..e6e98bf --- /dev/null +++ b/UDL_Equations.tex @@ -0,0 +1,2229 @@ +\documentclass[letterpaper,twoside,openany, titlepage,oldfontcommands,titles,dvipsnames]{memoir} + +\usepackage{amsmath} +\usepackage{amsfonts} +\usepackage{color} +\newcommand*\mystrut[1]{\vrule width0pt height0pt depth#1\relax} + +\setstocksize{11in}{8.5in} +\settrimmedsize{9in}{8in}{*} +\setlength{\trimtop}{1in} +\setlength{\trimedge}{0.25in} +\setlength{\textheight}{7.5in} +\setlength{\textwidth}{5.3125in} +\setlrmargins{*}{*}{1.5} +\setulmargins{0.905in}{*}{*} +\setlength{\marginparsep}{0mm} +\setlength{\marginparwidth}{1mm} + + +\title{Understanding Deep Learning Equations} + + + + +\begin{document} +\maketitle + +\chapter{Introduction} + +\chapter{Supervised Learning} + + +\begin{eqnarray} + \mathbf{y} = \mbox{\bf f}[\mathbf{x}]. + \end{eqnarray} + +\begin{eqnarray} + \mathbf{y} = \mbox{\bf f}[\mathbf{x}, \boldsymbol\phi]. + \end{eqnarray} + +\begin{eqnarray} + \hat{\boldsymbol\phi} = \mathop{\rm argmin}_{\boldsymbol\phi}\Bigl[L\left[\boldsymbol\phi\right] \Bigr]. + \end{eqnarray} + +\begin{eqnarray}\label{eq:sl_linear_regression} + y &=& \mbox{f}[x,\boldsymbol\phi]\nonumber \\ + &=&\phi_{0}+\phi_{1}x. + \end{eqnarray} + +\begin{eqnarray}\ + L[\boldsymbol\phi] &=& \sum_{i=1}^{I} \left(\mbox{f}[x_{i}, \boldsymbol\phi]-y_{i}\right)^{2}\nonumber \\ + &=& \sum_{i=1}^{I} \left(\phi_{0}+\phi_{1}x_i-y_{i}\right)^{2}.\label{eq:sl_loss_function} + \end{eqnarray} + +\begin{eqnarray} + \hat{\boldsymbol\phi} &=& \mathop{\rm argmin}_{\boldsymbol\phi}\Bigl[L[\boldsymbol\phi]\Bigr]\nonumber \\ + &=& \mathop{\rm argmin}_{\boldsymbol\phi}\left[\sum_{i=1}^{I} \left(\mbox{f}[x_{i}, \boldsymbol\phi]-y_{i}\right)^{2}\right]\nonumber \\ + &=& \mathop{\rm argmin}_{\boldsymbol\phi}\left[\sum_{i=1}^{I} \left(\phi_{0}+\phi_{1}x_i-y_{i}\right)^{2}\right]. + \end{eqnarray} + +\chapter{Shallow neural networks} + +\begin{eqnarray}\label{eq:snn_simple_eq} + y &=& \mbox{f}[x,\boldsymbol\phi]\nonumber \\ + &=&\phi_{0}+\phi_{1}\mbox{a}[\theta_{10} + \theta_{11}x]+\phi_{2}\mbox{a}[\theta_{20} + \theta_{21}x]+\phi_{3}\mbox{a}[\theta_{30} + \theta_{31}x]. + \end{eqnarray} + +\begin{eqnarray}\label{eq:snn_relu} + \mbox{a}[z] = \mbox{ReLU}[z] = \begin{cases} 0 & \quad z <0 \\ z & \quad z\geq 0\end{cases}. + \end{eqnarray} + +\begin{eqnarray}\label{eq:snn_simple_eq1b} + h_{1} &=& \mbox{a}[\theta_{10} + \theta_{11}x] \nonumber \\ + h_{2} &=& \mbox{a}[\theta_{20} + \theta_{21}x] \nonumber \\ + h_{3} &=& \mbox{a}[\theta_{30} + \theta_{31}x], + \end{eqnarray} + +\begin{eqnarray}\label{eq:snn_simple_eq2} + y = \phi_{0}+\phi_{1}h_{1}+\phi_{2}h_{2}+\phi_{3}h_{3}. + \end{eqnarray} + +\begin{eqnarray} + h_{d} = \mbox{a}[\theta_{d0} + \theta_{d1}x], + \end{eqnarray} + +\begin{eqnarray}\label{eq:snn_many_hidden} + y = \phi_{0}+\sum_{d=1}^{D}\phi_{d}h_{d}. + \end{eqnarray} + +\begin{eqnarray}\label{eq:snn_multiple_out2} + h_{1} &=& \mbox{a}[\theta_{10} + \theta_{11}x] \nonumber \\ + h_{2} &=& \mbox{a}[\theta_{20} + \theta_{21}x] \nonumber \\ + h_{3} &=& \mbox{a}[\theta_{30} + \theta_{31}x] \nonumber \\ + h_{4} &=& \mbox{a}[\theta_{40} + \theta_{41}x], + \end{eqnarray} + +\begin{eqnarray}\label{eq:snn_multiple_out1} + y_1 &=& \phi_{10}+\phi_{11}h_{1}+\phi_{12}h_{2}+\phi_{13}h_{3}+\phi_{14}h_{4}\nonumber \\ + y_2 &=& \phi_{20}+\phi_{21}h_{1}+\phi_{22}h_{2}+\phi_{23}h_{3}+\phi_{24}h_{4}. + \end{eqnarray} + +\begin{eqnarray}\label{eq:snn_multiple3} + h_{1} &=& \mbox{a}[\theta_{10} + \theta_{11}x_1+ \theta_{12}x_2] \nonumber \\ + h_{2} &=& \mbox{a}[\theta_{20} + \theta_{21}x_1+\theta_{22}x_2] \nonumber \\ + h_{3} &=& \mbox{a}[\theta_{30} + \theta_{31}x_1+\theta_{32}x_2], + \end{eqnarray} + +\begin{eqnarray}\label{eq:snn_multiple4} + y = \phi_{0}+\phi_{1}h_{1}+\phi_{2}h_{2}+\phi_{3}h_{3}. + \end{eqnarray} + +\begin{eqnarray}\label{eq:snn_general_1} + h_{d} = \mbox{a}\left[\theta_{d0} + \sum_{i=1}^{D_{i}}\theta_{di}x_i\right], + \end{eqnarray} + +\begin{eqnarray}\label{eq:snn_general_2} + y_j = \phi_{j0}+\sum_{d=1}^{D}\phi_{jd}h_{d}, + \end{eqnarray} + +\begin{eqnarray}\label{eq:snn_harswish} + \mbox{HardSwish}[z] = \begin{cases} 0 & \quad z <-3 \\ z(z+3)/6 & \quad -3\leq z\leq 3 \\ z &\quad z>3 \end{cases}. + \end{eqnarray} + +\begin{eqnarray} + \mbox{ReLU}[\alpha \cdot z] = \alpha \cdot \mbox{ReLU}[z]. + \end{eqnarray} + +\begin{eqnarray} + \mbox{heaviside}[z] = \begin{cases} 0 & \quad z <0 \\ 1 & \quad z\geq 0\end{cases} \hspace{2cm} \mbox{rect}[z] = \begin{cases} 0 & \quad z < 0 \\ 1 & \quad 0 \leq z\leq 1 \\ 0 & \quad z > 1\end{cases}. + \end{eqnarray} + + \chapter{Deep neural networks} + +\begin{eqnarray}\label{eq:dnn_comp_in} + h_{1} &=& \mbox{a}[\theta_{10} + \theta_{11}x] \nonumber \\ + h_{2} &=& \mbox{a}[\theta_{20} + \theta_{21}x] \nonumber \\ + h_{3} &=& \mbox{a}[\theta_{30} + \theta_{31}x], + \end{eqnarray} + +\begin{eqnarray}\label{eq:dnn_comp} + y = \phi_{0}+\phi_{1}h_{1}+\phi_{2}h_{2}+\phi_{3}h_{3}. + \end{eqnarray} + +\begin{eqnarray}\label{eq:dnn_comp_2} + h'_{1} &=& \mbox{a}[\theta'_{10} + \theta'_{11}y] \nonumber \\ + h'_{2} &=& \mbox{a}[\theta'_{20} + \theta'_{21}y] \nonumber \\ + h'_{3} &=& \mbox{a}[\theta'_{30} + \theta'_{31}y], + \end{eqnarray} + +\begin{eqnarray} \label{eq:dnn_comp_out} + y' = \phi'_{0}+\phi'_{1}h'_{1}+\phi'_{2}h'_{2}+\phi'_{3}h'_{3}. + \end{eqnarray} + +\begin{eqnarray}\label{eq:dnn_deep_linear} + h'_{1} &=\quad \mbox{a}[\theta'_{10} + \theta'_{11}y] &=\quad \mbox{a}[\theta'_{10} + \theta'_{11}\phi_{0}+\theta'_{11}\phi_{1}h_{1}+\theta'_{11}\phi_{2}h_{2}+\theta'_{11}\phi_{3}h_{3}] \nonumber \\ + h'_{2} &= \quad\mbox{a}[\theta'_{20} + \theta'_{21}y] &=\quad \mbox{a}[\theta'_{20} + \theta'_{21}\phi_{0}+\theta'_{21}\phi_{1}h_{1}+\theta'_{21}\phi_{2}h_{2}+\theta'_{21}\phi_{3}h_{3}] \nonumber \\ + h'_{3} &=\quad \mbox{a}[\theta'_{30} + \theta'_{31}y] &=\quad \mbox{a}[\theta'_{30} + \theta'_{31}\phi_{0}+\theta'_{31}\phi_{1}h_{1}+\theta'_{31}\phi_{2}h_{2}+\theta'_{31}\phi_{3}h_{3}], + \end{eqnarray} + +\begin{eqnarray}\label{eq:dnn_three_layer_middle} + h'_{1} &=& \mbox{a}[\psi_{10} + \psi_{11}h_{1}+ \psi_{12}h_{2}+ \psi_{13}h_{3}] \nonumber \\ + h'_{2} &=& \mbox{a}[\psi_{20} + \psi_{21}h_{1}+ \psi_{22}h_{2}+ \psi_{23}h_{3}] \nonumber \\ + h'_{3} &=& \mbox{a}[\psi_{30} + \psi_{31}h_{1}+ \psi_{32}h_{2}+ \psi_{33}h_{3}], + \end{eqnarray} + +\begin{eqnarray}\label{eq:dnn_three_layer_in} + h_{1} &=& \mbox{a}[\theta_{10} + \theta_{11}x] \nonumber \\ + h_{2} &=& \mbox{a}[\theta_{20} + \theta_{21}x] \nonumber \\ + h_{3} &=& \mbox{a}[\theta_{30} + \theta_{31}x], + \end{eqnarray} + +\begin{eqnarray}\label{eq:dnn_three_layer_middle2} + h'_{1} &=& \mbox{a}[\psi_{10} + \psi_{11}h_{1}+ \psi_{12}h_{2}+ \psi_{13}h_{3}] \nonumber \\ + h'_{2} &=& \mbox{a}[\psi_{20} + \psi_{21}h_{1}+ \psi_{22}h_{2}+ \psi_{23}h_{3}] \nonumber \\ + h'_{3} &=& \mbox{a}[\psi_{30} + \psi_{31}h_{1}+ \psi_{32}h_{2}+ \psi_{33}h_{3}], + \end{eqnarray} + +\begin{eqnarray}\label{eq:dnn_three_layer_out} + y' = \phi'_{0}+\phi'_{1}h'_{1}+\phi'_{2}h'_{2}+\phi'_{3}h'_{3}. + \end{eqnarray} + +\begin{eqnarray}\label{eq:dnn_expanded} + y' &=& \phi'_{0}+\phi'_{1}\mbox{a}\left[\psi_{10} + \psi_{11}\mbox{a}[\theta_{10} + \theta_{11}x] + \psi_{12}\mbox{a}[\theta_{20} + \theta_{21}x]+ \psi_{13}\mbox{a}[\theta_{30} + \theta_{31}x]\right]\nonumber \\ + &&\hspace{0.55cm}+\phi'_{2}\mbox{a}[\psi_{20} + \psi_{21}\mbox{a}[\theta_{10} + \theta_{11}x] + \psi_{22}\mbox{a}[\theta_{20} + \theta_{21}x]+ \psi_{23}\mbox{a}[\theta_{30} + \theta_{31}x]] \nonumber\\ + &&\hspace{0.55cm}+\phi'_{3}\mbox{a}[\psi_{30} + \psi_{31}\mbox{a}[\theta_{10} + \theta_{11}x] + \psi_{32}\mbox{a}[\theta_{20} + \theta_{21}x]+ \psi_{33}\mbox{a}[\theta_{30} + \theta_{31}x]],\nonumber \\ + \end{eqnarray} + +\begin{eqnarray} + \begin{bmatrix} + h_{1} \\ h_{2} \\ h_{3} + \end{bmatrix} + = \mbox{\bf a}\left[\begin{bmatrix}\theta_{10}\\ \theta_{20}\\ \theta_{30} \end{bmatrix}+\begin{bmatrix}\theta_{11}\\\theta_{21}\\\theta_{31}\end{bmatrix}x\right], + \end{eqnarray} + +\begin{eqnarray} + \begin{bmatrix} + h'_{1} \\ h'_{2} \\h'_{3} + \end{bmatrix} + =\mbox{\bf a}\left[\begin{bmatrix}\psi_{10} \\ \psi_{20}\\ \psi_{30}\end{bmatrix} + \begin{bmatrix}\psi_{11} &\psi_{12} & \psi_{13} \\\psi_{21} &\psi_{22} & \psi_{23} \\\psi_{31} &\psi_{32} & \psi_{33} \end{bmatrix} \begin{bmatrix} + h_{1} \\ h_{2} \\ h_{3} + \end{bmatrix} \right], + \end{eqnarray} + +\begin{eqnarray} + y' = \phi'_{0} + \begin{bmatrix} \phi'_{1} & \phi'_{2} & \phi'_{3} \end{bmatrix}\begin{bmatrix}h'_{1} \\ h'_{2} \\h'_{3} \end{bmatrix}, + \end{eqnarray} + +\begin{eqnarray} + \mathbf{h} &=& \mbox{\bf a}\left[\boldsymbol\theta_{0}+\boldsymbol\theta x\right] \nonumber\\ + \mathbf{h}' &=& \mbox{\bf a}\left[\boldsymbol\psi_{0}+\boldsymbol\Psi \mathbf{h}\right] \nonumber \\ + y' &=& \phi'_{0} + \boldsymbol\phi' \mathbf{h}', + \end{eqnarray} + +\begin{eqnarray}\label{eq:dnn_la1} + \mathbf{h}_{1} &=& \mathbf{a}[\boldsymbol\beta_{0} +\boldsymbol\Omega_{0}\mathbf{x}]\nonumber \\ + \mathbf{h}_{2} &=& \mathbf{a}[\boldsymbol\beta_{1} +\boldsymbol\Omega_{1}\mathbf{h}_{1}]\nonumber \\ + \mathbf{h}_{3} &=& \mathbf{a}[\boldsymbol\beta_{2} +\boldsymbol\Omega_{2}\mathbf{h}_{2}]\nonumber \\ + &\vdots&\nonumber\\ + \mathbf{h}_{K} &=& \mathbf{a}[\boldsymbol\beta_{K-1} +\boldsymbol\Omega_{K-1}\mathbf{h}_{K-1}] \nonumber\\ + \mathbf{y} &=& \boldsymbol\beta_{K} +\boldsymbol\Omega_{K}\mathbf{h}_{K}. + \end{eqnarray} + +\begin{eqnarray}\label{eq:dnn_la2} + \mathbf{y}\!\! &\!\!=\!\!& \!\!\boldsymbol\beta_{K} +\boldsymbol\Omega_{K}\mathbf{a}\left[\boldsymbol\beta_{K-1} +\boldsymbol\Omega_{K-1}\mathbf{a}\left[\ldots + \boldsymbol\beta_{2} +\boldsymbol\Omega_{2}\mathbf{a}\left[\boldsymbol\beta_{1} +\boldsymbol\Omega_{1}\mathbf{a}\left[\boldsymbol\beta_{0} +\boldsymbol\Omega_{0}\mathbf{x}\right]\right]\ldots\right]\right].\nonumber \\ + \end{eqnarray} + +\begin{eqnarray}\label{eq:dnn_deep_param_calc} + N_{r} = \left(\frac{D}{D_{i}}+1\right)^{D_{i}(K-1)}\cdot\sum_{j=0}^{D_{i}}\binom{D}{j}. + \end{eqnarray} + + +\begin{eqnarray} + \mbox{ReLU}\Bigl[\boldsymbol\beta_{1}\!+\!\lambda_1\!\cdot\!\boldsymbol\Omega_{1}\mbox{ReLU}\left[\boldsymbol\beta_{0}\!+\!\lambda_{0}\cdot\boldsymbol\Omega_{0}\mathbf{x}\right] \Bigr]\!=\! \lambda_0\lambda_{1}\cdot \mbox{ReLU}\left[\frac{1}{\lambda_0\lambda_1}\boldsymbol\beta_{1}\!+\!\boldsymbol\Omega_{1}\mbox{ReLU}\left[\frac{1}{\lambda_0}\boldsymbol\beta_{0}\!+\!\boldsymbol\Omega_{0}\mathbf{x}\right]\right], + \end{eqnarray} + +\chapter{Loss functions} + + +\begin{eqnarray}\label{eq:loss_max_like1} + \hat{\boldsymbol\phi} &=& \mathop{\rm argmax}_{\boldsymbol\phi}\left[\prod_{i=1}^{I} Pr(\mathbf{y}_{i}|\mathbf{x}_{i})\right]\nonumber\\ + &=& \mathop{\rm argmax}_{\boldsymbol\phi}\left[\prod_{i=1}^{I} Pr(\mathbf{y}_{i}|\boldsymbol\theta_{i})\right] \nonumber \\ &=& \mathop{\rm argmax}_{\boldsymbol\phi}\left[\prod_{i=1}^{I} Pr(\mathbf{y}_{i}|\mbox{\bf f}[\mathbf{x}_{i},\boldsymbol\phi])\right]. + \end{eqnarray} + +\begin{eqnarray} + Pr(\mathbf{y}_{1},\mathbf{y}_{2},\ldots, \mathbf{y}_{I}|\mathbf{x}_{1},\mathbf{x}_{2},\ldots,\mathbf{x}_{I}) = \prod_{i=1}^{I} Pr(\mathbf{y}_{i}|\mathbf{x}_{i}). + \end{eqnarray} + +\begin{eqnarray}\label{eq:loss_max_like2} + \hat{\boldsymbol\phi} &=& \mathop{\rm argmax}_{\boldsymbol\phi}\left[\prod_{i=1}^{I} Pr(\mathbf{y}_{i}|\mbox{\bf f}[\mathbf{x}_{i},\boldsymbol\phi])\right] \nonumber \\ + &=& \mathop{\rm argmax}_{\boldsymbol\phi}\left[\log\left[\prod_{i=1}^{I} Pr(\mathbf{y}_{i}|\mbox{\bf f}[\mathbf{x}_{i},\boldsymbol\phi])\right]\right]\nonumber \\ + &=& \mathop{\rm argmax}_{\boldsymbol\phi}\left[\sum_{i=1}^{I} \log\Bigl[Pr(\mathbf{y}_{i}|\mbox{\bf f}[\mathbf{x}_{i},\boldsymbol\phi])\Bigr]\right]. + \end{eqnarray} + +\begin{eqnarray} + \hat{\boldsymbol\phi} + &=& \mathop{\rm argmin}_{\boldsymbol\phi}\left[-\sum_{i=1}^{I} \log\Bigl[Pr(\mathbf{y}_{i}|\mbox{\bf f}[\mathbf{x}_{i},\boldsymbol\phi])\Bigr]\right]\nonumber \\ + &=& \mathop{\rm argmin}_{\boldsymbol\phi}\Bigl[L[\boldsymbol\phi]\Bigr], + \end{eqnarray} + +\begin{eqnarray} + \hat{\mathbf{y}} = \mathop{\rm argmax}_{\mathbf{y}}\Bigl[Pr(\mathbf{y}|\mbox{\bf f}[\mathbf{x},\hat{\boldsymbol\phi}])\Bigr]. + \end{eqnarray} + +\begin{eqnarray} + \hat{\boldsymbol\phi} = \mathop{\rm argmin}_{\boldsymbol\phi}\Bigl[L[\boldsymbol\phi]\Bigr] = \mathop{\rm argmin}_{\boldsymbol\phi}\left[-\sum_{i=1}^{I} \log\Bigl[Pr(\mathbf{y}_{i}|\mbox{\bf f}[\mathbf{x}_{i},\boldsymbol\phi])\Bigr]\right]. + \end{eqnarray} + +\begin{eqnarray} + Pr(y|\mu,\sigma^2) = \frac{1}{\sqrt{2\pi\sigma^{2}}}\exp\left[-\frac{(y-\mu)^{2}}{2\sigma^{2}}\right]. + \end{eqnarray} + +\begin{eqnarray}\label{eq:loss_pdf_uni_reg} + Pr(y|\mbox{f}[\mathbf{x},\boldsymbol\phi],\sigma^2) = \frac{1}{\sqrt{2\pi\sigma^{2}}}\exp\left[-\frac{(y-\mbox{f}[\mathbf{x},\boldsymbol\phi])^{2}}{2\sigma^{2}}\right]. + \end{eqnarray} + +\begin{eqnarray}\label{eq:loss_normal_full} + L[\boldsymbol\phi] &=& -\sum_{i=1}^{I} \log\left[Pr(y_{i}|\mbox{f}[\mathbf{x}_{i},\boldsymbol\phi],\sigma^{2})\right]\nonumber \\ + &=&-\sum_{i=1}^{I} \log\left[\frac{1}{\sqrt{2\pi\sigma^{2}}}\exp\left[-\frac{(y_i-\mbox{f}[\mathbf{x}_i,\boldsymbol\phi])^{2}}{2\sigma^{2}}\right]\right]. + \end{eqnarray} + +\begin{eqnarray}\label{eq:loss_normal_full2} + \hat{\boldsymbol\phi} &=& \mathop{\rm argmin}_{\boldsymbol\phi}\left[-\sum_{i=1}^{I} \log\left[\frac{1}{\sqrt{2\pi\sigma^{2}}}\exp\left[-\frac{(y_i-\mbox{f}[\mathbf{x}_i,\boldsymbol\phi])^{2}}{2\sigma^{2}}\right]\right]\right] + \nonumber \\ + &=&\mathop{\rm argmin}_{\boldsymbol\phi}\left[-\sum_{i=1}^{I} \left(\log\left[\frac{1}{\sqrt{2\pi\sigma^{2}}}\right] -\frac{(y_i-\mbox{f}[\mathbf{x}_i,\boldsymbol\phi])^{2}}{2\sigma^{2}}\right)\right]\nonumber \\ + &=& \mathop{\rm argmin}_{\boldsymbol\phi}\left[-\sum_{i=1}^{I} -\frac{(y_i-\mbox{f}[\mathbf{x}_i,\boldsymbol\phi])^{2}}{2\sigma^{2}}\right]\nonumber \\ + &=& \mathop{\rm argmin}_{\boldsymbol\phi}\left[\sum_{i=1}^{I} (y_i-\mbox{f}[\mathbf{x}_i,\boldsymbol\phi])^{2}\right], + \end{eqnarray} + +\begin{eqnarray}\label{eq:loss_least_squares} + L[\boldsymbol\phi] = \sum_{i=1}^{I} \bigl(y_i-\mbox{f}[\mathbf{x}_i,\boldsymbol\phi]\bigr)^{2}. + \end{eqnarray} + +\begin{eqnarray} + \hat{y} = \mathop{\rm argmax}_{y}\left[Pr(y|\mbox{f}[\mathbf{x},\hat{\boldsymbol\phi}],\sigma^2)\right]. + \end{eqnarray} + +\begin{eqnarray} + \hat{\boldsymbol\phi},\hat{\sigma}^{2} = \mathop{\rm argmin}_{\boldsymbol\phi,\sigma^{2}}\left[-\sum_{i=1}^{I} \log\left[\frac{1}{\sqrt{2\pi\sigma^{2}}}\exp\left[-\frac{(y_i-\mbox{f}[\mathbf{x}_i,\boldsymbol\phi])^{2}}{2\sigma^{2}}\right]\right]\right]. + \end{eqnarray} + +\begin{eqnarray} + \mu &=& \mbox{f}_1[\mathbf{x},\boldsymbol\phi] \nonumber \\ + \sigma^2 &=& \mbox{f}_2[\mathbf{x},\boldsymbol\phi]^2, + \end{eqnarray} + +\begin{eqnarray} + \hat{\boldsymbol\phi} = \mathop{\rm argmin}_{\boldsymbol\phi}\left[-\sum_{i=1}^{I} \biggl(\log\left[\frac{1}{\sqrt{2\pi\mbox{f}_2[\mathbf{x}_i,\boldsymbol\phi]^2}}\right] -\frac{(y_i-\mbox{f}_1[\mathbf{x}_i,\boldsymbol\phi])^{2}}{2\mbox{f}_2[\mathbf{x}_i,\boldsymbol\phi]^2}\biggr)\right]. + \end{eqnarray} + +\begin{eqnarray} + Pr(y|\lambda) = \begin{cases} 1-\lambda & \quad y =0 \\ \lambda & \quad y= 1\end{cases}, + \end{eqnarray} + +\begin{eqnarray} + Pr(y|\lambda) = (1-\lambda)^{1-y}\cdot \lambda^{y}. + \end{eqnarray} + +\begin{eqnarray}\label{eq:logistic_sigmoid} + \mbox{sig}[z] = \frac{1}{1+\exp[-z]}. + \end{eqnarray} + +\begin{eqnarray}\label{eq:loss_binary} + Pr(y|\mathbf{x}) = (1-\mbox{sig}[\mbox{f}[\mathbf{x},\boldsymbol\phi]])^{1-y}\cdot \mbox{sig}[\mbox{f}[\mathbf{x},\boldsymbol\phi]]^y. + \end{eqnarray} + +\begin{eqnarray}\label{eq:loss_binary_cross_entropy} + L[\boldsymbol\phi] = \sum_{i=1}^{I}-(1-y_{i})\log\Bigl[1-\mbox{sig}[\mbox{f}[\mathbf{x}_i,\boldsymbol\phi]]\Bigr] - y_{i}\log\Bigl[\mbox{sig}[\mbox{f}[\mathbf{x}_i,\boldsymbol\phi]]\Bigr]. + \end{eqnarray} + +\begin{eqnarray} + Pr(y=k) = \lambda_{k}. + \end{eqnarray} + +\begin{eqnarray} + \mbox{softmax}_{k}[\mathbf{z}] = \frac{\exp[z_{k}]}{\sum_{k'=1}^{K}\exp[z_{k'}]}, + \end{eqnarray} + +\begin{eqnarray} + Pr(y=k|\mathbf{x}) = \mbox{softmax}_{k}\Bigr[\mbox{\bf f}[\mathbf{x},\boldsymbol\phi]\Bigl]. + \end{eqnarray} + +\begin{eqnarray}\label{eq:loss_multiclass_class} + L[\boldsymbol\phi] &=& -\sum_{i=1}^{I}\log\left[\mbox{softmax}_{y_{i}}\Bigl[\mbox{\bf f}\left[\mathbf{x}_i,\boldsymbol\phi\right]\Bigr]\right]\nonumber\\ + &=& -\sum_{i=1}^{I}\left(\mbox{f}_{y_{i}}\left[\mathbf{x}_i,\boldsymbol\phi\right]-\log\left[\sum_{k'=1}^{K}\exp\left[\mbox{ f}_{k'}\left[\mathbf{x}_i,\boldsymbol\phi\right]\right]\right]\right), + \end{eqnarray} + +\begin{eqnarray}\label{eq:loss_multiple} + Pr(\mathbf{y}|\mbox{\bf f}[\mathbf{x},\boldsymbol\phi])= \prod_{d}Pr(y_{d}|\mbox{\bf f}_d[\mathbf{x},\boldsymbol\phi]), + \end{eqnarray} + +\begin{eqnarray} + L[\boldsymbol\phi] = -\sum_{i=1}^{I}\log\Bigl[Pr(\mathbf{y}_i|\mbox{\bf f}[\mathbf{x}_{i},\boldsymbol\phi])\Bigr]= -\sum_{i=1}^{I}\sum_{d}\log\Bigl[Pr(y_{id}|\mbox{\bf f}_d[\mathbf{x}_{i},\boldsymbol\phi])\Bigr]. + \end{eqnarray} + +\begin{eqnarray} + D_{KL}\bigl[q||p\bigr] = \int_{-\infty}^{\infty}q(z) \log\bigl[q(z)\bigr]dz - \int_{-\infty}^{\infty}q(z) \log\bigl[p(z)\bigr]dz. + \end{eqnarray} + +\begin{eqnarray}\label{eq:loss_cross_entropy_empirical} + q(y) = \frac{1}{I} \sum_{i=1}^{I} \delta[y-y_{i}], + \end{eqnarray} + +\begin{eqnarray} + \hat{\boldsymbol\theta} &=& \mathop{\rm argmin}_{\boldsymbol\theta}\left[ \int_{-\infty}^{\infty}q(y) \log\bigl[q(y)\bigr]dy - \int_{-\infty}^{\infty}q(y) \log\bigl[Pr(y|\boldsymbol\theta)\bigr] dy\right] \nonumber \\ + &=&\mathop{\rm argmin}_{\boldsymbol\theta}\left[- \int_{-\infty}^{\infty}q(y) \log\bigl[Pr(y|\boldsymbol\theta)\bigr] dy\right], + \end{eqnarray} + +\begin{eqnarray}\label{eq:loss_cross_deriv} + \hat{\boldsymbol\theta} &=& \mathop{\rm argmin}_{\theta}\left[-\int_{-\infty}^{\infty}\left(\frac{1}{I} \sum_{i=1}^{I} \delta[y-y_{i}]\right)\log\bigl[Pr(y|\boldsymbol\theta)\bigr]dy\right] \nonumber\\ + &=&\mathop{\rm argmin}_{\boldsymbol\theta}\left[-\frac{1}{I} \sum_{i=1}^{I} \log\bigl[Pr(y_i|\boldsymbol\theta)\bigr]\right]\nonumber\\ + &=&\mathop{\rm argmin}_{\boldsymbol\theta}\left[-\sum_{i=1}^{I} \log\bigl[Pr(y_i|\boldsymbol\theta)\bigr]\right]. + \end{eqnarray} + +\begin{eqnarray} + \hat{\boldsymbol\phi} = \mathop{\rm argmin}_{\boldsymbol\phi}\left[-\sum_{i=1}^{I} \log\bigl[Pr(y_{i}|\mbox{\bf f}[\mathbf{x}_{i},\boldsymbol\phi])\bigr]\right]. + \end{eqnarray} + +\begin{eqnarray}\label{eq:loss_prob_sigmoid} + \mbox{sig}[z] = \frac{1}{1+\exp[-z]}. + \end{eqnarray} + +\begin{eqnarray} + L = -(1-y)\log\Bigl[1-\mbox{sig}[\mbox{f}[\mathbf{x},\boldsymbol\phi]]\Bigr] - y\log\Bigl[\mbox{sig}[\mbox{f}[\mathbf{x},\boldsymbol\phi]]\Bigr], + \end{eqnarray} + +\begin{eqnarray} + Pr(y|\mu,\kappa) = \frac{\exp\bigl[\kappa \cos[y-\mu]\bigr]}{2\pi \cdot \mbox{Bessel}_{0}[\kappa]} , + \end{eqnarray} + +\begin{eqnarray} + Pr(y|\lambda,\mu_1,\mu_2,\sigma^2_1,\sigma^2_2) = \frac{\lambda}{\sqrt{2\pi\sigma_1^{2}}}\exp\left[\frac{-(y-\mu_1)^{2}}{2\sigma_1^{2}}\right]+\frac{1-\lambda}{\sqrt{2\pi\sigma_2^{2}}}\exp\left[\frac{-(y-\mu_2)^{2}}{2\sigma_2^{2}}\right], + \end{eqnarray} + +\begin{eqnarray} + Pr(y=k)=\frac{\lambda^{k}e^{-\lambda}}{k!}. + \end{eqnarray} + +\chapter{Fitting models} + +\begin{eqnarray} + \hat{\boldsymbol\phi} = \mathop{\rm argmin}_{\boldsymbol\phi}\Bigl[L[\boldsymbol\phi]\Bigr]. + \end{eqnarray} + +\begin{eqnarray} + \frac{\partial L}{\partial \boldsymbol\phi} = \begin{bmatrix}\frac{\partial L}{\partial \phi_{0}} \\[5pt] \frac{\partial L}{\partial \phi_{1}}\\ \vdots \\[5pt] \frac{\partial L}{\partial \phi_{N}}\end{bmatrix}. + \end{eqnarray} + +\begin{eqnarray}\label{eq:train_gradient_descent} + \boldsymbol\phi\longleftarrow\boldsymbol\phi - \alpha \cdot \frac{\partial L}{\partial \boldsymbol\phi}, + \end{eqnarray} + +\begin{eqnarray} + y &=& \mbox{f}[x,\boldsymbol\phi]\nonumber \\ + &=& \phi_{0} + \phi_{1}x. + \end{eqnarray} + +\begin{eqnarray}\label{eq:train_lr_cost_function} + L[\boldsymbol\phi] \hspace{0.2cm}= \hspace{0.2cm}\sum_{i=1}^{I} \ell_{i} &=& \sum_{i=1}^{I} \left(\mbox{f}[x_i,\boldsymbol\phi]-y_{i}\right)^{2}\nonumber \\ + &=& \sum_{i=1}^{I} \left(\phi_{0}+\phi_{1}x_{i}-y_{i}\right)^{2}, + \end{eqnarray} + +\begin{eqnarray}\label{eq:train_linear_deriv1} + \frac{\partial L}{\partial \boldsymbol\phi} = \frac{\partial}{\partial \boldsymbol\phi}\sum_{i=1}^{I} \ell_{i} = \sum_{i=1}^{I} \frac{\partial \ell_{i}}{\partial \boldsymbol\phi}, + \end{eqnarray} + +\begin{eqnarray}\label{eq:train_linear_deriv2} + \frac{\partial \ell_{i}}{\partial \boldsymbol\phi} = \begin{bmatrix}\frac{\partial \ell_{i}}{\partial \phi_{0}}\\\vspace{-0.2cm}\\\frac{\partial \ell_{i}}{\partial \phi_{1}}\end{bmatrix} = \begin{bmatrix}2(\phi_{0}+\phi_{1}x_{i}-y_{i})\\ \vspace{-0.2cm}\\ 2x_{i}(\phi_{0}+\phi_{1}x_{i}-y_{i})\end{bmatrix}. + \end{eqnarray} + +\begin{eqnarray}\label{eq:train_gabor} + \mbox{f}[x,\boldsymbol\phi] = \sin[\phi_0 + 0.06\cdot\phi_{1}x]\cdot \exp\left(-\frac{(\phi_0+0.06\cdot\phi_{1}x)^2}{32.0}\right). + \end{eqnarray} + +\begin{eqnarray} + L[\boldsymbol\phi] = \sum_{i=1}^{I}\left(\mbox{f}[x_{i},\boldsymbol\phi]-y_{i}\right)^2. + \end{eqnarray} + +\begin{eqnarray}\label{eq:train_sgd} + \boldsymbol\phi_{t+1}\longleftarrow\boldsymbol\phi_{t} - \alpha \cdot \sum_{i\in\mathcal{B}_{t}}\frac{\partial \ell_{i}[\boldsymbol\phi_{t}]}{\partial \boldsymbol\phi}, + \end{eqnarray} + +\begin{eqnarray}\label{eq:train_momentum} + \mathbf{m}_{t+1} &\leftarrow& \beta \cdot \mathbf{m}_t + (1-\beta) \sum_{i\in\mathcal{B}_t}\frac{\partial \ell_{i}[\boldsymbol\phi_{t}]}{\partial \boldsymbol\phi}\nonumber \\ + \boldsymbol\phi_{t+1} &\leftarrow&\boldsymbol\phi_{t} - \alpha \cdot \mathbf{m}_{t+1}, + \end{eqnarray} + +\begin{eqnarray}\label{eq:train_nesterov} + \mathbf{m}_{t+1} &\leftarrow& \beta \cdot \mathbf{m}_{t} + (1-\beta) \sum_{i\in\mathcal{B}_{t}}\frac{\partial \ell_{i}[\boldsymbol\phi_{t}-\alpha\beta\cdot\mathbf{m}_{t}]}{\partial \boldsymbol\phi}\nonumber \\ + \boldsymbol\phi_{t+1} &\leftarrow&\boldsymbol\phi_{t} - \alpha\cdot \mathbf{m}_{t+1}, + \end{eqnarray} + +\begin{eqnarray} + \mathbf{m}_{t+1} &\leftarrow& \frac{\partial L[\boldsymbol\phi_{t}]}{\partial \boldsymbol\phi}\nonumber \\ + \mathbf{v}_{t+1} &\leftarrow& \left(\frac{\partial L[\boldsymbol\phi_{t}]}{\partial \boldsymbol\phi}\right)^2. + \end{eqnarray} + +\begin{eqnarray} + \boldsymbol\phi_{t+1} &\leftarrow & \boldsymbol\phi_{t} - \alpha \cdot \frac{\mathbf{m}_{t+1}}{\sqrt{\mathbf{v}_{t+1}}+\epsilon}, + \end{eqnarray} + +\begin{eqnarray}\label{eq:train_adam} + \mathbf{m}_{t+1} &\leftarrow& \beta \cdot \mathbf{m}_{t} + (1-\beta) \frac{\partial L[\boldsymbol\phi_{t}]}{\partial \boldsymbol\phi}\nonumber \\ + \mathbf{v}_{t+1} &\leftarrow& \gamma \cdot \mathbf{v}_{t} + (1-\gamma) \left(\frac{\partial L[\boldsymbol\phi_{t}]}{\partial \boldsymbol\phi}\right)^2, + \end{eqnarray} + +\begin{eqnarray}\label{eq:train_adam_modify} + \tilde{\mathbf{m}}_{t+1} &\leftarrow & \frac{\mathbf{m}_{t+1}}{1-\beta^{t+1}}\nonumber \\ + \tilde{\mathbf{v}}_{t+1} &\leftarrow & \frac{\mathbf{v}_{t+1}}{1-\gamma^{t+1}}. + \end{eqnarray} + +\begin{eqnarray} + \boldsymbol\phi_{t+1} &\leftarrow & \boldsymbol\phi_{t} - \alpha\cdot\frac{ \tilde{\mathbf{m}}_{t+1}}{\sqrt{\tilde{\mathbf{v}}_{t+1}}+\epsilon}. + \end{eqnarray} + +\begin{eqnarray}\label{eq:train_adam_final} + \mathbf{m}_{t+1} &\leftarrow& \beta \cdot \mathbf{m}_{t} + (1-\beta) \sum_{i\in\mathcal{B}_{t}}\frac{\partial \ell_i[\boldsymbol\phi_{t}]}{\partial \boldsymbol\phi}\nonumber \\ + \mathbf{v}_{t+1} &\leftarrow& \gamma \cdot \mathbf{v}_{t} + (1-\gamma) \left(\sum_{i\in\mathcal{B}_{t}}\frac{\partial \ell_i[\boldsymbol\phi_{t}]}{\partial \boldsymbol\phi}\right)^2, + \end{eqnarray} + +\begin{eqnarray} + \mathbf{H}[\boldsymbol\phi] = \begin{bmatrix} \frac{\partial^{2} L}{\partial\phi_{0}^{2}} & \frac{\partial^{2} L}{\partial\phi_{0}\partial\phi_{1}} &\hdots &\frac{\partial^{2} L}{\partial\phi_{0}\partial\phi_{N}}\\ + \frac{\partial^{2} L}{\partial\phi_{1}\partial\phi_{0}} &\frac{\partial^{2} L}{\partial\phi_{1}^{2}} &\hdots&\frac{\partial^{2} L}{\partial\phi_{1}\partial\phi_{N}} \\ \vdots &\vdots & \ddots &\vdots\\ + \frac{\partial^{2} L}{\partial\phi_{N}\partial\phi_{0}} &\frac{\partial^{2} L}{\partial\phi_{N}\partial\phi_{1}} &\hdots&\frac{\partial^{2} L}{\partial\phi_{N}^2} \end{bmatrix}. + \end{eqnarray} + +\begin{eqnarray} + \mathbf{H}[\boldsymbol\phi] = \begin{bmatrix} \frac{\partial^{2} L}{\partial\phi_{0}^{2}} & \frac{\partial^{2} L}{\partial\phi_{0}\partial\phi_{1}}\\ + \frac{\partial^{2} L}{\partial\phi_{1}\partial\phi_{0}} &\frac{\partial^{2} L}{\partial\phi_{1}^{2}} \end{bmatrix}, + \end{eqnarray} + +\begin{eqnarray} + Pr(y=1|x) = \mbox{sig}[\phi_{0}+\phi_{1}x], + \end{eqnarray} + +\begin{eqnarray} + \mbox{sig}[z] = \frac{1}{1+\exp[-z]}. + \end{eqnarray} + +\begin{eqnarray} + \mbox{f}[x,\boldsymbol\phi] = \phi_{0}+\phi_{1}\mbox{a}[\theta_{10} + \theta_{11}x]+\phi_{2}\mbox{a}[\theta_{20} + \theta_{21}x]+\phi_{3}\mbox{a}[\theta_{30} + \theta_{31}x]. + \end{eqnarray} + + \chapter{Gradients and initialization} + +\begin{eqnarray} + \mathbf{h}_{1} &=& \mathbf{a}[\boldsymbol\beta_{0} +\boldsymbol\Omega_{0}\mathbf{x}]\nonumber \\ + \mathbf{h}_{2} &=& \mathbf{a}[\boldsymbol\beta_{1} +\boldsymbol\Omega_{1}\mathbf{h}_{1}] \nonumber\\ + \mathbf{h}_{3} &=& \mathbf{a}[\boldsymbol\beta_{2} +\boldsymbol\Omega_{2}\mathbf{h}_{2}] \nonumber\\ + \mbox{\bf f}[\mathbf{x},\boldsymbol\phi] &=& \boldsymbol\beta_{3} +\boldsymbol\Omega_{3}\mathbf{h}_{3}, + \end{eqnarray} + +\begin{eqnarray} + L[\boldsymbol\phi]= \sum_{i=1}^{I} \ell_{i}. + \end{eqnarray} + +\begin{eqnarray}\label{eq:train2_sgd} + \boldsymbol\phi_{t+1}\longleftarrow\boldsymbol\phi_{t} - \alpha \sum_{i\in\mathcal{B}_{t}}\frac{\partial \ell_{i}[\boldsymbol\phi_{t}]}{\partial \boldsymbol\phi}, + \end{eqnarray} + +\begin{eqnarray} + \frac{\partial \ell_{i}}{\partial\boldsymbol\beta_{k}} \quad\quad \mbox{and} \quad\quad \frac{\partial \ell_{i}}{\partial\boldsymbol\Omega_{k}}, + \end{eqnarray} + +\begin{eqnarray} + \mbox{f}[x,\boldsymbol\phi] = \beta_3+\omega_3\cdot\cos\Bigl[\beta_2+\omega_2\cdot\exp\bigl[\beta_1+\omega_1\cdot\sin[\beta_0+\omega_0\cdot x]\bigr]\Bigr], + \end{eqnarray} + +\begin{eqnarray} + \ell_i = (\mbox{f}[x_i,\boldsymbol\phi]-y_i)^2, + \end{eqnarray} + +\begin{eqnarray} + \frac{\partial \ell_i}{\partial \beta_{0}}, \quad \frac{\partial \ell_i}{\partial \omega_0}, \quad \frac{\partial \ell_i}{\partial \beta_{1}}, \quad \frac{\partial \ell_i}{\partial \omega_1}, \quad + \frac{\partial \ell_i}{\partial \beta_{2}}, \quad \frac{\partial \ell_i}{\partial \omega_{2}}, \quad \frac{\partial \ell_i}{\partial \beta_{3}}, \quad\mbox{and} \quad \frac{\partial \ell_i}{\partial \omega_{3}}. + \end{eqnarray} + +\begin{eqnarray}\label{eq:train2_complicated_deriv} + \frac{\partial \ell_i}{\partial \omega_{0}} &=& -2 \left( \beta_3+\omega_3\cdot\cos\Bigl[\beta_2+\omega_2\cdot\exp\bigl[\beta_1+\omega_1\cdot\sin[\beta_0+\omega_0\cdot x_i]\bigr]\Bigr]-y_i\right)\nonumber \\ + &&\hspace{0.5cm}\cdot \omega_1\omega_2\omega_3\cdot x_i\cdot\cos[\beta_0+\omega_0 \cdot x_i]\cdot\exp\Bigl[\beta_1 + \omega_1 \cdot \sin[\beta_0+\omega_0\cdot x_i]\Bigr]\nonumber\\ + && \hspace{1cm}\cdot \sin\biggl[\beta_2+\omega_2\cdot \exp\Bigl[\beta_1 + \omega_1 \cdot \sin[\beta_0+\omega_0\cdot x_i]\Bigr]\biggr]. + \end{eqnarray} + +\begin{eqnarray} + f_{0} &=& \beta_{0} + \omega_{0}\cdot x_i\nonumber\\ + h_{1} &=& \sin[f_{0}]\nonumber\\ + f_{1} &=& \beta_{1} + \omega_{1}\cdot h_{1}\nonumber\\ + h_{2} &=& \exp[f_{1}]\nonumber\\ + f_{2} &=& \beta_{2} + \omega_{2} \cdot h_{2}\nonumber\\ + h_{3} &=& \cos[f_{2}]\nonumber\\ + f_{3} &=& \beta_{3} + \omega_{3}\cdot h_{3}\nonumber\\ + \ell_{i} &=& (f_{3}-y_{i})^2. + \end{eqnarray} + +\begin{eqnarray} + \frac{\partial \ell_i}{\partial f_{3}}, \quad \frac{\partial \ell_i}{\partial h_3}, \quad \frac{\partial \ell_i}{\partial f_2}, \quad + \frac{\partial \ell_i}{\partial h_2}, \quad \frac{\partial \ell_i}{\partial f_1}, \quad \frac{\partial \ell_i}{\partial h_1}, \quad\mbox{and} \quad \frac{\partial \ell_i}{\partial f_0}. + \end{eqnarray} + +\begin{eqnarray} + \frac{\partial \ell_i}{\partial f_{3}} = 2(f_3-y_i). + \end{eqnarray} + +\begin{eqnarray} + \frac{\partial \ell_i}{\partial h_{3}} =\frac{\partial f_{3}}{\partial h_{3}} \frac{\partial \ell_i}{\partial f_{3}} . + \end{eqnarray} + +\begin{eqnarray} + \frac{\partial \ell_i}{\partial f_{2}} &=& \frac{\partial h_{3}}{\partial f_{2}}\left( + \frac{\partial f_{3}}{\partial h_{3}}\frac{\partial \ell_i}{\partial f_{3}} \right) + \nonumber \\ + \frac{\partial \ell_i}{\partial h_{2}} &=& \frac{\partial f_{2}}{\partial h_{2}}\left(\frac{\partial h_{3}}{\partial f_{2}}\frac{\partial f_{3}}{\partial h_{3}}\frac{\partial \ell_i}{\partial f_{3}}\right)\nonumber \\ + \frac{\partial \ell_i}{\partial f_{1}} &=& \frac{\partial h_{2}}{\partial f_{1}}\left( \frac{\partial f_{2}}{\partial h_{2}}\frac{\partial h_{3}}{\partial f_{2}}\frac{\partial f_{3}}{\partial h_{3}}\frac{\partial \ell_i}{\partial f_{3}} \right)\nonumber \\ + \frac{\partial \ell_i}{\partial h_{1}} &=& \frac{\partial f_{1}}{\partial h_{1}}\left(\frac{\partial h_{2}}{\partial f_{1}} \frac{\partial f_{2}}{\partial h_{2}}\frac{\partial h_{3}}{\partial f_{2}}\frac{\partial f_{3}}{\partial h_{3}}\frac{\partial \ell_i}{\partial f_{3}} \right)\nonumber \\ + \frac{\partial \ell_i}{\partial f_{0}} &=& \frac{\partial h_{1}}{\partial f_{0}}\left(\frac{\partial f_{1}}{\partial h_{1}}\frac{\partial h_{2}}{\partial f_{1}} \frac{\partial f_{2}}{\partial h_{2}}\frac{\partial h_{3}}{\partial f_{2}}\frac{\partial f_{3}}{\partial h_{3}}\frac{\partial \ell_i}{\partial f_{3}} \right).\label{eq:train2_simple_chain} + \end{eqnarray} + +\begin{eqnarray} + \frac{\partial \ell_i}{\partial \beta_{k}} &=& \frac{\partial f_{k}}{\partial \beta_{k}}\frac{\partial \ell_i}{\partial f_{k}}\nonumber \\ + \frac{\partial \ell_i}{\partial \omega_{k}} &=& \frac{\partial f_{k}}{\partial \omega_{k}}\frac{\partial \ell_i}{\partial f_{k}}. + \end{eqnarray} + +\begin{eqnarray} + \frac{\partial f_{k}}{\partial \beta_{k}} = 1 \quad\quad\mbox{and}\quad \quad \frac{\partial f_{k}}{\partial \omega_{k}} &=& h_{k}. + \end{eqnarray} + +\begin{eqnarray} + \frac{\partial f_{0}}{\partial \beta_{0}} = 1 \quad\quad\mbox{and}\quad \quad \frac{\partial f_{0}}{\partial \omega_{0}} &=& x_{i}. + \end{eqnarray} + +\begin{eqnarray} + \mathbf{f}_{0} &=& \boldsymbol\beta_{0} +\boldsymbol\Omega_{0}\mathbf{x}_i\nonumber \\ + \mathbf{h}_{1} &=& \mathbf{a}[\mathbf{f}_{0}]\nonumber \\ + \mathbf{f}_{1} &=& \boldsymbol\beta_{1} +\boldsymbol\Omega_{1}\mathbf{h}_{1}\nonumber \\ + \mathbf{h}_{2} &=& \mathbf{a}[\mathbf{f}_{1}]\nonumber \\ + \mathbf{f}_{2} &=& \boldsymbol\beta_{2} +\boldsymbol\Omega_{2}\mathbf{h}_{2}\nonumber \\ + \mathbf{h}_{3} &=& \mathbf{a}[\mathbf{f}_{2}]\nonumber \\ + \mathbf{f}_{3}&=& \boldsymbol\beta_{3} +\boldsymbol\Omega_{3}\mathbf{h}_{3}\nonumber \\ + \ell_{i} &=& \mbox{l}[\mathbf{f}_{3},y_{i}], + \end{eqnarray} + +\begin{eqnarray}\label{eq:train2_backward1} + \frac{\partial \ell_{i}}{\partial \mathbf{f}_{2}}=\frac{\partial \mathbf{h}_{3}}{\partial \mathbf{f}_{2}}\frac{\partial \mathbf{f}_3}{\partial \mathbf{h}_{3}} \frac{\partial \ell_{i}}{\partial \mathbf{f}_3}. + \end{eqnarray} + +\begin{eqnarray}\label{eq:train2_backward2} + \frac{\partial \ell_{i}}{\partial \mathbf{f}_{1}}&=& \frac{\partial \mathbf{h}_{2}}{\partial \mathbf{f}_{1}}\frac{\partial \mathbf{f}_{2}}{\partial \mathbf{h}_{2}} + \left(\frac{\partial \mathbf{h}_{3}}{\partial \mathbf{f}_{2}}\frac{\partial \mathbf{f}_3}{\partial \mathbf{h}_{3}} \frac{\partial \ell_{i}}{\partial \mathbf{f}_3}\right) \\ + \frac{\partial \ell_{i}}{\partial \mathbf{f}_{0}}&=&\frac{\partial \mathbf{h}_{1}}{\partial \mathbf{f}_{0}}\frac{\partial \mathbf{f}_{1}}{\partial \mathbf{h}_{1}}\left(\frac{\partial \mathbf{h}_{2}}{\partial \mathbf{f}_{1}}\frac{\partial \mathbf{f}_{2}}{\partial \mathbf{h}_{2}} + \frac{\partial \mathbf{h}_{3}}{\partial \mathbf{f}_{2}}\frac{\partial \mathbf{f}_3}{\partial \mathbf{h}_{3}} \frac{\partial \ell_{i}}{\partial \mathbf{f}_3}\right).\label{eq:train2_backward2a} + \end{eqnarray} + +\begin{eqnarray} + \frac{\partial \mathbf{f}_3}{\partial \mathbf{h}_{3}} = \frac{\partial}{\partial \mathbf{h}_{3}}\left(\boldsymbol\beta_{3} +\boldsymbol\Omega_{3}\mathbf{h}_{3}\right) = \boldsymbol\Omega_{3}^{T}. + \end{eqnarray} + +\begin{eqnarray} + \frac{\partial \ell_{i}}{\partial \boldsymbol\beta_k} &=& \frac{\partial \mathbf{f}_{k}}{\partial \boldsymbol\beta_k} \frac{\partial \ell_{i}}{\partial \mathbf{f}_{k}} \nonumber\\ + &=& \frac{\partial}{\partial \boldsymbol\beta_k}\left(\boldsymbol\beta_{k} +\boldsymbol\Omega_{k}\mathbf{h}_{k}\right) \frac{\partial \ell_{i}}{\partial \mathbf{f}_{k}} \nonumber \\ + &=& \frac{\partial \ell_{i}}{\partial \mathbf{f}_{k}}, + \end{eqnarray} + +\begin{eqnarray} + \frac{\partial \ell_{i}}{\partial \boldsymbol\Omega_k} &=& \frac{\partial \mathbf{f}_{k}}{\partial \boldsymbol\Omega_k} \frac{\partial \ell_{i}}{\partial \mathbf{f}_{k}} \nonumber\\ + &=& \frac{\partial}{\partial \boldsymbol\Omega_k}\left(\boldsymbol\beta_{k} +\boldsymbol\Omega_{k}\mathbf{h}_{k}\right) \frac{\partial \ell_{i}}{\partial \mathbf{f}_{k}} \nonumber \\ + &=& \frac{\partial \ell_{i}}{\partial \mathbf{f}_{k}}\mathbf{h}_k^{T}. + \end{eqnarray} + +\begin{eqnarray} + \mathbf{f}_{0} &=& \boldsymbol\beta_{0} +\boldsymbol\Omega_{0}\mathbf{x}_i\nonumber \\ + \mathbf{h}_{k} &=& \mathbf{a}[\mathbf{f}_{k-1}]\hspace{2.76cm} k\in\{1,2,\ldots, K\}\nonumber \\ + \mathbf{f}_{k} &=& \boldsymbol\beta_{k} +\boldsymbol\Omega_{k}\mathbf{h}_{k}.\hspace{2cm} k\in\{1,2,\ldots, K\} + \end{eqnarray} + +\begin{eqnarray}\label{eq:train2_bp_backward_summary} + \frac{\partial \ell_{i}}{\partial \boldsymbol\beta_k} &=& \frac{\partial \ell_{i}}{\partial \mathbf{f}_k} \hspace{4.1cm} k\in\{K,K-1,\ldots, 1\}\nonumber\\ + \frac{\partial \ell_{i}}{\partial \boldsymbol\Omega_k} &=& \frac{\partial \ell_{i}}{\partial \mathbf{f}_k}\mathbf{h}_{k}^{T}\hspace{3.65cm} k\in\{K,K-1,\ldots, 1\}\nonumber\\ + \frac{\partial \ell_{i}}{\partial \mathbf{f}_{k-1}} &=& \mathbb{I}[\mathbf{f}_{k-1}>0]\odot \left(\boldsymbol\Omega_{k}^{T}\frac{\partial \ell_{i}}{\partial \mathbf{f}_{k}}\right),\hspace{0.82cm} k\in\{K,K-1,\ldots, 1\} + \end{eqnarray} + +\begin{eqnarray} + \frac{\partial \ell_{i}}{\partial \boldsymbol\beta_0} &=& \frac{\partial \ell_{i}}{\partial \mathbf{f}_0} \nonumber\\ + \frac{\partial \ell_{i}}{\partial \boldsymbol\Omega_0} &=& \frac{\partial \ell_{i}}{\partial \mathbf{f}_0}\mathbf{x}_{i}^{T}. + \end{eqnarray} + +\begin{eqnarray} + \mathbf{f}_{k} &=& \boldsymbol\beta_{k} +\boldsymbol\Omega_{k}\mathbf{h}_{k}\nonumber\\ + &=& \boldsymbol\beta_{k} +\boldsymbol\Omega_{k}\textbf{a}[\mathbf{f}_{k-1}], + \end{eqnarray} + +\begin{eqnarray} + \mathbf{h} &=& \mbox{\bf a}[\mathbf{f}],\nonumber \\ + \mathbf{f}' &=& \boldsymbol\beta +\boldsymbol\Omega\mathbf{h} + \end{eqnarray} + +\begin{eqnarray} + \mathbb{E}[f'_{i}] &=& \mathbb{E}\left[\beta_{i} + \sum_{j=1}^{D_h}\Omega_{ij}h_{j}\right]\nonumber \\ + &=& \mathbb{E}\left[\beta_{i}\right] +\sum_{j=1}^{D_h}\mathbb{E}\left[\Omega_{ij}h_{j}\right] \nonumber\\ + &=& \mathbb{E}\left[\beta_{i}\right] +\sum_{j=1}^{D_h}\mathbb{E}\left[\Omega_{ij}\right]\mathbb{E}\left[h_{j}\right] \nonumber\\ + &=& 0 + \sum_{j=1}^{D_h} 0\cdot\mathbb{E}\left[h_{j}\right] = 0, + \end{eqnarray} + +\begin{eqnarray} + \sigma^{2}_{f'_i} &=& \mathbb{E}[f_{i}^{\prime 2}]-\mathbb{E}[f'_{i}]^{2} \nonumber \\ + &=& \mathbb{E}\left[\left(\beta_{i}+\sum_{j=1}^{D_h}\Omega_{ij}h_{j}\right)^2\right]-0\nonumber\\ + &=& \mathbb{E}\left[\left(\sum_{j=1}^{D_h}\Omega_{ij}h_{j}\right)^2\right]\nonumber\\ + &=& \sum_{j=1}^{D_h}\mathbb{E}\left[\Omega_{ij}^2\right]\mathbb{E}\left[h_{j}^2\right]\nonumber \\ + &=&\sum_{j=1}^{D_h} \sigma_\Omega^2 \mathbb{E}\left[h_{j}^2\right] = \sigma_\Omega^2 \sum_{j=1}^{D_h} \mathbb{E}\left[h_{j}^2\right], + \end{eqnarray} + +\begin{eqnarray} + \sigma^{2}_{f'_{i}} = \sigma_\Omega^2 \sum_{j=1}^{D_h} \frac{\sigma_{f}^2}{2} = \frac{1}{2}D_{h} \sigma_\Omega^2 \sigma_{f}^2. + \end{eqnarray} + +\begin{eqnarray}\label{eq:train2_init_forward} + \sigma_\Omega^2 = \frac{2}{D_h}, + \end{eqnarray} + +\begin{eqnarray}\label{eq:train2_init_back} + \sigma_\Omega^2 = \frac{2}{D_{h'}}, + \end{eqnarray} + +\begin{eqnarray} + \sigma_\Omega^2 = \frac{4}{D_{h}+D_{h'}}. + \end{eqnarray} + +\begin{eqnarray} + y &=& \phi_{0}+\phi_{1}\mbox{a}\Bigl[\psi_{01} + \psi_{11}\mbox{a}[\theta_{01} + \theta_{11}x] + \psi_{21}\mbox{a}[\theta_{02} + \theta_{12}x]\Bigr]\nonumber \\ + &&\hspace{0.55cm}+\phi_{2}\mbox{a}\Bigl[\psi_{02} + \psi_{12}\mbox{a}[\theta_{01} + \theta_{11}x] + \psi_{22}\mbox{a}[\theta_{02} + \theta_{12}x]\Bigr], + \end{eqnarray} + +\begin{eqnarray} + \ell_i = (y_i-\mbox{f}[\mathbf{x}_i,\boldsymbol\phi])^2. + \end{eqnarray} + +\begin{eqnarray} + \ell_{i} = -(1-y_{i})\log\Bigl[1-\mbox{sig}\bigl[\mbox{f}[\mathbf{x}_i,\boldsymbol\phi]\bigr]\Bigr] - y_{i}\log\Bigl[\mbox{sig}\bigl[\mbox{f}[\mathbf{x}_i,\boldsymbol\phi]\bigr]\Bigr], + \end{eqnarray} + +\begin{eqnarray}\label{eq:train2_logistic} + \mbox{sig}[z] = \frac{1}{1+\exp[-z]}. + \end{eqnarray} + +\begin{eqnarray} + \frac{\partial \mathbf{z}}{\partial \mathbf{h}} = \boldsymbol\Omega^{T}, + \end{eqnarray} + +\begin{eqnarray} + \mbox{Heaviside}[z] = \begin{cases} 0 & \quad z <0 \\ 1 & \quad z\geq 0\end{cases}, + \end{eqnarray} + +\begin{eqnarray} + \mbox{rect}[z] = \begin{cases} 0 & \quad z < 0 \\ 1 & \quad 0 \leq z\leq 1 \\ 0 & \quad z > 1\end{cases}. + \end{eqnarray} + +\begin{eqnarray} + \frac{\partial \ell}{\partial \boldsymbol\Omega} = \frac{\partial \ell}{\partial \mathbf{f}}\mathbf{h}^{T}. + \end{eqnarray} + +\begin{eqnarray}\label{eq:train2_prob_leaky_relu} + \mbox{a}[z] = \mbox{ReLU}[z] = \begin{cases} \alpha \cdot z & \quad z <0 \\ z & \quad z\geq 0\end{cases}, + \end{eqnarray} + +\begin{eqnarray}\label{eq:prob_comp_graph}\index{reverse-mode differentiation}\index{differentiation!reverse mode}\index{backpropagation!on acyclic graph} + y = \exp\left[\exp[x]+\exp[x]^2\right]+ \sin[\exp[x]+\exp[x]^2]. + \end{eqnarray} + +\begin{eqnarray} + f_{1} &=& \exp[x]\nonumber \\ + f_{2} &=& f_{1}^2\nonumber \\ + f_{3} &=& f_{1}+f_{2}\nonumber \\ + f_{4} &=& \exp[f_{3}]\nonumber \\ + f_{5} &=& \sin[f_{3}]\nonumber \\ + y &=& f_{4}+f_{5}. + \end{eqnarray} + +\begin{eqnarray} + \frac{\partial y}{\partial f_{5}}, \frac{\partial y}{\partial f_{4}}, \frac{\partial y}{\partial f_{3}}, + \frac{\partial y}{\partial f_{2}}, \frac{\partial y}{\partial f_{1}} \mbox{ and } \frac{\partial y}{\partial x}, + \end{eqnarray} + +\begin{eqnarray} + \frac{\partial f_{1}}{\partial x}, \frac{\partial f_{2}}{\partial x}, \frac{\partial f_{3}}{\partial x}, \frac{\partial f_4}{\partial x}, + \frac{\partial f_{5}}{\partial x}, \mbox{ and } \frac{\partial y}{\partial x}, + \end{eqnarray} + +\begin{eqnarray} + b = \mbox{ReLU}[a] = \begin{cases} 0 & \quad a <0 \\ a& \quad a\geq 0\end{cases}, + \end{eqnarray} + +\chapter{Measuring performance} + +\begin{eqnarray} + \mu[x] = \mathbb{E}_{y}[y[x]] = \int y[x] Pr(y|x) dy, + \end{eqnarray} + +\begin{eqnarray} + L[x] &=& \bigl(\mbox{f}[x,\boldsymbol\phi]-y[x]\bigr)^2 \\ + &=& \Bigl(\bigl(\mbox{f}[x,\boldsymbol\phi]-\mu[x]\bigr)+\bigl(\mu[x]-y[x]\bigr)\Bigr)^2\nonumber \\ + &=& \bigl(\mbox{f}[x,\boldsymbol\phi]-\mu[x]\bigr)^2 + 2\bigl(\mbox{f}[x,\boldsymbol\phi]-\mu[x]\bigr)\bigl(\mu[x]-y[x]\bigr) + \bigl(\mu[x]-y[x]\bigr)^2,\nonumber + \end{eqnarray} + +\begin{eqnarray}\label{eq:perf_biasvariance_plus_noise} + \mathbb{E}_{y}\bigl[L[x]\bigr] + &=&\mathbb{E}_{y}\Bigl[\bigl(\mbox{f}[x,\boldsymbol\phi]\!-\!\mu[x]\bigr)^2+2\bigl(\mbox{f}[x,\boldsymbol\phi]\!-\!\mu[x]\bigr)\bigl(\mu[x]\!-\!y[x]\bigr)+\bigl(\mu[x]\!-\!y[x]\bigr)^2\Bigr]\nonumber \\ + &=& \bigl(\mbox{f}[x,\boldsymbol\phi]\!-\!\mu[x]\bigr)^2+2\bigl(\mbox{f}[x,\boldsymbol\phi]-\mu[x]\bigr)\bigl(\mu[x]\!-\!\mathbb{E}_y\left[y[x]\right]\bigr)+\mathbb{E}_y\left[(\mu[x]\!-\!y[x])^2\right]\nonumber\\ + &=&\bigl(\mbox{f}[x,\boldsymbol\phi]\!-\!\mu[x]\bigr)^2+2\bigl(\mbox{f}[x,\boldsymbol\phi]\!-\!\mu[x]\bigr)\cdot 0 +\mathbb{E}_y\left[\bigl(\mu[x]\!-\!y[x]\bigr)^2\right]\nonumber \\ + &=&\bigl(\mbox{f}[x,\boldsymbol\phi]-\mu[x]\bigr)^2+\sigma^{2}, + \end{eqnarray} + +\begin{eqnarray} + \mbox{f}_{\mu}[x] = \mathbb{E}_{\mathcal{D}}\Bigl[ \mbox{f}\bigl[x,\boldsymbol\phi[\mathcal{D}]\bigr] \Bigr]. + \end{eqnarray} + +\begin{eqnarray} + \bigl(\mbox{f}[x,\boldsymbol\phi[\mathcal{D}]]\!-\!\mu[x]\bigr)^2 &&\\ + &&\hspace{-2.5cm}=\Bigl(\bigl(\mbox{f}[x,\boldsymbol\phi[\mathcal{D}]]\!-\! \mbox{f}_{\mu}[x]\bigr)+\bigl( \mbox{f}_{\mu}[x]-\mu[x]\bigr)\Bigr)^2 \nonumber \\ + &&\hspace{-2.5cm}=\bigl(\mbox{f}[x,\boldsymbol\phi[\mathcal{D}]]\!-\! \mbox{f}_{\mu}[x]\bigr)^2+2\bigl(\mbox{f}[x,\boldsymbol\phi[\mathcal{D}]]\!-\! \mbox{f}_{\mu}[x]\bigr)\bigl(\mbox{f}_{\mu}[x]\!-\!\mu[x]\bigr)+\bigl(\mbox{f}_{\mu}[x]\!-\!\mu[x]\bigr)^2.\nonumber + \end{eqnarray} + +\begin{eqnarray}\label{eq:perf_bv2} + \mathbb{E}_{\mathcal{D}}\Bigl[\bigl(\mbox{f}[x,\boldsymbol\phi[\mathcal{D}]]-\mu[x]\bigr)^2\Bigr] = \mathbb{E}_\mathcal{D}\Bigl[\bigl(\mbox{f}[x,\boldsymbol\phi[\mathcal{D}]]- \mbox{f}_{\mu}[x]\bigr)^2\Bigr]+\bigl(\mbox{f}_{\mu}[x]-\mu[x]\bigr)^2, + \end{eqnarray} + +\begin{eqnarray}\label{eq:perf_bias_variance_final} + \mathbb{E}_{\mathcal{D}}\Bigl[\mathbb{E}_{y}[L[x]]\Bigr] = + \begingroup\color{red} + \underbrace{\color{black}\mystrut{2.0ex}\mathbb{E}_\mathcal{D}\Bigl[\bigl(\mbox{f}[x,\boldsymbol\phi[\mathcal{D}]]- \mbox{f}_{\mu}[x]\bigr)^2\Bigr]}_{\mbox{variance}} + \endgroup + + + \begingroup\color{red} + \underbrace{\color{black}\mystrut{2.0ex}\bigl(\mbox{f}_{\mu}[x]\!-\!\mu[x]\bigr)^2}_{\mbox{bias}} + \endgroup + + + \begingroup\color{red} + \underbrace{\color{black}\mystrut{2.0ex}\sigma^2.}_{\mbox{noise}} + \endgroup + \end{eqnarray} + +\begin{eqnarray} + \mbox{Vol}[r] = \frac{r^{D}\pi^{D/2}}{\Gamma[D/2+1]}, + \end{eqnarray} + +\chapter{Regularization} + +\begin{eqnarray} + \hat{\boldsymbol\phi} &=& \mathop{\rm argmin}_{\boldsymbol\phi}\bigl[L[\boldsymbol\phi]\bigr]\nonumber \\ + &=& \mathop{\rm argmin}_{\boldsymbol\phi}\left[\sum_{i=1}^{I}\ell_{i}[\mathbf{x}_{i},\mathbf{y}_{i}]\right], + \end{eqnarray} + +\begin{eqnarray} + \hat{\boldsymbol\phi} = \mathop{\rm argmin}_{\boldsymbol\phi}\left[\sum_{i=1}^{I}\ell_{i}[\mathbf{x}_{i},\mathbf{y}_{i}] + \lambda \cdot \mbox{g}[\boldsymbol\phi]\right], + \end{eqnarray} + +\begin{eqnarray} + \hat{\boldsymbol\phi} = \mathop{\rm argmax}_{\boldsymbol\phi}\left[\prod_{i=1}^{I}Pr(\mathbf{y}_{i}|\mathbf{x}_{i},\boldsymbol\phi)\right]. + \end{eqnarray} + +\begin{eqnarray} + \hat{\boldsymbol\phi} = \mathop{\rm argmax}_{\boldsymbol\phi}\left[\prod_{i=1}^{I}Pr(\mathbf{y}_{i}|\mathbf{x}_{i},\boldsymbol\phi)Pr(\boldsymbol\phi)\right]. + \end{eqnarray} + +\begin{eqnarray}\label{eq:reg_l2} + \hat{\boldsymbol\phi} = \mathop{\rm argmin}_{\boldsymbol\phi}\left[\sum_{i=1}^{I}\ell_{i}[\mathbf{x}_{i},\mathbf{y}_{i}] + \lambda \sum_{j}\phi_{j}^{2}\right], + \end{eqnarray} + +\begin{eqnarray}\label{eq:reg_implicit} + \frac{d\boldsymbol\phi}{dt} = -\frac{\partial L}{\partial\boldsymbol\phi}. + \end{eqnarray} + +\begin{eqnarray} + \boldsymbol\phi_{t+1} = \boldsymbol\phi_{t}-\alpha \frac{\partial L[\boldsymbol\phi_{t}]}{\partial\boldsymbol\phi}, + \end{eqnarray} + +\begin{eqnarray}\label{eq:reg_imp_gd} + \tilde{L}_{GD}[\boldsymbol\phi] = L[\boldsymbol\phi] + \frac{\alpha}{4} \left\lVert\frac{\partial L}{\partial \boldsymbol\phi} \right\rVert^{2}. + \end{eqnarray} + +\begin{eqnarray}\label{eq:reg_imp_sgd} + \tilde{L}_{SGD}[\boldsymbol\phi] &=& \tilde{L}_{GD}[\boldsymbol\phi]+\frac{\alpha}{4 B}\sum_{b=1}^{B}\left\lVert\frac{\partial L_{b}}{\partial \boldsymbol\phi} - \frac{\partial L}{\partial \boldsymbol\phi} \right\rVert^{2}\nonumber \\ + &=& L[\boldsymbol\phi] + \frac{\alpha}{4} \left\lVert\frac{\partial L}{\partial \boldsymbol\phi} \right\rVert^{2}+\frac{\alpha}{4 B}\sum_{b=1}^{B}\left\lVert\frac{\partial L_{b}}{\partial \boldsymbol\phi} - \frac{\partial L}{\partial \boldsymbol\phi} \right\rVert^{2}. + \end{eqnarray} + +\begin{eqnarray} + L = \frac{1}{I} \sum_{i=1}^{I}\ell_{i}[\mathbf{x}_{i},y_{i}] \quad \quad \mbox{and}\quad\quad L_{b} = \frac{1}{|\mathcal{B}|} \sum_{i\in \mathcal{B}_{b}}\ell_{i}[\mathbf{x}_{i},y_{i}]. + \end{eqnarray} + +\begin{eqnarray} + Pr(\boldsymbol\phi|\{\mathbf{x}_{i},\mathbf{y}_{i}\}) = \frac{\prod_{i=1}^{I} Pr(\mathbf{y}_{i}|\mathbf{x}_{i},\boldsymbol\phi) Pr(\boldsymbol\phi)}{\int \prod_{i=1}^{I} Pr(\mathbf{y}_{i}|\mathbf{x}_{i},\boldsymbol\phi) Pr(\boldsymbol\phi)d\boldsymbol\phi } , + \end{eqnarray} + +\begin{eqnarray} + Pr(\mathbf{y}|\mathbf{x}, \{\mathbf{x}_{i},\mathbf{y}_{i}\}) = \int Pr(\mathbf{y}|\mathbf{x},\boldsymbol\phi) Pr(\boldsymbol\phi |\{\mathbf{x}_{i},\mathbf{y}_{i}\}) d\boldsymbol\phi. + \end{eqnarray} + +\begin{eqnarray}\label{eq:reg_weight_decay} + \boldsymbol\phi \longleftarrow (1-\lambda') \boldsymbol\phi - \alpha \frac{\partial L}{\partial \boldsymbol\phi}, + \end{eqnarray} + +\begin{eqnarray}\label{eq:reg_imp_discrete} + \boldsymbol\phi_{1} = \boldsymbol\phi_{0} + \alpha \cdot\mbox{\bf g}[\boldsymbol\phi_0], + \end{eqnarray} + +\begin{eqnarray} + \frac{d\boldsymbol\phi}{dt} =\mbox{\bf g}[\boldsymbol\phi]. + \end{eqnarray} + +\begin{eqnarray}\label{eq:reg_imp_correction} + \frac{d\boldsymbol\phi}{dt} \approx \mbox{\bf g}[\boldsymbol\phi] +\alpha \mbox{\bf g}_{1}[\boldsymbol\phi] + \ldots, + \end{eqnarray} + +\begin{eqnarray} + \boldsymbol\phi[\alpha] &\approx& \left.\boldsymbol\phi + \alpha \frac{d\boldsymbol\phi}{dt} + \frac{\alpha^2}{2}\frac{d^2\boldsymbol\phi }{dt^2}\right|_{\boldsymbol\phi=\boldsymbol\phi_{0}}\nonumber \\ + &\approx& \left.\boldsymbol\phi + \alpha \left(\mbox{\bf g}[\boldsymbol\phi] +\alpha \mbox{\bf g}_{1}[\boldsymbol\phi] \right) + \frac{\alpha^2}{2}\left(\frac{\partial \mbox{\bf g}[\boldsymbol\phi]}{\partial \boldsymbol\phi}\frac{d\boldsymbol\phi}{dt} +\alpha \frac{\partial \mbox{\bf g}_{1}[\boldsymbol\phi]}{\partial \boldsymbol\phi}\frac{d\boldsymbol\phi}{dt}\right)\right|_{\boldsymbol\phi=\boldsymbol\phi_{0}}\nonumber \\ + &=& \left.\boldsymbol\phi + \alpha \left(\mbox{\bf g}[\boldsymbol\phi] +\alpha \mbox{\bf g}_{1}[\boldsymbol\phi]\right) + \frac{\alpha^2}{2}\left(\frac{\partial \mbox{\bf g}[\boldsymbol\phi]}{\partial \boldsymbol\phi}\mbox{\bf g}[\boldsymbol\phi] +\alpha \frac{\partial \mbox{\bf g}_{1}[\boldsymbol\phi]}{\partial \boldsymbol\phi}\mbox{\bf g}[\boldsymbol\phi]\right)\right|_{\boldsymbol\phi=\boldsymbol\phi_{0}}\nonumber \\ + &\approx& \left.\boldsymbol\phi + \alpha \mbox{\bf g}[\boldsymbol\phi] + \alpha^2\left(\mbox{\bf g}_{1}[\boldsymbol\phi] +\frac{1}{2}\frac{\partial \mbox{\bf g}[\boldsymbol\phi]}{\partial \boldsymbol\phi}\mbox{\bf g}[\boldsymbol\phi]\right)\right|_{\boldsymbol\phi=\boldsymbol\phi_{0}}, + \end{eqnarray} + +\begin{eqnarray} + \mbox{\bf g}_{1}[\boldsymbol\phi] = -\frac{1}{2}\frac{\partial \mbox{\bf g}[\boldsymbol\phi]}{\partial \boldsymbol\phi}\mbox{\bf g}[\boldsymbol\phi]. + \end{eqnarray} + +\begin{eqnarray}\label{eq:reg_imp_proof1} + \frac{d\boldsymbol\phi}{dt} &\approx& \mbox{\bf g}[\boldsymbol\phi] +\alpha \mbox{\bf g}_{1}[\boldsymbol\phi] \nonumber \\ + &=& -\frac{\partial L}{\partial \boldsymbol\phi}-\frac{\alpha}{2}\left(\frac{\partial^2 L}{\partial \boldsymbol\phi^2}\right)\frac{\partial L}{\partial \boldsymbol\phi}. + \end{eqnarray} + +\begin{eqnarray}\label{eq:reg_imp_proof2} + L_{GD}[\boldsymbol\phi] = L[\boldsymbol\phi] + \frac{\alpha}{4} \left\lVert\frac{\partial L}{\partial \boldsymbol\phi} \right\rVert^{2}, + \end{eqnarray} + +\begin{eqnarray} + Pr(\boldsymbol\phi) = \prod_{j=1}^{J}\mbox{Norm}_{\phi_j}[0, \sigma^{2}_{\boldsymbol\phi}], + \end{eqnarray} + +\begin{eqnarray} + \boldsymbol\phi \longleftarrow (1-\lambda) \boldsymbol\phi - \alpha \frac{\partial L}{\partial \boldsymbol\phi}, + \end{eqnarray} + +\begin{eqnarray} + \tilde{L}[\boldsymbol\phi] = L[\boldsymbol\phi] + \frac{\lambda}{2\alpha} \sum_{k}\phi_{k}^2, + \end{eqnarray} + +\chapter{Convolutional networks} + +\begin{eqnarray} + \mbox{\bf f}\bigl[\mbox{\bf t}[\mathbf{x}]\bigr] = \mbox{\bf f}[\mathbf{x}]. + \end{eqnarray} + +\begin{eqnarray} + \mbox{\bf f}\bigl[\mbox{\bf t}[\mathbf{x}]\bigr] = \mbox{\bf t}\bigl[\mbox{\bf f}[\mathbf{x}]\bigr]. + \end{eqnarray} + +\begin{eqnarray}\label{eq:conv_kernel_3a} + z_{i} = \omega_{1}x_{i-1}+\omega_{2}x_{i} + \omega_{3}x_{i+1}, + \end{eqnarray} + +\begin{eqnarray}\label{eq:conv_kernel_3} + h_{i} &=& \mbox{a}\left[\beta + \omega_{1}x_{i-1}+\omega_{2}x_{i} + \omega_{3}x_{i+1}\right]\nonumber\\ + &=& \mbox{a}\left[\beta + \sum_{j=1}^{3} \omega_{j} x_{i+j-2}\right], + \end{eqnarray} + +\begin{eqnarray}\label{eq:conv_fully} + h_{i} &=& \mbox{a}\left[\beta_{i} + \sum_{j=1}^{D} \omega_{ij} x_{j}\right]. + \end{eqnarray} + +\begin{eqnarray}\label{eq:conv_kernel_2d} + h_{ij} &=& \mbox{a}\left[\beta + \sum_{m=1}^{3}\sum_{n=1}^{3} \omega_{mn} x_{i+m-2,j+n-2}\right], + \end{eqnarray} + + \chapter{Residual networks} + +\begin{eqnarray}\label{eq:residual_sequential} + \mbox{\bf h}_1 &=& \mbox{\bf f}_{1}[\mathbf{x}, + \boldsymbol\phi_{1}]\nonumber \\ + \mbox{\bf h}_2 &=& \mbox{\bf f}_{2}[\mathbf{h}_{1},\boldsymbol\phi_{2}]\nonumber \\ + \mbox{\bf h}_3 &=& \mbox{\bf f}_{3}[\mathbf{h}_{2},\boldsymbol\phi_{3}]\nonumber \\ + \mathbf{y} &=& \mbox{\bf f}_{4}[\mathbf{h}_{3},\boldsymbol\phi_{4}], + \end{eqnarray} + +\begin{eqnarray} + \mathbf{y}= \mbox{\bf f}_{4}\biggl[\mbox{\bf f}_{3}\Bigl[\mbox{\bf f}_{2}\bigl[\mbox{\bf f}_{1}[\mathbf{x},\boldsymbol\phi_{1}],\boldsymbol\phi_{2}\bigr],\boldsymbol\phi_{3}\Bigr],\boldsymbol\phi_{4}\biggr]. + \end{eqnarray} + +\begin{eqnarray}\label{eq:residual_chain} + \frac{\partial \mathbf{y}}{\partial \mathbf{f}_1} = \frac{\partial \mathbf{f}_{2}}{\partial \mathbf{f}_{1}}\frac{\partial \mathbf{f}_{3}}{\partial \mathbf{f}_{2}}\frac{\partial \mathbf{f}_{4}}{\partial \mathbf{f}_{3}}. + \end{eqnarray} + +\begin{eqnarray}\label{eq:residual_example} + \mathbf{h}_{1} &=& \mathbf{x} + \mbox{\bf f}_{1}[\mathbf{x},\boldsymbol\phi_{1}]\nonumber \\ + \mathbf{h}_{2} &=& \mathbf{h}_{1} + \mbox{\bf f}_{2}[\mathbf{h}_{1},\boldsymbol\phi_{2}]\nonumber \\ + \mathbf{h}_3 &=& \mathbf{h}_{2} + \mbox{\bf f}_{3}[\mathbf{h}_{2},\boldsymbol\phi_{3}]\nonumber \\ + \mathbf{y} &=& \mathbf{h}_{3} + \mbox{\bf f}_{4}[\mathbf{h}_{3}, \boldsymbol\phi_{4}], + \end{eqnarray} + +\begin{eqnarray}\label{eq:residual_substitute} + &\mathbf{y} = \mathbf{x} &\hspace{-2mm}+\hspace{2mm} \mbox{\bf f}_{1}[\mathbf{x}] \\ + &&\hspace{-2mm}+\hspace{2mm} \mbox{\bf f}_{2}\bigl[\mathbf{x} + \mbox{\bf f}_{1}[\mathbf{x}]\bigr] \nonumber \\ + &&\hspace{-2mm}+\hspace{2mm} \mbox{\bf f}_{3}\Bigl[\mathbf{x} + \mbox{\bf f}_{1}[\mathbf{x}] + \mbox{\bf f}_{2}\bigl[\mathbf{x} + \mbox{\bf f}_{1}[\mathbf{x}]\bigr]\Bigr]\nonumber\\ + &&\hspace{-2mm}+\hspace{2mm}\mbox{\bf f}_{4}\biggl[\mathbf{x} + \mbox{\bf f}_{1}[\mathbf{x}] + \mbox{\bf f}_{2}\bigl[\mathbf{x} + \mbox{\bf f}_{1}[\mathbf{x}]\bigr]+ \mbox{\bf f}_{3}\Bigl[\mathbf{x} + \mbox{\bf f}_{1}[\mathbf{x}] + \mbox{\bf f}_{2}\bigl[\mathbf{x} + \mbox{\bf f}_{1}[\mathbf{x}]\bigr]\Bigr]\biggr],\nonumber + \end{eqnarray} + +\begin{eqnarray}\label{eq:residual_deriv} + \frac{\partial \mathbf{y}}{\partial \mbox{\bf f}_{1}} = \mathbf{I} + \frac{\partial \mathbf{f}_{2}}{\partial \mathbf{f}_1} + \left(\frac{\partial \mathbf{f}_{3}}{\partial \mathbf{f}_1}+ \frac{\partial \mathbf{f}_{2}}{\partial \mathbf{f}_1}\frac{\partial \mathbf{f}_{3}}{\partial \mathbf{f}_2} \right) + + \left(\frac{\partial \mathbf{f}_{4}}{\partial \mathbf{f}_1} + + \frac{\partial \mathbf{f}_{2}}{\partial \mathbf{f}_1}\frac{\partial \mathbf{f}_{4}}{\partial \mathbf{f}_2} + + \frac{\partial \mathbf{f}_{3}}{\partial \mathbf{f}_1}\frac{\partial \mathbf{f}_{4}}{\partial \mathbf{f}_3} + + \frac{\partial \mathbf{f}_{2}}{\partial \mathbf{f}_1}\frac{\partial \mathbf{f}_{3}}{\partial \mathbf{f}_2}\frac{\partial \mathbf{f}_{4}}{\partial \mathbf{f}_3}\right), + \end{eqnarray} + +\begin{eqnarray} + m_{h} &=& \frac{1}{|\mathcal{B}|} \sum_{i\in\mathcal{B}} h_{i}\nonumber \\ + s_{h} &=& \sqrt{\frac{1}{|\mathcal{B}|} \sum_{i\in\mathcal{B}} (h_{i}-m_{h})^2}, + \end{eqnarray} + +\begin{eqnarray}\label{eq:residual_bn_apply} + h_i \leftarrow \frac{h_i-m_{h}}{s_{h}+\epsilon}\hspace{2cm}\forall i\in\mathcal{B}, + \end{eqnarray} + +\begin{eqnarray} + h_i \leftarrow \gamma h_i + \delta\hspace{2cm}\forall i\in\mathcal{B}. + \end{eqnarray} + +\begin{eqnarray}\label{eq:residual_prob_forward} + \begin{split} + f_{1} &= \mathbb{E}[z_{i}]\\ + f_{2i} &= z_{i} - f_{1}\\ + f_{3i} &= f_{2i}^2\\ + f_{4} &= \mathbb{E}[f_{3i}] \\ + \end{split} + \qquad\qquad\qquad + \begin{split} + f_{5} &= \sqrt{f_{4}+\epsilon} \\ + f_{6} &= 1/f_{5}\\ + f_{7i} &= f_{2i}\times f_{6} \\ + z_{i}' &= f_{7i} \times \gamma + \delta, + \end{split} + \end{eqnarray} + +\chapter{Transformers} + + +\begin{eqnarray} + \mbox{{\bf f}}[\mathbf{x}] = \mbox{{\bf ReLU}}[\boldsymbol\beta +\boldsymbol\Omega\mathbf{x}], + \end{eqnarray} + + + +\begin{eqnarray}\label{eq:transformer_values} + \mathbf{v}_{m} = \boldsymbol\beta_{v}+\boldsymbol\Omega_{v}\mathbf{x}_{m}, + \end{eqnarray} + +\begin{eqnarray}\label{eq:transformer_sattention1} + \mbox{{\bf sa}}_{n}[\mathbf{x}_{1},\ldots, \mathbf{x}_{N}] = \sum_{m=1}^{N}a[\mathbf{x}_{m}, \mathbf{x}_{n}]\mathbf{v}_{m}. + \end{eqnarray} + +\begin{eqnarray} + \mathbf{q}_{n} &=& \boldsymbol\beta_{q}+\boldsymbol\Omega_{q}\mathbf{x}_{n}\nonumber \\ + \mathbf{k}_{m} &=& \boldsymbol\beta_{k}+\boldsymbol\Omega_{k}\mathbf{x}_{m}, + \end{eqnarray} + +\begin{eqnarray}\label{eq:transformer_sattention2} + a[\mathbf{x}_{m},\mathbf{x}_{n}] &=& \mbox{softmax}_{m}\left[\mathbf{k}_{\bullet}^{T}\mathbf{q}_{n}\right]\nonumber\\ + &=& \frac{\exp\left[\mathbf{k}_{m}^{T}\mathbf{q}_{n}\right]}{\sum_{m'=1}^{N}\exp\left[\mathbf{k}_{m'}^{T}\mathbf{q}_{n} \right]}, + \end{eqnarray} + +\begin{eqnarray} + \mathbf{V}[\mathbf{X}] &=& \boldsymbol\beta_{v}\mathbf{1}^{T}+\boldsymbol\Omega_{\textit{v}}\mathbf{X}\nonumber \\ + \mathbf{Q}[\mathbf{X}] &=& \boldsymbol\beta_{q}\mathbf{1}^{T}+\boldsymbol\Omega_{\textit{q}}\mathbf{X}\nonumber \\ + \mathbf{K}[\mathbf{X}] &=& \boldsymbol\beta_{k}\mathbf{1}^{T}+\boldsymbol\Omega_{\textit{k}}\mathbf{X}, + \end{eqnarray} + +\begin{eqnarray} + \mbox{{\bf Sa}}[\mathbf{X}] =\mathbf{V}[\mathbf{X}]\cdot\mbox{\bf Softmax}\Bigl[\mathbf{K}[\mathbf{X}]^{T}\mathbf{Q}[\mathbf{X}]\Bigr], + \end{eqnarray} + +\begin{eqnarray}\label{eq:transformer_sa_matrix} + \mbox{{\bf Sa}}[\mathbf{X}] =\mathbf{V}\cdot\mbox{\bf Softmax}\Bigl[\mathbf{K}^{T}\mathbf{Q}\Bigr]. + \end{eqnarray} + +\begin{eqnarray} + \mbox{{\bf Sa}}[\mathbf{X}] =\mathbf{V}\cdot\mbox{\bf Softmax}\left[\frac{\mathbf{K}^{T}\mathbf{Q}}{\sqrt{D}_{q}}\right]. + \end{eqnarray} + +\begin{eqnarray} + \mathbf{V}_h &=& \boldsymbol\beta_{vh}\mathbf{1}^{T}+\boldsymbol\Omega_{\mathit{vh}}\mathbf{X}\nonumber \\ + \mathbf{Q}_h &=& \boldsymbol\beta_{qh}\mathbf{1}^{T}+\boldsymbol\Omega_{\mathit{qh}}\mathbf{X}\nonumber \\ + \mathbf{K}_h &=& \boldsymbol\beta_{kh}\mathbf{1}^{T}+\boldsymbol\Omega_{\mathit{kh}}\mathbf{X}. + \end{eqnarray} + +\begin{eqnarray} + \mbox{{\bf Sa}}_{h}[\mathbf{X}] =\mathbf{V}_h\cdot\mbox{\bf Softmax}\left[\frac{\mathbf{K}_h^{T}\mathbf{Q}_h}{\sqrt{D}_{q}}\right], + \end{eqnarray} + +\begin{eqnarray} + \mbox{{\bf MhSa}}[\mathbf{X}] = \boldsymbol\Omega_{c}\Bigl[\mbox{{\bf Sa}}_{1}[\mathbf{X}]^T,\mbox{{\bf Sa}}_{2}[\mathbf{X}]^T,\ldots,\mbox{{\bf Sa}}_{H}[\mathbf{X}]^T \Bigr]^T. + \end{eqnarray} + +\begin{eqnarray} + \mathbf{X} &\leftarrow& \mathbf{X} + \mbox{\bf{MhSa}}[\mathbf{X}] \nonumber \\ + \mathbf{X} &\leftarrow& \mbox{\bf{LayerNorm}}[\mathbf{X}] \hspace{3cm}\nonumber\\ + \mathbf{x}_{n} &\leftarrow& \mathbf{x}_{n}+\mbox{\bf{mlp}}[\mathbf{x}_{n}] \hspace{3.6cm}\forall\; n\in\{1,\ldots, N\}\nonumber\\ + \mathbf{X} &\leftarrow& \mbox{\bf{LayerNorm}}[\mathbf{X}], + \end{eqnarray} + +\begin{eqnarray} + Pr(\mbox{\textcolor{red}{It takes great courage to let yourself appear weak}}) &=&\nonumber\\ + &&\hspace{-8cm}Pr(\mbox{\textcolor{red}{It}})\times Pr(\mbox{\textcolor{red}{takes}}|\mbox{\textcolor{red}{It}})\times Pr(\mbox{\textcolor{red}{great}}|\mbox{\textcolor{red}{It takes}})\times Pr(\mbox{\textcolor{red}{courage}}|\mbox{\textcolor{red}{It takes great}})\times\nonumber \\ + &&\hspace{-8cm}Pr(\mbox{\textcolor{red}{to}}|\mbox{\textcolor{red}{It takes great courage}})\times Pr(\mbox{\textcolor{red}{let}}|\mbox{\textcolor{red}{It takes great courage to}})\times\nonumber\\ + &&\hspace{-8cm}Pr(\mbox{\textcolor{red}{yourself}}|\mbox{\textcolor{red}{It takes great courage to let}})\times\nonumber\\ + &&\hspace{-8cm}Pr(\mbox{\textcolor{red}{appear}}|\mbox{\textcolor{red}{It takes great courage to let yourself}})\times\nonumber\\ + &&\hspace{-8cm}Pr(\mbox{\textcolor{red}{weak}}|\mbox{\textcolor{red}{It takes great courage to let yourself appear}}). + \end{eqnarray} + +\begin{eqnarray}\label{eq:transformer_autoregressive} + Pr(t_{1},t_{2},\ldots, t_{N}) = Pr(t_{1})\prod_{n=2}^{N}Pr(t_{n}|t_{1},\ldots, t_{n-1}). + \end{eqnarray} + +\begin{eqnarray}\label{eq:transformer_position_quad} + \mbox{{\bf Sa}}[\mathbf{X}] =\mathbf{V}\cdot\mbox{\bf Softmax}\left[\frac{\mathbf{K}^{T}\mathbf{Q}}{\sqrt{D}_{q}}\right], + \end{eqnarray} + +\begin{eqnarray} + \mathbf{V} &=& \boldsymbol\beta_{v}\mathbf{1}^{T}+\boldsymbol\Omega_{\textit{v}}\mathbf{X}\nonumber \\ + \mathbf{Q} &=& \boldsymbol\beta_{q}\mathbf{1}^{T}+\boldsymbol\Omega_{\textit{q}}(\mathbf{X}+\boldsymbol\Pi)\nonumber \\ + \mathbf{K} &=& \boldsymbol\beta_{k}\mathbf{1}^{T}+\boldsymbol\Omega_{\textit{k}}(\mathbf{X}+\boldsymbol\Pi). + \end{eqnarray} + + + +\begin{eqnarray} + \mbox{\bf Sa}[\mathbf{X}\mathbf{P}] = \mbox{\bf Sa}[\mathbf{X}]\mathbf{P}. + \end{eqnarray} + +\begin{eqnarray} + y_{i} = \mbox{softmax}_{i}[\mathbf{z}] = \frac{\exp[z_{i}]}{\sum_{j=1}^{5}\exp[z_{j}]}, + \end{eqnarray} + +\begin{eqnarray} + a[\mathbf{x}_{m},\mathbf{x}_{n}] = \mbox{softmax}_{m}\left[\mathbf{k}_{\bullet}^T\mathbf{q}_{n}\right] + = \frac{\exp\left[\mathbf{k}_m^T\mathbf{q}_{n}\right]}{\sum_{m'=1}^{N}\exp\left[\mathbf{k}_{m'}^{T}\mathbf{q}_{n} \right]}. + \end{eqnarray} + +\chapter{Graph neural networks} + +\begin{eqnarray} + \mathbf{X}' &=& \mathbf{X}\mathbf{P}\nonumber \\ + \mathbf{A}' &=& \mathbf{P}^T\mathbf{A}\mathbf{P}, + \end{eqnarray} + +\begin{eqnarray}\label{eq:graph_graph_level} + Pr(y=1|\mathbf{X},\mathbf{A}) = \mbox{sig}\left[\beta_{K}+\boldsymbol\omega_{K}\mathbf{H}_{K}\mathbf{1}/N\right], + \end{eqnarray} + +\begin{eqnarray}\label{eq:graph_node_level} + Pr(y^{(n)}=1|\mathbf{X},\mathbf{A}) = \mbox{sig}\left[\beta_{K}+\boldsymbol\omega_{K}\mathbf{h}^{(n)}_{K}\right]. + \end{eqnarray} + +\begin{eqnarray} + Pr(y^{(mn)}=1|\mathbf{X},\mathbf{A}) = \mbox{sig}\left[\mathbf{h}^{(m)T}\mathbf{h}^{(n)}\right]. + \end{eqnarray} + +\begin{eqnarray} + \mathbf{H}_1 &=& \mbox{\bf F}[\mathbf{X},\mathbf{A},\boldsymbol\phi_0]\nonumber \\ + \mathbf{H}_2 &=& \mbox{\bf F}[\mathbf{H}_1,\mathbf{A},\boldsymbol\phi_1]\nonumber \\ + \mathbf{H}_3 &=& \mbox{\bf F}[\mathbf{H}_2,\mathbf{A},\boldsymbol\phi_2]\nonumber \\ + \vdots &=& \vdots \nonumber \\ + \mathbf{H}_K &=& \mbox{\bf F}[\mathbf{H}_{K-1},\mathbf{A},\boldsymbol\phi_{K-1}], + \end{eqnarray} + +\begin{eqnarray} + \mathbf{H}_{k+1}\mathbf{P} =\mbox{\bf F}[\mathbf{H}_k\mathbf{P},\mathbf{P}^{T}\mathbf{A}\mathbf{P},\boldsymbol\phi_k]. + \end{eqnarray} + +\begin{eqnarray}\label{eq:graph_output_invar} + y = \mbox{sig}\left[\beta_{K}+\boldsymbol\omega_{K}\mathbf{H}_{K}\mathbf{1}/N\right] = \mbox{sig}\left[\beta_{K}+\boldsymbol\omega_{K}\mathbf{H}_{K}\mathbf{P}\mathbf{1}/N\right], + \end{eqnarray} + +\begin{eqnarray} + \mbox{\bf agg}[n,k] = \sum_{m\in\mbox{ne}[n]}\mathbf{h}_k^{(m)}, + \end{eqnarray} + +\begin{eqnarray} + \mathbf{h}^{(n)}_{k+1} = \mbox{\bf a}\left[\boldsymbol\beta_{k} + \boldsymbol\Omega_{k} \cdot \mathbf{h}^{(n)}_{k} +\boldsymbol\Omega_{k} \cdot \mbox{\bf agg}[n,k]\right]. + \end{eqnarray} + +\begin{eqnarray}\label{eq:graph_concat_update} + \mathbf{H}_{k+1} &=& \mbox{\bf a}\left[\boldsymbol\beta_{k}\mathbf{1}^{T} + \boldsymbol\Omega_{k}\mathbf{H}_{k} +\boldsymbol\Omega_{k} \mathbf{H}_{k}\mathbf{A} \right]\nonumber \\ + & =& \mbox{\bf a}\left[\boldsymbol\beta_{k}\mathbf{1}^{T} +\boldsymbol\Omega_{k} \mathbf{H}_{k}(\mathbf{A}+\mathbf{I}) \right], + \end{eqnarray} + +\begin{eqnarray}\label{eq:graph_simple_example} + \mathbf{H}_{1} &=& \mbox{\bf a}\left[\boldsymbol\beta_{0}\mathbf{1}^{T}+\boldsymbol\Omega_{0}\mathbf{X}(\mathbf{A}+\mathbf{I})\right]\nonumber \\ + \mathbf{H}_{2} &=& \mbox{\bf a}\left[\boldsymbol\beta_{1}\mathbf{1}^{T} + \boldsymbol\Omega_{1}\mathbf{H}_{1}(\mathbf{A}+\mathbf{I}) \right]\nonumber \\ + \vdots &=& \vdots \nonumber \\ + \mathbf{H}_{K} &=& \mbox{\bf a}\left[\boldsymbol\beta_{K-1}\mathbf{1}^{T} + \boldsymbol\Omega_{K-1}\mathbf{H}_{k-1}(\mathbf{A}+\mathbf{I}) \right]\nonumber \\ + \mbox{f}[\mathbf{X},\mathbf{A},\boldsymbol\Phi] &=& \mbox{sig}\left[\beta_{K}+\boldsymbol\omega_{K}\mathbf{H}_{K}\mathbf{1}/N\right], + \end{eqnarray} + +\begin{eqnarray}\label{eq:graph_node_matrix} + \mbox{\bf f}[\mathbf{X},\mathbf{A},\boldsymbol\Phi] = \mbox{\bf sig} + \left[\beta_{K}\mathbf{1}^{T}+\boldsymbol\omega_K\mathbf{H}_K\right], + \end{eqnarray} + +\begin{eqnarray} + \mathbf{H}_{k+1} = \mbox{\bf a}\Bigl[\boldsymbol\beta_{k}\mathbf{1}^{T} + \boldsymbol\Omega_{k} \mathbf{H}_{k}(\mathbf{A}+\mathbf{I}) \Bigr]. + \end{eqnarray} + +\begin{eqnarray} + \mathbf{H}_{k+1} = \mbox{\bf a}\Bigl[\boldsymbol\beta_{k}\mathbf{1}^{T} + \boldsymbol\Omega_{k} \mathbf{H}_{k}(\mathbf{A}+ (1+\epsilon_{k})\mathbf{I}) \Bigr]. + \end{eqnarray} + +\begin{eqnarray} + \mathbf{H}_{k+1} &=& \mbox{\bf a}\left[\boldsymbol\beta_{k}\mathbf{1}^{T} + \boldsymbol\Omega_{k} \mathbf{H}_{k}\mathbf{A} + \boldsymbol\Psi_{k}\mathbf{H}_k \right]\nonumber \\ + &=& \mbox{\bf a}\left[\boldsymbol\beta_{k}\mathbf{1}^{T} + \begin{bmatrix}\boldsymbol\Omega_{k}&\boldsymbol\Psi_{k}\end{bmatrix} \begin{bmatrix}\mathbf{H}_{k}\mathbf{A}\\ \mathbf{H}_{k}\end{bmatrix} \right]\nonumber \\ + &=& \mbox{\bf a}\left[\boldsymbol\beta_{k}\mathbf{1}^{T} +\boldsymbol\Omega_{k}' \begin{bmatrix}\mathbf{H}_{k}\mathbf{A}\\\ \mathbf{H}_{k}\end{bmatrix} \right], + \end{eqnarray} + +\begin{eqnarray}\label{eq:graph_residual} + \mathbf{H}_{k+1} = \begin{bmatrix}\mbox{\bf a}\left[\boldsymbol\beta_{k}\mathbf{1}^{T} + \boldsymbol\Omega_{k} \mathbf{H}_{k}\mathbf{A} \right]\\\mathbf{H}_{k}\end{bmatrix}. + \end{eqnarray} + +\begin{eqnarray}\label{eq:graph_mean_agg} + \mbox{\bf agg}[n] = \frac{1}{|\mbox{ne}[n]|}\sum_{m\in\mbox{ne}[n]}\mathbf{h}_{m}, + \end{eqnarray} + +\begin{eqnarray} + \mathbf{H}_{k+1} = \mbox{\bf a}\Bigl[\boldsymbol\beta_{k}\mathbf{1}^{T} + \boldsymbol\Omega_{k} \mathbf{H}_{k}(\mathbf{A}\mathbf{D}^{-1}+\mathbf{I}) \Bigr]. + \end{eqnarray} + +\begin{eqnarray}\label{eq:graph_kipf_normalization} + \mbox{\bf agg}[n] = \sum_{m\in\mbox{ne}[n]}\frac{\mathbf{h}_{m}}{\sqrt{|\mbox{ne}[n]||\mbox{ne}[m]|}}, + \end{eqnarray} + +\begin{eqnarray} + \mathbf{H}_{k+1} = \mbox{\bf a}\Bigl[\boldsymbol\beta_{k}\mathbf{1}^{T} + \boldsymbol\Omega_{k} \mathbf{H}_{k}(\mathbf{D}^{-1/2}\mathbf{A}\mathbf{D}^{-1/2}+\mathbf{I}) \Bigr]. + \end{eqnarray} + +\begin{eqnarray} + \mbox{\bf agg}[n] =\mathop{\rm \bf max}_{m\in\mbox{ne}[n]}\bigl[\mathbf{h}_{m}\bigr], + \end{eqnarray} + +\begin{eqnarray} + \mathbf{H}'_{k} = \boldsymbol\beta_k\mathbf{1}^{T} + \boldsymbol\Omega_k\mathbf{H}_k. + \end{eqnarray} + +\begin{eqnarray} + s_{mn} = \mbox{a}\left[\boldsymbol\phi_k^{T}\begin{bmatrix}\mathbf{h}'_{m}\\\mathbf{h}'_{n}\end{bmatrix}\right]. + \end{eqnarray} + +\begin{eqnarray} + \mathbf{H}_{k+1} = \mbox{\bf a}\Bigl[\mathbf{H}'_{k}\cdot\mbox{\bf Softmask}[\mathbf{S}, \mathbf{A}+\mathbf{I}] \Bigr], + \end{eqnarray} + +\begin{eqnarray}\label{eq:gnn} + \mathbf{h}_{n} \leftarrow \mbox{\bf f}\left[\mathbf{x}_{n},\mathbf{x}_{m\in\mbox{ne}[n]},\mathbf{e}_{e\in\mbox{nee}[n]},\mathbf{h}_{m\in\mbox{ne}[n]},\boldsymbol\phi\right], + \end{eqnarray} + +\begin{eqnarray} + \mathbf{h}_{k+1}^{(n)} = \mbox{\bf mlp}\left[\left(1+\epsilon_{k}\right) \mathbf{h}_{k}^{(n)} +\sum_{m\in\mbox{ne}[n]}\mathbf{h}_{k}^{(m)} \right]. + \end{eqnarray} + +\begin{eqnarray*} + \mathbf{A}_{1} = \begin{bmatrix} + 0 & 1 & 1 & 0 & 0 & 0 & 0\\ + 1 & 0 & 0 & 1 & 1 & 1 & 0\\ + 1 & 0 & 0 & 0 & 0 & 1 & 1\\ + 0 & 1 & 0 & 0 & 0 & 1 & 1\\ + 0 & 1 & 0 & 0 & 0 & 0 & 1\\ + 0 & 1 & 1 & 1 & 0 & 0 & 0\\ + 0 & 0 & 1 & 1 & 1 & 0 & 0 + \end{bmatrix} \quad\quad \mbox{and} \quad\quad \mathbf{A}_{2} = \begin{bmatrix} + 0 & 0 & 1 & 1 & 0 & 0 & 1 \\ + 0 & 0 & 1 & 1 & 1 & 0 & 0\\ + 1 & 1 & 0 & 0 & 0 & 0 & 0\\ + 1 & 1 & 0 & 0 & 1 & 1 & 1\\ + 0 & 1 & 0 & 1 & 0 & 0 & 1\\ + 0 & 0 & 0 & 1 & 0 & 0 & 1\\ + 1 & 0 & 0 & 1 & 1 & 1 & 0 + \end{bmatrix}. + \end{eqnarray*} + +\begin{eqnarray} + \mbox{sig}\left[\beta_{K}+\boldsymbol\omega_{K}\mathbf{H}_{K}\mathbf{1}\right] = \mbox{sig}\left[\beta_{K}+\boldsymbol\omega_{K}\mathbf{H}_{K}\mathbf{P}\mathbf{1}\right], + \end{eqnarray} + +\begin{eqnarray} + \mathbf{H}_{k+1} &=& \mbox{GraphLayer}[\mathbf{H}_{k},\mathbf{A}] \nonumber \\ + &=& \mbox{\bf a}\left[\boldsymbol\beta_{k}\mathbf{1}^{T} + \boldsymbol\Omega_{k}\begin{bmatrix}\mathbf{H}_{k}\\ \mathbf{H}_{k}\mathbf{A} \end{bmatrix}\right], + \end{eqnarray} + +\begin{eqnarray} + \mbox{GraphLayer}[\mathbf{H}_{k},\mathbf{A}]\mathbf{P} = \mbox{GraphLayer}[\mathbf{H}_{k}\mathbf{P},\mathbf{P}^{T}\mathbf{A}\mathbf{P}], + \end{eqnarray} + +\begin{eqnarray} + \mbox{\bf agg}[n] = \frac{1}{1+|\mbox{ne}[n]|}\left(\mathbf{h}_{n}+\sum_{m\in\mbox{ne}[n]}\mathbf{h}_{m}\right). + \end{eqnarray} + + \chapter{Unsupervised learning} + +\begin{eqnarray} + L[\boldsymbol\phi] = -\sum_{i=1}^{I}\log\Bigl[Pr(\mathbf{x}_{i}|\boldsymbol\phi)\Bigr]. + \end{eqnarray} + +\begin{eqnarray} + IS &=& \exp\left[\frac{1}{I}\sum_{i=1}^{I}D_{KL}\Bigl[Pr(y|\mathbf{x}^*_{i})||Pr(y) \Bigr]\right], + \end{eqnarray} + +\begin{eqnarray} + Pr(y) = \frac{1}{I}\sum_{i=1}^I Pr(y|\mathbf{x}^*_i). + \end{eqnarray} + +\chapter{Generative adversarial networks} + +\begin{eqnarray} + x_j^{*}= \mbox{g}[z_j,\theta] = z_j+\theta, + \end{eqnarray} + +\begin{eqnarray} + \hat{\boldsymbol\phi} = \mathop{\rm argmin}_{\boldsymbol\phi} \left[\sum_{i}-(1-y_{i})\log\Bigl[1-\mbox{sig}[\mbox{f}[\mathbf{x}_i,\boldsymbol\phi]]\Bigr] - y_{i}\log\Bigl[\mbox{sig}[\mbox{f}[\mathbf{x}_i,\boldsymbol\phi]]\Bigr]\right], + \end{eqnarray} + +\begin{eqnarray}\label{eq:gan_cost_fun_sum0} + \hat{\boldsymbol\phi} = \mathop{\rm argmin}_{\boldsymbol\phi} \left[\sum_{j}-\log\Bigl[1-\mbox{sig}[\mbox{f}[\mathbf{x}^*_j,\boldsymbol\phi]]\Bigr] - \sum_{i}\log\Bigl[\mbox{sig}[\mbox{f}[\mathbf{x}_i,\boldsymbol\phi]]\Bigr]\right], + \end{eqnarray} + +\begin{eqnarray}\label{eq:gan_cost_fn_sum} + \hat{\boldsymbol\theta} = \mathop{\rm argmax}_{\boldsymbol\theta}\!\left[\min_{\boldsymbol\phi}\!\left[\sum_{j}-\log\Bigl[1\!-\!\mbox{sig}[\mbox{f}[\mbox{\bf g}[\mathbf{z}_{j},\boldsymbol\theta] ,\boldsymbol\phi]]\Bigr] \!- \!\sum_{i}\log\Bigl[\mbox{sig}[\mbox{f}[\mathbf{x}_i,\boldsymbol\phi]]\Bigr]\right]\right]\!\!. + \end{eqnarray} + +\begin{eqnarray}\label{eq:gan_cost_fun_sumtwo} + L[\boldsymbol\phi] &=& \sum_{j} -\log\Bigl[1-\mbox{sig}[\mbox{f}[\mbox{\bf g}[\mathbf{z}_{j},\boldsymbol\theta] ,\boldsymbol\phi]]\Bigr] -\sum_{i}\log\Bigl[\mbox{sig}[\mbox{f}[\mathbf{x}_i,\boldsymbol\phi]]\Bigr]\nonumber\\ + L[\boldsymbol\theta] &=& \sum_{j} \log\Bigl[1-\mbox{sig}[\mbox{f}[\mbox{\bf g}[\mathbf{z}_{j},\boldsymbol\theta] ,\boldsymbol\phi]]\Bigr], + \end{eqnarray} + +\begin{eqnarray}\label{eq:gan_integral} + L[\boldsymbol\phi] &\!=\!& -\frac{1}{J}\sum_{j=1}^J\biggr(\log\Bigl[1-\mbox{sig}[\mbox{f}[\mathbf{x}^{*}_j,\boldsymbol\phi]]\Bigr]\biggr) -\frac{1}{I}\sum_{i=1}^I\biggl(\log\Bigl[\mbox{sig}[\mbox{f}[\mathbf{x}_i,\boldsymbol\phi]]\Bigr]\biggr) \\ + &\!\approx\!&-\mathbb{E}_{\mathbf{x}^*}\biggl[\log\Bigl[1-\mbox{sig}[\mbox{f}[\mathbf{x}^* ,\boldsymbol\phi]]\Bigr]\biggr] -\mathbb{E}_{\mathbf{x}}\biggl[\log\Bigl[\mbox{sig}[\mbox{f}[\mathbf{x},\boldsymbol\phi]]\Bigr]\biggr]\nonumber\\ + &\!=\!&-\int Pr(\mathbf{x}^{*})\log\Bigl[1-\mbox{sig}[\mbox{f}[\mathbf{x}^* ,\boldsymbol\phi]]\Bigr]d\mathbf{x}^* - \int Pr(\mathbf{x})\log\Bigl[\mbox{sig}[\mbox{f}[\mathbf{x},\boldsymbol\phi]]\Bigr]d\mathbf{x}\nonumber, + \end{eqnarray} + +\begin{eqnarray} + Pr(\mbox{real}|\tilde{\mathbf{x}}) = \mbox{sig}\bigl[\mbox{f}[\tilde{\mathbf{x}},\boldsymbol\phi]\bigr] = \frac{Pr(\tilde{\mathbf{x}}|\mbox{real}) }{Pr(\tilde{\mathbf{x}}|\mbox{generated})+Pr(\tilde{\mathbf{x}}|\mbox{real})} =\frac{Pr(\mathbf{x})}{Pr(\mathbf{x}^{*})+Pr(\mathbf{x})}, + \end{eqnarray} + +\begin{eqnarray}\label{eq:gan_integral_opt} + L[\boldsymbol\phi]\!\!&\!=\!&\!\!-\int \!Pr(\mathbf{x}^{*})\log\Bigl[1-\mbox{sig}[\mbox{f}[\mathbf{x}^* ,\boldsymbol\phi]]\Bigr]d\mathbf{x}^* - \int \!Pr(\mathbf{x})\log\Bigl[\mbox{sig}[\mbox{f}[\mathbf{x},\boldsymbol\phi]]\Bigr]d\mathbf{x}\\ + \!\!&\!=\!&\!\!-\int \!Pr(\mathbf{x}^{*})\log\left[1-\frac{Pr(\mathbf{x})}{Pr(\mathbf{x}^{*})+Pr(\mathbf{x})}\right]d\mathbf{x}^* - \int\! Pr(\mathbf{x})\log\left[\frac{Pr(\mathbf{x})}{Pr(\mathbf{x}^{*})+Pr(\mathbf{x})}\right]d\mathbf{x}\nonumber \\ + \!\!&\!=\!&\!\!-\int \!Pr(\mathbf{x}^{*})\log\left[\frac{Pr(\mathbf{x}^{*})}{Pr(\mathbf{x}^{*})+Pr(\mathbf{x})}\right]d\mathbf{x}^* - \int \!Pr(\mathbf{x})\log\left[\frac{Pr(\mathbf{x})}{Pr(\mathbf{x}^{*})+Pr(\mathbf{x})}\right]d\mathbf{x}.\nonumber + \end{eqnarray} + +\begin{eqnarray}\label{eq:gan_js_dist} + D_{JS}\Bigl[Pr(\mathbf{x}^{*})\left|\left|\right.\right.Pr(\mathbf{x})\Bigr] &&\\ &&\hspace{-3cm}=\frac{1}{2}D_{KL}\left[Pr(\mathbf{x}^{*})\left|\left|\frac{Pr(\mathbf{x}^{*})+Pr(\mathbf{x})}{2}\right.\right.\right]+\frac{1}{2}D_{KL}\left[Pr(\mathbf{x})\left|\left|\frac{Pr(\mathbf{x}^{*})+Pr(\mathbf{x})}{2}\right.\right.\right]\nonumber \\ + &&\hspace{-3cm}=\frac{1}{2}\int + \begingroup\color{red} + \underbrace{\color{black} Pr(\mathbf{x}^{*})\log\left[\frac{2Pr(\mathbf{x}^{*})}{Pr(\mathbf{x}^{*})+Pr(\mathbf{x})}\right]d\mathbf{x}^*}_{\mbox{quality}} \endgroup + +\begingroup\color{red} + \underbrace{\color{black}\frac{1}{2}\int Pr(\mathbf{x})\log\left[\frac{2Pr(\mathbf{x})}{Pr(\mathbf{x}^{*})+Pr(\mathbf{x})}\right]d\mathbf{x}}_{\mbox{coverage}}\endgroup .\nonumber + \end{eqnarray} + +\begin{eqnarray}\label{eq:gan_was_primal_discrete} + D_{w}\Bigl[Pr(x)||q(x)\Bigr] = \min_{\mathbf{P}} \left[\sum_{i,j} P_{ij}\cdot|i-j|\right], + \end{eqnarray} + +\begin{eqnarray}\label{eq:gan_primal_constraints} + \begin{tabular}{rrrclcl} + &&$\sum_{j}P_{ij}\!\!$ &$\!\!=\!\!$&~$\!\!\!Pr(x=i)$ &&\mbox{initial distribution of~$Pr(x)$}\\ + &&$\sum_{i}P_{ij}\!\!$ &$\!\!=\!\!$&~$\!\!q(x=j)$ &&\mbox{initial distribution of~$q(x)$}\\ + &&$P_{ij}\!\!$ &\!\!~$\geq\!\!$ &~$\!\!0$ &\hspace{1cm} &\mbox{non-negative masses}. + \end{tabular} + \end{eqnarray} + +\begin{eqnarray}\label{eq:gan_was_dual_discrete} + D_{w}\Bigl[Pr(x)||q(x)\Bigr] = \max_{\mathbf{f}}\left[\sum_{i} Pr(x=i)f_{i} - \sum_{j}q(x=j)f_{j}\right], + \end{eqnarray} + +\begin{eqnarray} + |f_{i+1}-f_{i}| < 1. + \end{eqnarray} + +\begin{eqnarray} + D_{w}\Bigl[Pr(\mathbf{x}),q(\mathbf{x})\Bigr] = \min_{\pi[\bullet,\bullet]}\left[\int\!\!\int \pi(\mathbf{x}_{1},\mathbf{x}_{2})\cdot||\mathbf{x}_{1}-\mathbf{x}_{2}||d\mathbf{x}_{1}d\mathbf{x}_{2}\right], + \end{eqnarray} + +\begin{eqnarray} + D_{w}\Bigl[Pr(\mathbf{x}),q(\mathbf{x})\Bigr] = \max_{\mbox{f}[\mathbf{x}]}\left[\int Pr(\mathbf{x})\mbox{f}[\mathbf{x}]d\mathbf{x} - \int q(\mathbf{x})\mbox{f}[\mathbf{x}]d\mathbf{x}\right], + \end{eqnarray} + +\begin{eqnarray} + L[\boldsymbol\phi] &=& \sum_{j} \mbox{f}[\mathbf{x}_{j}^{*},\boldsymbol\phi] -\sum_{i}\mbox{f}[\mathbf{x}_i,\boldsymbol\phi]\nonumber \\ + &=&\sum_{j} \mbox{f}[\mbox{\bf g}[\mathbf{z}_{j},\boldsymbol\theta],\boldsymbol\phi] -\sum_{i}\mbox{f}[\mathbf{x}_i,\boldsymbol\phi], + \end{eqnarray} + +\begin{eqnarray} + \left| \frac{\partial \mbox{f}[\mathbf{x},\boldsymbol\phi]}{\partial\mathbf{x}}\right| < 1. + \end{eqnarray} + +\begin{eqnarray} + \mathbf{b} =\bigl[Pr(x\!=\!1),Pr(x\!=\!2),Pr(x\!=\!3),Pr(x\!=\!4),q(x\!=\!1),q(x\!=\!2),q(x\!=\!3),q(x\!=\!4)\bigr]^T. + \end{eqnarray} + + + +\begin{eqnarray} + Pr(z) = \begin{cases} 0 & \quad z <0 \\ 1 & \quad 0\leq z\leq 1 \\ 0 & \quad z> 1 \end{cases},\quad \mbox{and} \quad Pr(z) = \begin{cases} 0 & \quad z a \end{cases}. + \end{eqnarray} + + +\begin{eqnarray} + D_{kl} = \log\left[\frac{\sigma_2}{\sigma_1}\right]+ \frac{\sigma_1^2 + (\mu_1 - \mu_2)^2}{2 \sigma_2^2} - \frac{1}{2}, + \end{eqnarray} + +\begin{eqnarray} + D_{w} = (\mu_{1}-\mu_{2})^2 + \sigma_{1}+\sigma_{2}-2\sqrt{\sigma_{1}\sigma_{2}}, + \end{eqnarray} + +\chapter{Normalizing flows} + +\begin{eqnarray}\label{eq:flows_likelihood1d} + Pr(x|\boldsymbol\phi) &=& \left| \frac{\partial \mbox{f}[z,\boldsymbol\phi]}{\partial z}\right|^{-1} \cdot Pr(z), + \end{eqnarray} + + +\begin{eqnarray} + \hat{\boldsymbol\phi} &=& \mathop{\rm argmax}_{\boldsymbol\phi}\left[\prod_{i=1}^{I} Pr(x_{i}|\boldsymbol\phi)\right]\nonumber \\ + &=& \mathop{\rm argmin}_{\boldsymbol\phi}\left[\sum_{i=1}^{I} -\log\Bigl[Pr(x_{i}|\boldsymbol\phi)\Bigr]\right]\nonumber \\ + &=& \mathop{\rm argmin}_{\boldsymbol\phi}\left[\sum_{i=1}^{I} \log\Biggl[\biggl| \frac{\partial \mbox{f}[z_i,\boldsymbol\phi]}{\partial z_i}\biggr|\Biggr]-\log\bigl[Pr(z_i)\bigr] \right], + \end{eqnarray} + +\begin{eqnarray} + Pr(\mathbf{x}|\boldsymbol\phi) = \left| \frac{\partial \mbox{\bf f}[\mathbf{z},\boldsymbol\phi]}{\partial \mathbf{z}}\right|^{-1}\cdot Pr(\mathbf{z}), + \end{eqnarray} + + + +\begin{eqnarray} + \mathbf{x} = \mbox{\bf f}[\mathbf{z},\boldsymbol\phi] =\mbox{\bf f}_{K}\biggl[\mbox{\bf f}_{K-1}\Bigl[\ldots \mbox{\bf f}_{2}\bigl[\mbox{\bf f}_{1}[\mathbf{z},\boldsymbol\phi_{1}],\boldsymbol\phi_{2}\bigr],\ldots\boldsymbol\phi_{K-1}\Bigr],\boldsymbol\phi_{K}\biggr]. + \end{eqnarray} + +\begin{eqnarray}\label{eq:flows_inverse_compositional} + \mathbf{z} = \mbox{\bf f}^{-1}[\mathbf{x},\boldsymbol\phi] =\mbox{\bf f}^{-1}_{1}\biggl[\mbox{\bf f}^{-1}_{2}\Bigl[\ldots \mbox{\bf f}^{-1}_{K-1}\bigl[\mbox{\bf f}^{-1}_{K}[\mathbf{x},\boldsymbol\phi_{K}],\boldsymbol\phi_{K-1}\bigr],\ldots\boldsymbol\phi_{2}\Bigr],\boldsymbol\phi_{1}\biggr]. + \end{eqnarray} + +\begin{eqnarray}\label{eq:flows_network_jac} + \frac{\partial \mbox{\bf f}[\mathbf{z},\boldsymbol\phi]}{\partial \mathbf{z}} = + \frac{\partial \mathbf{f}_{K}[\mathbf{f}_{K-1},\boldsymbol\phi_{K}]}{\partial \mathbf{f}_{K-1}}\cdot + \frac{\partial \mathbf{f}_{K-1}[\mathbf{f}_{K-2},\boldsymbol\phi_{K-1}]}{\partial \mathbf{f}_{K-2}}\ldots + \frac{\partial \mathbf{f}_{2}[\mathbf{f}_{1},\boldsymbol\phi_2]}{\partial \mathbf{f}_{1}}\cdot + \frac{\partial\mathbf{f}_{1}[\mathbf{z},\boldsymbol\phi_1]}{\partial \mathbf{z}}, + \end{eqnarray} + +\begin{eqnarray}\label{eq:flows_network_det} + \left|\frac{\partial \mbox{\bf f}[\mathbf{z},\boldsymbol\phi]}{\partial \mathbf{z}}\right|= + \left|\frac{\partial \mathbf{f}_{K}[\mathbf{f}_{K-1},\boldsymbol\phi_K]}{\partial \mathbf{f}_{K-1}}\right|\cdot + \left|\frac{\partial \mathbf{f}_{K-1}[\mathbf{f}_{K-2},\boldsymbol\phi_{K-1}]}{\partial \mathbf{f}_{K-2}}\right|\ldots + \left|\frac{\partial \mathbf{f}_{2}[\mathbf{f}_{1},\boldsymbol\phi_2]}{\partial \mathbf{f}_{1}}\right|\cdot + \left|\frac{\partial \mathbf{f}_{1}[\mathbf{z},\boldsymbol\phi_1]}{\partial \mathbf{z}}\right|. + \end{eqnarray} + +\begin{eqnarray} + \hat{\boldsymbol\phi} &=& \mathop{\rm argmax}_{\boldsymbol\phi}\Biggl[\prod_{i=1}^{I}Pr(\mathbf{z}_i) \cdot \biggl| \frac{\partial \mbox{\bf f}[\mathbf{z}_i,\boldsymbol\phi]}{\partial \mathbf{z}_i}\biggr|^{-1} \Biggr]\nonumber \\ + &=& \mathop{\rm argmin}_{\boldsymbol\phi}\Biggl[\sum_{i=1}^{I} \log\Biggl[\biggl| \frac{\partial \mbox{\bf f}[\mathbf{z}_i,\boldsymbol\phi]}{\partial \mathbf{z}_i}\biggr| \Biggr] -\log\bigl[Pr(\mathbf{z}_i)\bigr]\Biggr], + \end{eqnarray} + +\begin{eqnarray} + \boldsymbol\Omega = \mathbf{P}\mathbf{L}(\mathbf{U}+\mathbf{D}), + \end{eqnarray} + +\begin{eqnarray} + \mbox{\bf f}[\mathbf{h}] = \Bigl[\mbox{f}[h_{1},\boldsymbol\phi], \mbox{f}[h_{2},\boldsymbol\phi], \ldots \mbox{f}[h_{D},\boldsymbol\phi]\Bigr]^{T}. + \end{eqnarray} + +\begin{eqnarray} + \left|\frac{\partial \mbox{\bf f}[\mathbf{h}]}{\partial \mathbf{h}} \right| = \prod_{d=1}^{D}\left|\frac{\partial \mbox{f}[h_d]}{\partial h_d} \right|. + \end{eqnarray} + +\begin{eqnarray}\label{eq:flow_elementwise} + \mbox{f}[h,\boldsymbol\phi] = \left(\sum_{k=1}^{b-1}\phi_{k}\right)+ (hK-b+1)\phi_{b}, + \end{eqnarray} + +\begin{eqnarray} + \mathbf{h}'_{1}&=& \mathbf{h}_{1}\nonumber \\ + \mathbf{h}'_{2} &=& \mbox{\bf g}\Bigl[\mathbf{h}_{2},\boldsymbol\phi[\mathbf{h}_{1}]\Bigr]. + \end{eqnarray} + +\begin{eqnarray} + \mathbf{h}_{1}&=& \mathbf{h}'_{1}\nonumber \\ + \mathbf{h}_{2}&=& \mbox{\bf g}^{-1}\Bigl[\mathbf{h}'_{2},\boldsymbol\phi[\mathbf{h}_{1}]\Bigr]. + \end{eqnarray} + +\begin{eqnarray} + h'_{d} = \mbox{g}\Bigl[h_{d},\boldsymbol\phi[\mathbf{h}_{1:d-1}]\Bigr]. + \end{eqnarray} + +\begin{eqnarray} + h'_{1} &=& \mbox{g}\Bigl[h_{1},\boldsymbol\phi\Bigr]\nonumber \\ + h'_{2} &=& \mbox{g}\Bigl[h_{2},\boldsymbol\phi[h_{1}]\Bigr]\nonumber \\ + h'_{3} &=& \mbox{g}\Bigl[h_{3},\boldsymbol\phi[h_{1:2}]\Bigr]\nonumber \\ + h'_{4} &=& \mbox{g}\Bigl[h_{4},\boldsymbol\phi[h_{1:3}]\Bigr]. + \end{eqnarray} + +\begin{eqnarray} + h_{1} &=& \mbox{g}^{-1}\Bigl[h'_{1},\boldsymbol\phi\Bigr]\nonumber \\ + h_{2} &=& \mbox{g}^{-1}\Bigl[h'_{2},\boldsymbol\phi[h_{1}]\Bigr]\nonumber \\ + h_{3} &=& \mbox{g}^{-1}\Bigl[h'_{3},\boldsymbol\phi[h_{1:2}]\Bigr]\nonumber \\ + h_{4} &=& \mbox{g}^{-1}\Bigl[h'_{4},\boldsymbol\phi[h_{1:3}]\Bigr]. + \end{eqnarray} + +\begin{eqnarray} + \mathbf{h}'_{1} &=& \mathbf{h}_{1}+ \mbox{\bf f}_{1}[\mathbf{h}_{2},\boldsymbol\phi_{1}]\nonumber \\ + \mathbf{h}'_{2} &=& \mathbf{h}_{2}+ \mbox{\bf f}_{2}[\mathbf{h}'_{1},\boldsymbol\phi_{2}], + \end{eqnarray} + +\begin{eqnarray} + \mathbf{h}_{2}&=& \mathbf{h}'_{2}-\mbox{\bf f}_{2}[\mathbf{h}'_{1},\boldsymbol\phi_{2}]\nonumber \\ + \mathbf{h}_{1}&=& \mathbf{h}'_{1}-\mbox{\bf f}_{1}[\mathbf{h}_{2},\boldsymbol\phi_{1}]. + \end{eqnarray} + +\begin{eqnarray}\label{eq:flow_contraction} + \mbox{dist}\Bigl[\mbox{f}[z^{\prime}],\mbox{f}[z]\Bigr] < \beta\cdot \mbox{dist}\Bigl[z^{\prime}, z\Bigr]\quad\quad\quad\quad \forall\; z,z^{\prime}, + \end{eqnarray} + +\begin{eqnarray} + y = z+ \mbox{f}[z] + \end{eqnarray} + +\begin{eqnarray}\label{eq:flow_irevnet1} + \log\Biggl[\biggl|\mathbf{I} + \frac{\partial \mbox{\bf f}[\mathbf{h},\boldsymbol\phi]}{\partial \mathbf{h}}\biggr|\Biggr] &=& \mbox{trace}\Biggl[\log\biggl[\mathbf{I} + \frac{\partial \mbox{\bf f}[\mathbf{h},\boldsymbol\phi]}{\partial \mathbf{h}}\biggr]\Biggr]\nonumber \\ + &=& \sum_{k=1}^{\infty}\frac{(-1)^{k-1}}{k}\mbox{trace}\Biggl[\frac{\partial \mbox{\bf f}[\mathbf{h},\boldsymbol\phi]}{\partial \mathbf{h}}\Biggr]^{k}, + \end{eqnarray} + +\begin{eqnarray} + \mbox{trace}[\mathbf{A}] &=& \mbox{trace}\left[\mathbf{A}\mathbb{E}\left[\boldsymbol\epsilon\boldsymbol\epsilon^{T}\right]\right]\nonumber \\ + &=& \mbox{trace}\left[\mathbb{E}\left[\mathbf{A}\boldsymbol\epsilon\boldsymbol\epsilon^{T}\right]\right]\nonumber \\ + &=& \mathbb{E}\left[\mbox{trace}\left[\mathbf{A}\boldsymbol\epsilon\boldsymbol\epsilon^{T}\right]\right]\nonumber \\ + &=& \mathbb{E}\left[\mbox{trace}\left[\boldsymbol\epsilon^{T}\mathbf{A}\boldsymbol\epsilon\right]\right]\nonumber \\ + &=& \mathbb{E}\left[\boldsymbol\epsilon^{T}\mathbf{A}\boldsymbol\epsilon\right], + \end{eqnarray} + +\begin{eqnarray} + \mbox{trace}[\mathbf{A}] &=& \mathbb{E}\left[\boldsymbol\epsilon^{T}\mathbf{A}\boldsymbol\epsilon\right]\nonumber \\ + &\approx& \frac{1}{I}\sum_{i=1}^{I}\boldsymbol\epsilon_{i}^{T}\mathbf{A}\boldsymbol\epsilon_{i}. + \end{eqnarray} + +\begin{eqnarray}\label{eq:flow_reverse_KL} + \hat{\boldsymbol\phi} = \mathop{\rm argmin}_{\boldsymbol\phi}\left[\mbox{KL}\left[\frac{1}{I}\sum_{i=1}^I \delta\bigl[\mathbf{x}-\mbox{f}[\mathbf{z}_{i},\boldsymbol\phi]\bigr]\biggl|\biggr|q(\mathbf{x})\right]\right]. + \end{eqnarray} + +\begin{eqnarray} + \hat{\boldsymbol\phi} = \mathop{\rm argmin}_{\boldsymbol\phi}\left[\mbox{KL}\left[\frac{1}{I}\sum_{i=1}^{I}\delta[\mathbf{x}-\mathbf{x}_{i}]\biggl|\biggr|Pr(\mathbf{x}_i,\boldsymbol\phi)\right]\right]. + \end{eqnarray} + + + +\begin{eqnarray}\label{eq:prob_flow_transform2} + Pr(z) = \frac{1}{\sqrt{2\pi}}\exp\left[\frac{-z^2}{2}\right], + \end{eqnarray} + +\begin{eqnarray} + x = \mbox{f}[z] = \frac{1}{1+\exp[-z]}. + \end{eqnarray} + +\begin{eqnarray} + \boldsymbol\Omega_{1} = \begin{bmatrix}2 & 0 & 0 & 0 \\ 0 & -5 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 2\end{bmatrix}\hspace{2cm}\boldsymbol\Omega_{2} = \begin{bmatrix}1 & 0 & 0 & 0 \\ 2 & 4 & 0 & 0 \\ 1 & -1 & 2 & 0 \\ 4 & -2 & -2 & 1 \end{bmatrix}. + \end{eqnarray} + +\begin{eqnarray} + Pr(\mathbf{x}) = \mbox{Pr}(\mathbf{z}) \cdot \left| \frac{\partial \mbox{\bf f}[\mathbf{z}]}{\partial \mathbf{z}}\right|^{-1}. + \end{eqnarray} + +\begin{eqnarray} + \mbox{LReLU}[z] = \begin{cases} 0.1z & \quad z <0 \\ z & \quad z\geq 0\end{cases}. + \end{eqnarray} + +\begin{eqnarray} + \mbox{\bf f}[\mathbf{z}] = \Bigl[\mbox{LReLU}[z_{1}], \mbox{LReLU}[z_{2}], \ldots, \mbox{LReLU}[z_{D}]\Bigr]^{T}. + \end{eqnarray} + +\begin{eqnarray} + \mathbf{h}' = \mbox{f}[h,\boldsymbol\phi] = \sqrt{\left[Kh-b+1\right]\phi_{b}} + \sum_{k=1}^{b-1}\sqrt{\phi_{k}}, + \end{eqnarray} + + \chapter{Variational autoencoders} + +\begin{eqnarray} + Pr(\mathbf{x}) = \int Pr(\mathbf{x}, \mathbf{z}) d\mathbf{z}. + \end{eqnarray} + +\begin{eqnarray}\label{eq:vae_latent_abstract} + Pr(\mathbf{x}) = \int Pr(\mathbf{x} | \mathbf{z}) Pr(\mathbf{z}) d\mathbf{z}. + \end{eqnarray} + +\begin{eqnarray} + Pr(z=n) &=& \lambda_{n}\nonumber \\ + Pr(x |z = n) &=& \mbox{Norm}_{x}\bigl[\mu_{n},\sigma^2_{n}\bigr].\label{eq:vae_mog_like_prior} + \end{eqnarray} + +\begin{eqnarray}\label{eq:vae_mog_marg} + Pr(x) &=& \sum_{n=1}^{N} Pr(x, z=n) \nonumber \\ + &=& \sum_{n=1}^{N} Pr(x| z=n) \cdot Pr(z=n)\nonumber \\ + &=& \sum_{n=1}^{N} \lambda_{n}\cdot \mbox{Norm}_{x}\bigl[\mu_{n},\sigma^2_{n}\bigr]. + \end{eqnarray} + +\begin{eqnarray} + Pr(\mathbf{z}) = \mbox{Norm}_{\mathbf{z}}[\mathbf{0},\mathbf{I}]. + \end{eqnarray} + +\begin{eqnarray}\label{eq:vae_intro_likelihood} + Pr(\mathbf{x} |\mathbf{z},\boldsymbol\phi) = \mbox{Norm}_{\mathbf{x}}\Bigl[\mathbf{f}[\mathbf{z},\boldsymbol\phi],\sigma^{2}\mathbf{I}\Bigr]. + \end{eqnarray} + +\begin{eqnarray}\label{eq:vae_nonlin_like_int} + Pr(\mathbf{x}|\boldsymbol\phi) &=& \int Pr(\mathbf{x}, \mathbf{z}|\boldsymbol\phi) d\mathbf{z} \nonumber \\ + &=& \int Pr(\mathbf{x}| \mathbf{z},\boldsymbol\phi) \cdot Pr(\mathbf{z})d\mathbf{z}\nonumber \\ + &=& \int \mbox{Norm}_{\mathbf{x}}\Bigl[\mathbf{f}[\mathbf{z},\boldsymbol\phi],\sigma^{2}\mathbf{I}\Bigr]\cdot \mbox{Norm}_{\mathbf{z}}\left[\mathbf{0},\mathbf{I}\right]d\mathbf{z}. + \end{eqnarray} + +\begin{eqnarray} + \hat{\boldsymbol\phi} &=& \mathop{\rm argmax}_{\boldsymbol\phi} \left[\sum_{i=1}^{I}\log\Bigl[Pr(\mathbf{x}_{i}|\boldsymbol\phi) \Bigr]\right],\label{eq:vae_log_like} + \end{eqnarray} + +\begin{eqnarray} + Pr(\mathbf{x}_i|\boldsymbol\phi) &=& \int \mbox{Norm}_{\mathbf{x}_i}[\mathbf{f}[\mathbf{z},\boldsymbol\phi],\sigma^{2}\mathbf{I}]\cdot \mbox{Norm}_{\mathbf{z}}[\mathbf{0},\mathbf{I}]d\mathbf{z}. + \end{eqnarray} + + + +\begin{eqnarray} + g[\mathbb{E}[y]] \geq \mathbb{E}\bigl[g[y]\bigr]. + \end{eqnarray} + +\begin{eqnarray}\label{eq:vae_Jensen} + \log\bigl[\mathbb{E}[y]\bigr]\geq\mathbb{E}\bigl[\log[y]\bigr], + \end{eqnarray} + +\begin{eqnarray}\label{eq:vae_JensenInt} + \log\left[\int Pr(y) y dy\right]\geq \int Pr(y)\log[y]dy. + \end{eqnarray} + +\begin{eqnarray}\label{eq:vae_JensenInt2} + \log\left[\int Pr(y) h[y] dy\right]\geq \int Pr(y)\log[h[y]]dy. + \end{eqnarray} + + + +\begin{eqnarray} + \log[Pr(\mathbf{x}|\boldsymbol\phi)] &=& \log\left[\int Pr(\mathbf{x},\mathbf{z}|\boldsymbol\phi)d\mathbf{z} \right]\nonumber\\ + &=& \log\left[\int q(\mathbf{z}) \frac{ Pr(\mathbf{x},\mathbf{z}|\boldsymbol\phi)}{q(\mathbf{z})}d\mathbf{z} \right], + \end{eqnarray} + +\begin{eqnarray} + \log\left[\int q(\mathbf{z}) \frac{ Pr(\mathbf{x},\mathbf{z}|\boldsymbol\phi)}{q(\mathbf{z})}d\mathbf{z} \right] + &\geq& \int q(\mathbf{z}) \log\left[\frac{ Pr(\mathbf{x},\mathbf{z}|\boldsymbol\phi)}{q(\mathbf{z})} \right]d\mathbf{z},\label{eq:vae_ELBOJensen} + \end{eqnarray} + +\begin{eqnarray}\label{eq:vae_ELBO} + \mbox{ELBO}[\boldsymbol\theta, \boldsymbol\phi] = \int q(\mathbf{z}|\boldsymbol\theta) \log\left[\frac{ Pr(\mathbf{x},\mathbf{z}|\boldsymbol\phi)}{q(\mathbf{z}|\boldsymbol\theta)} \right]d\mathbf{z}. + \end{eqnarray} + +\begin{eqnarray}\label{eq:vae_bound_tight} + \mbox{ELBO}[\boldsymbol\theta, \boldsymbol\phi] &=& \int q(\mathbf{z}|\boldsymbol\theta) \log\left[\frac{ Pr(\mathbf{x},\mathbf{z}|\boldsymbol\phi)}{q(\mathbf{z}|\boldsymbol\theta)} \right]d\mathbf{z}\nonumber \\ + &=& \int q(\mathbf{z}|\boldsymbol\theta) \log\left[\frac{ Pr(\mathbf{z}|\mathbf{x},\boldsymbol\phi)Pr(\mathbf{x}|\boldsymbol\phi)}{q(\mathbf{z}|\boldsymbol\theta)} \right]d\mathbf{z}\nonumber \\ + &=& \int q(\mathbf{z}|\boldsymbol\theta) + \log\bigl[Pr(\mathbf{x}|\boldsymbol\phi)\bigr]d\mathbf{z} +\int q(\mathbf{z}|\boldsymbol\theta) \log\left[\frac{ Pr(\mathbf{z}|\mathbf{x},\boldsymbol\phi)}{q(\mathbf{z}|\boldsymbol\theta)} \right]d\mathbf{z} \nonumber\nonumber \\ + &=& \log\bigl[Pr(\mathbf{x} |\boldsymbol\phi)\bigr] +\int q(\mathbf{z}|\boldsymbol\theta) \log\left[\frac{ Pr(\mathbf{z}|\mathbf{x},\boldsymbol\phi)}{q(\mathbf{z}|\boldsymbol\theta)} \right]d\mathbf{z} \nonumber \\ + &=& \log\bigl[Pr(\mathbf{x} |\boldsymbol\phi)\bigr] -\mbox{D}_{KL}\Bigl[ q(\mathbf{z}|\boldsymbol\theta) \Bigl|\Bigr|Pr(\mathbf{z}|\mathbf{x},\boldsymbol\phi)\Bigr].\label{eq:vae_ELBOEvidenceKL} + \end{eqnarray} + +\begin{eqnarray}\label{eq:vae_elbo_reform2} + \mbox{ELBO}[\boldsymbol\theta, \boldsymbol\phi] &=& \int q(\mathbf{z}|\boldsymbol\theta) \log\left[\frac{ Pr(\mathbf{x},\mathbf{z}|\boldsymbol\phi)}{q(\mathbf{z}|\boldsymbol\theta)} \right]d\mathbf{z}\nonumber \\ + &=& \int q(\mathbf{z}|\boldsymbol\theta) \log\left[\frac{ Pr(\mathbf{x}|\mathbf{z},\boldsymbol\phi)Pr(\mathbf{z})}{q(\mathbf{z}|\boldsymbol\theta)} \right]d\mathbf{z}\nonumber \\ + &=& \int q(\mathbf{z}|\boldsymbol\theta) \log\left[ Pr(\mathbf{x}|\mathbf{z},\boldsymbol\phi) \right]d\mathbf{z} + + \int q(\mathbf{z}|\boldsymbol\theta) \log\left[\frac{Pr(\mathbf{z})}{q(\mathbf{z}|\boldsymbol\theta)}\right]d\mathbf{z} + \nonumber \\ + &=& \int q(\mathbf{z}|\boldsymbol\theta) \log\bigl[ Pr(\mathbf{x}|\mathbf{z},\boldsymbol\phi) \bigr]d\mathbf{z} + - \mbox{D}_{KL}\Bigl[ q(\mathbf{z}|\boldsymbol\theta)\Bigl|\Bigr|Pr(\mathbf{z})\Bigr], + \end{eqnarray} + +\begin{eqnarray}\label{eq:vae_NonLinearLVMBayes} + Pr(\mathbf{z}|\mathbf{x},\boldsymbol\phi) = \frac{Pr(\mathbf{x}|\mathbf{z},\boldsymbol\phi)Pr(\mathbf{z})}{Pr(\mathbf{x}|\boldsymbol\phi)}, + \end{eqnarray} + +\begin{eqnarray}\label{eq:vae_posterior_pred} + q(\mathbf{z}|\mathbf{x},\boldsymbol\theta) = \mbox{Norm}_{\mathbf{z}}\Bigl[\mbox{\bf g}_{\boldsymbol\mu}[\mathbf{x},\boldsymbol\theta], \mbox{\bf g}_{\boldsymbol\Sigma}[\mathbf{x},\boldsymbol\theta]\Bigr], + \end{eqnarray} + +\begin{eqnarray}\label{eq:vae_VAECrit} + \mbox{ELBO}[\boldsymbol\theta, \boldsymbol\phi] + = \int q(\mathbf{z}|\mathbf{x},\boldsymbol\theta) \log\bigl[ Pr(\mathbf{x}|\mathbf{z},\boldsymbol\phi) \bigr]d\mathbf{z} + - \mbox{D}_{KL}\Bigl[ q(\mathbf{z}|\mathbf{x},\boldsymbol\theta)\Bigl|\Bigr|Pr(\mathbf{z})\Bigr], + \end{eqnarray} + +\begin{eqnarray}\label{eq:VAE_approx_expectation} + \mathbb{E}_{\mathbf{z}}\bigl[\mbox{a}[\mathbf{z}]\bigr] = \int \mbox{a}[\mathbf{z}] q(\mathbf{z}|\mathbf{x},\boldsymbol\theta) d\mathbf{z} \approx \frac{1}{N}\sum_{n=1}^{N}\mbox{a}[\mathbf{z}^{*}_n], + \end{eqnarray} + +\begin{eqnarray}\label{eq:vae_ELBO_Sample} + \mbox{ELBO}[\boldsymbol\theta, \boldsymbol\phi] &\approx& \log\bigl[ Pr(\mathbf{x}|\mathbf{z}^{*},\boldsymbol\phi) \bigr]- \mbox{D}_{KL}\Bigl[ q(\mathbf{z}|\mathbf{x},\boldsymbol\theta)\Bigl|\Bigr|Pr(\mathbf{z})\Bigr]. + \end{eqnarray} + +\begin{eqnarray} + \mbox{D}_{KL}\Bigl[ q(\mathbf{z}|\mathbf{x},\boldsymbol\theta)\Bigl|\Bigr|Pr(\mathbf{z})\Bigr] = \frac{1}{2}\biggl(\mbox{Tr}[\boldsymbol\Sigma] + \boldsymbol\mu^T\boldsymbol\mu - D_{\mathbf{z}} - \log\Bigl[\mbox{det}[\boldsymbol\Sigma]\Bigr]\biggr). + \end{eqnarray} + +\begin{eqnarray} + \mathbf{z}^{*} = \boldsymbol\mu + \boldsymbol\Sigma^{1/2}\boldsymbol\epsilon^{*}, + \end{eqnarray} + +\begin{eqnarray} + Pr(\mathbf{x}) &=& \int Pr(\mathbf{x}|\mathbf{z}) Pr(\mathbf{z})d\mathbf{z}\nonumber\\ + &=& \mathbb{E}_{\mathbf{z}} \Bigl[Pr(\mathbf{x}|\mathbf{z}) \Bigr] \nonumber \\ + &=& \mathbb{E}_{\mathbf{z}}\Bigl[\mbox{Norm}_{\mathbf{x}}[\mathbf{f}[\mathbf{z},\boldsymbol\phi],\sigma^{2}\mathbf{I}]\Bigr]. + \end{eqnarray} + +\begin{eqnarray} + Pr(\mathbf{x}) \approx \frac{1}{N} \sum_{n=1}^{N}Pr(\mathbf{x}|\mathbf{z}_{n}). + \end{eqnarray} + +\begin{eqnarray} + Pr(\mathbf{x}) &=& \int Pr(\mathbf{x}|\mathbf{z}) Pr(\mathbf{z}) d\mathbf{z}\nonumber \\ + &=& \int \frac{Pr(\mathbf{x}|\mathbf{z})Pr(\mathbf{z})}{q(\mathbf{z})} q(\mathbf{z})d\mathbf{z}\nonumber \\ + &=& \mathbb{E}_{q(\mathbf{z})}\biggl[\frac{Pr(\mathbf{x}|\mathbf{z})Pr(\mathbf{z})}{q(\mathbf{z})}\biggr]\nonumber \\ + &\approx& \frac{1}{N}\sum_{n=1}^{N}\frac{Pr(\mathbf{x}|\mathbf{z}_n)Pr(\mathbf{z}_n)}{q(\mathbf{z}_n)}, + \end{eqnarray} + +\begin{eqnarray} + L_{\mbox{new}} = -\mbox{ELBO}[\boldsymbol\theta, \boldsymbol\phi] + \lambda_{1} \mathbb{E}_{Pr(\mathbf{x})}\Bigl[\mbox{r}_{1}\bigl[q(\mathbf{z}|\mathbf{x},\boldsymbol\theta) \bigr]\Bigr] + \lambda_{2} \mbox{r}_{2}\bigl[q(\mathbf{z}|\boldsymbol\theta)\bigr]. + \end{eqnarray} + +\begin{eqnarray}\label{eq:vae_beta} + \mbox{ELBO}[\boldsymbol\theta, \boldsymbol\phi] &\approx& \log\bigl[ Pr(\mathbf{x}|\mathbf{z}^{*},\boldsymbol\phi) \bigr]- \beta\cdot \mbox{D}_{KL}\Bigl[ q(\mathbf{z}|\mathbf{x},\boldsymbol\theta)\Bigl|\Bigr|Pr(\mathbf{z})\Bigr], + \end{eqnarray} + +\begin{eqnarray} + g\bigl[\mathbb{E}[y]\bigr] \leq \mathbb{E}\bigl[g[y]\bigr]. + \end{eqnarray} + +\begin{eqnarray} + \mbox{D}_{KL}\Bigl[q(\mathbf{z}|\mathbf{x})\Bigl|\Bigr|Pr(\mathbf{z}|\mathbf{x},\boldsymbol\phi)\Bigr] = \int q(\mathbf{z}|\mathbf{x}) \log\left[\frac{q(\mathbf{z}|\mathbf{x})}{Pr(\mathbf{z}|\mathbf{x},\boldsymbol\phi)}\right]d\mathbf{z}. + \end{eqnarray} + +\begin{eqnarray} + \frac{\partial}{\partial \boldsymbol\phi} \mathbb{E}_{Pr(x|\boldsymbol\phi)}\bigl[\mbox{f}[x]\bigr], + \end{eqnarray} + +\begin{eqnarray} + \frac{\partial}{\partial \boldsymbol\phi} \mathbb{E}_{Pr(x|\boldsymbol\phi)}\bigl[\mbox{f}[x]\bigr] &=& \mathbb{E}_{Pr(x|\boldsymbol\phi)}\left[\mbox{f}[x]\frac{\partial}{\partial \boldsymbol\phi} \log\bigl[ Pr(x|\boldsymbol\phi)\bigr]\right]\nonumber \\ + &\approx & \frac{1}{I}\sum_{i=1}^{I}\mbox{f}[x_i]\frac{\partial}{\partial \boldsymbol\phi} \log\bigl[ Pr(x_i|\boldsymbol\phi)\bigr]. + \end{eqnarray} + +\chapter{Diffusion models} + +\begin{eqnarray}\label{eq:diffusion_process} + \mathbf{z}_{1} &=& \sqrt{1-\beta_{1}}\cdot \mathbf{x} + \sqrt{\beta_{1}}\cdot \boldsymbol\epsilon_{1} \\ + \mathbf{z}_{t} &=& \sqrt{1-\beta_{t}}\cdot \mathbf{z}_{t-1} + \sqrt{\beta_{t}}\cdot \boldsymbol\epsilon_{t} \quad\quad\quad \forall\hspace{1mm} t \in2,\ldots, T,\nonumber + \end{eqnarray} + +\begin{eqnarray}\label{eq:diffusion_process_prob} + q(\mathbf{z}_{1}|\mathbf{x}) &=& \mbox{Norm}_{\mathbf{z}_{1}}\left[\sqrt{1-\beta_{1}}\mathbf{x},\beta_{1}\mathbf{I}\right]\\ + q(\mathbf{z}_{t}|\mathbf{z}_{t-1}) &=& \mbox{Norm}_{\mathbf{z}_{t}}\left[\sqrt{1-\beta_{t}}\mathbf{z}_{t-1},\beta_{t}\mathbf{I}\right] \quad\quad\quad \forall \hspace{1mm} t \in\{2, \ldots, T\}.\nonumber + \end{eqnarray} + +\begin{eqnarray}\label{eq:diffusion_joint_forward} + q(\mathbf{z}_{1\ldots T}|\mathbf{x}) =q(\mathbf{z}_{1}|\mathbf{x}) \prod_{t=2}^{T}q(\mathbf{z}_{t}|\mathbf{z}_{t-1}). + \end{eqnarray} + +\begin{eqnarray} + \mathbf{z}_{1} &=& \sqrt{1-\beta_{1}}\cdot\mathbf{x} + \sqrt{\beta_{1}}\cdot\boldsymbol\epsilon_{1}\nonumber \\ + \mathbf{z}_{2} &=& \sqrt{1-\beta_{2}}\cdot\mathbf{z}_{1} + \sqrt{\beta_{2}}\cdot\boldsymbol\epsilon_{2}. + \end{eqnarray} + +\begin{eqnarray}\label{eq:diffusion_sample_step2} + \mathbf{z}_{2} &=& \sqrt{1-\beta_{2}}\left( \sqrt{1-\beta_{1}}\cdot\mathbf{x} + \sqrt{\beta_{1}}\cdot\boldsymbol\epsilon_{1}\right) + \sqrt{\beta_{2}}\cdot\boldsymbol\epsilon_{2}\\ + &=& \sqrt{1-\beta_{2}}\left( \sqrt{1-\beta_{1}}\cdot\mathbf{x} + \sqrt{1-(1-\beta_{1})}\cdot\boldsymbol\epsilon_{1}\right) + \sqrt{\beta_{2}}\cdot\boldsymbol\epsilon_{2}\nonumber \\ + &=& \sqrt{(1-\beta_{2})(1-\beta_{1})}\cdot \mathbf{x} + \sqrt{1-\beta_{2}-(1-\beta_{2})(1-\beta_{1})}\cdot\boldsymbol\epsilon_{1}+\sqrt{\beta_{2}}\cdot\boldsymbol\epsilon_{2}.\nonumber + \end{eqnarray} + +\begin{eqnarray} + \mathbf{z}_{2} &=& \sqrt{(1-\beta_{2})(1-\beta_{1})} \cdot \mathbf{x} + \sqrt{1-(1-\beta_{2})(1-\beta_{1})}\cdot\boldsymbol\epsilon, + \end{eqnarray} + +\begin{eqnarray} + \mathbf{z}_{t} = \sqrt{\alpha_t}\cdot \mathbf{x} + \sqrt{1-\alpha_t}\cdot\boldsymbol\epsilon, + \end{eqnarray} + +\begin{eqnarray} + q(\mathbf{z}_{t}|\mathbf{x}) =\mbox{Norm}_{\mathbf{z}_{t}}\Bigl[\sqrt{\alpha_t}\cdot\mathbf{x},(1-\alpha_{t})\mathbf{I}\Bigr]. + \end{eqnarray} + +\begin{eqnarray} + q(\mathbf{z}_{t}) &=& \int \!\!\int q(\mathbf{z}_{1\ldots t},\mathbf{x})d\mathbf{z}_{1\ldots t-1}d\mathbf{x}\nonumber \\ + &=&\int \!\!\int q(\mathbf{z}_{1\ldots t}|\mathbf{x})Pr(\mathbf{x})d\mathbf{z}_{1\ldots t-1}d\mathbf{x}, + \end{eqnarray} + +\begin{eqnarray} + q(\mathbf{z}_{t}) =\int q(\mathbf{z}_{t}|\mathbf{x})Pr(\mathbf{x})d\mathbf{x}. + \end{eqnarray} + + + +\begin{eqnarray} + q(\mathbf{z}_{t-1}|\mathbf{z}_{t}) = \frac{q(\mathbf{z}_{t}|\mathbf{z}_{t-1})q(\mathbf{z}_{t-1}) }{q(\mathbf{z}_{t})}. + \end{eqnarray} + +\begin{eqnarray}\label{eq:diffusion_bayes} + q(\mathbf{z}_{t-1}|\mathbf{z}_{t},\mathbf{x}) & = & \frac{q(\mathbf{z}_{t}|\mathbf{z}_{t-1},\mathbf{x})q(\mathbf{z}_{t-1}|\mathbf{x})}{q(\mathbf{z}_{t}|\mathbf{x})} \\ + &\propto& q(\mathbf{z}_{t}|\mathbf{z}_{t-1})q(\mathbf{z}_{t-1}|\mathbf{x})\nonumber \\ + &=& \mbox{Norm}_{\mathbf{z}_{t}}\left[\sqrt{1-\beta_{t}}\cdot\mathbf{z}_{t-1},\beta_{t}\mathbf{I}\right]\mbox{Norm}_{\mathbf{z}_{t-1}}\Bigl[\sqrt{\alpha_{t-1}}\cdot\mathbf{x},(1-\alpha_{t-1})\mathbf{I}\Bigr]\nonumber \\ + &\propto& \mbox{Norm}_{\mathbf{z}_{t-1}}\left[\frac{1}{\sqrt{1-\beta_{t}}}\mathbf{z}_{t},\frac{\beta_{t}}{1-\beta_{t}}\mathbf{I}\right]\mbox{Norm}_{\mathbf{z}_{t-1}}\Bigl[\sqrt{\alpha_{t-1}}\cdot\mathbf{x},(1-\alpha_{t-1})\mathbf{I}\Bigr] \nonumber + \end{eqnarray} + + +\begin{eqnarray}\label{eq:diffusion_gaussian_identity1} + \mbox{Norm}_{\mathbf{v}}\left[\mathbf{A}\mathbf{w},\mathbf{B}\right] \propto \mbox{Norm}_{\mathbf{w}}\Bigl[\bigl(\mathbf{A}^T\mathbf{B}^{-1}\mathbf{A}\bigr)^{-1}\mathbf{A}^T\mathbf{B}^{-1}\mathbf{v},\bigl(\mathbf{A}^T\mathbf{B}^{-1}\mathbf{A}\bigr)^{-1}\Bigr], + \end{eqnarray} + +\begin{eqnarray}\label{eq:diffusion_gaussian_identity2} + \mbox{Norm}_{\mathbf{w}}[\mathbf{a},\mathbf{A}]\cdot\mbox{Norm}_{\mathbf{w}}[\mathbf{b},\mathbf{B}] & + \propto &\\ + &&\hspace{-2cm} \mbox{Norm}_{\mathbf{w}}\Bigl[\bigl(\mathbf{A}^{-1}+\mathbf{B}^{-1}\bigr)^{-1}(\mathbf{A}^{-1}\mathbf{a}+\mathbf{B}^{-1}\mathbf{b}),\bigl(\mathbf{A}^{-1}+\mathbf{B}^{-1}\bigr)^{-1} \Bigr],\nonumber + \end{eqnarray} + + +\begin{eqnarray}\label{eq:diffusion_conditional_reverse} + q(\mathbf{z}_{t-1}|\mathbf{z}_{t},\mathbf{x}) = \mbox{Norm}_{\mathbf{z}_{t-1}}\left[\frac{(1-\alpha_{t-1})}{1-\alpha_{t}}\sqrt{1-\beta_{t}}\mathbf{z}_{t}+\frac{\sqrt{\alpha_{t-1}}\beta_{t}}{1-\alpha_{t}}\mathbf{x},\frac{\beta_{t}(1-\alpha_{t-1})}{1-\alpha_{t}}\mathbf{I} \right]. + \end{eqnarray} + +\begin{eqnarray}\label{eq:diffusion_decoder_prob} + Pr(\mathbf{z}_{T})&=& \mbox{Norm}_{\mathbf{z}_{T}}[\mathbf{0},\mathbf{I}]\nonumber \\ + Pr(\mathbf{z}_{t-1}|\mathbf{z}_{t},\boldsymbol\phi_{t}) &=& \mbox{Norm}_{\mathbf{z}_{t-1}}\Bigl[\mbox{\bf f}_{t}[\mathbf{z}_{t},\boldsymbol\phi_{t}],\sigma_{t}^{2}\mathbf{I}\Bigr]\nonumber\\ + Pr(\mathbf{x}|\mathbf{z}_{1},\boldsymbol\phi_{1}) & =&\mbox{Norm}_{\mathbf{x}}\Bigl[\mbox{\bf f}_{1}[\mathbf{z}_{1},\boldsymbol\phi_{1}],\sigma^{2}_{1}\mathbf{I}\Bigr], + \end{eqnarray} + +\begin{eqnarray}\label{eq:diffusion_joint_reverse} + Pr(\mathbf{x},\mathbf{z}_{1\ldots T}|\boldsymbol\phi_{1\ldots T}) = Pr(\mathbf{x}|\mathbf{z}_{1},\boldsymbol\phi_1)\prod_{t=2}^{T}Pr(\mathbf{z}_{t-1}|\mathbf{z}_{t},\boldsymbol\phi_t)\cdot Pr(\mathbf{z}_{T}). + \end{eqnarray} + +\begin{eqnarray}\label{eq:diffusion_marginalize} + Pr(\mathbf{x}|\boldsymbol\phi_{1\ldots T}) = \int Pr(\mathbf{x},\mathbf{z}_{1\ldots T}|\boldsymbol\phi_{1\ldots T}) d\mathbf{z}_{1\ldots T}. + \end{eqnarray} + +\begin{eqnarray} + \hat{\boldsymbol\phi}_{1\ldots T} = \mathop{\rm argmax}_{\boldsymbol\phi_{1\ldots T}} + \left[\sum_{i=1}^{I}\log\Bigl[Pr(\mathbf{x}_i|\boldsymbol\phi_{1\ldots T})\Bigr]\right]. + \end{eqnarray} + +\begin{eqnarray} + \log\left[Pr(\mathbf{x}|\boldsymbol\phi_{1\ldots T})\right] &=& \log\left[\int Pr(\mathbf{x},\mathbf{z}_{1\ldots T}|\boldsymbol\phi_{1\ldots T}) d\mathbf{z}_{1\ldots T}\right]\nonumber\\ + &=& \log\left[\int q(\mathbf{z}_{1\ldots T}|\mathbf{x}) + \frac{Pr(\mathbf{x},\mathbf{z}_{1\ldots T}|\boldsymbol\phi_{1\ldots T}) }{q(\mathbf{z}_{1\ldots T}|\mathbf{x})}d\mathbf{z}_{1\ldots T}\right]\nonumber \\ + &\geq& \int q(\mathbf{z}_{1\ldots T}|\mathbf{x})\log\left[ + \frac{Pr(\mathbf{x},\mathbf{z}_{1\ldots T}|\boldsymbol\phi_{1\ldots T}) }{q(\mathbf{z}_{1\ldots T}|\mathbf{x})}\right]d\mathbf{z}_{1\ldots T}. + \end{eqnarray} + +\begin{eqnarray}\label{eq:diffusion_loss_1} + \mbox{ELBO}\bigl[\boldsymbol\phi_{1\ldots T}\bigr] = \int q(\mathbf{z}_{1\ldots T}|\mathbf{x})\log\left[ + \frac{Pr(\mathbf{x},\mathbf{z}_{1\ldots T}|\boldsymbol\phi_{1\ldots T}) }{q(\mathbf{z}_{1\dots T}|\mathbf{x})}\right]d\mathbf{z}_{1\ldots T}. + \end{eqnarray} + + + +\begin{eqnarray}\label{eq:diffusion_simplify_1} + \!\log\left[ + \frac{Pr(\mathbf{x},\mathbf{z}_{1\ldots T}|\boldsymbol\phi_{1\ldots T}) }{q(\mathbf{z}_{1\ldots T}|\mathbf{x})}\right]\!\!&\!\!=\!\!&\!\! \log\left[ + \frac{ Pr(\mathbf{x}|\mathbf{z}_{1},\boldsymbol\phi_1)\prod_{t=2}^{T}Pr(\mathbf{z}_{t-1}|\mathbf{z}_{t},\boldsymbol\phi_t)\cdot Pr(\mathbf{z}_{T})}{q(\mathbf{z}_{1}|\mathbf{x})\prod_{t=2}^{T}q(\mathbf{z}_{t}|\mathbf{z}_{t-1})}\right] \\ + \!\!&\!\!=\!\!&\!\! \log\left[\!\frac{ Pr(\mathbf{x}|\mathbf{z}_{1},\boldsymbol\phi_1)}{q(\mathbf{z}_{1}|\mathbf{x})}\!\right]\!+\!\log\left[\!\frac{\prod_{t=2}^{T}Pr(\mathbf{z}_{t-1}|\mathbf{z}_{t},\boldsymbol\phi_t)}{\prod_{t=2}^{T}q(\mathbf{z}_{t}|\mathbf{z}_{t-1})}\!\right]\!+\!\log\Bigl[Pr(\mathbf{z}_{T})\Bigr]. \nonumber + \end{eqnarray} + +\begin{eqnarray} + q(\mathbf{z}_{t}|\mathbf{z}_{t-1}) = q(\mathbf{z}_{t}|\mathbf{z}_{t-1},\mathbf{x}) = \frac{q(\mathbf{z}_{t-1}|\mathbf{z}_{t},\mathbf{x})q(\mathbf{z}_{t}|\mathbf{x})}{q(\mathbf{z}_{t-1}|\mathbf{x})}, + \end{eqnarray} + +\begin{eqnarray}\label{eq:diffusion_simplify_2} + \log\left[ + \frac{Pr(\mathbf{x},\mathbf{z}_{1\ldots T}|\boldsymbol\phi_{1\ldots T}) }{q(\mathbf{z}_{1\ldots T}|\mathbf{x})}\right] &&\nonumber \\ + &&\hspace{-3cm}= \log\left[\frac{ Pr(\mathbf{x}|\mathbf{z}_{1},\boldsymbol\phi_1)}{q(\mathbf{z}_{1}|\mathbf{x})}\right]+\log\left[\frac{\prod_{t=2}^{T}Pr(\mathbf{z}_{t-1}|\mathbf{z}_{t},\boldsymbol\phi_t)\cdot q(\mathbf{z}_{t-1}|\mathbf{x})}{\prod_{t=2}^{T}q(\mathbf{z}_{t-1}|\mathbf{z}_{t},\mathbf{x})\cdot q(\mathbf{z}_{t}|\mathbf{x})}\right]+\log\Bigl[Pr(\mathbf{z}_{T})\Bigr]\nonumber \\ + &&\hspace{-3cm}=\log\left[Pr(\mathbf{x}|\mathbf{z}_{1},\boldsymbol\phi_1)\right]+\log\left[\frac{\prod_{t=2}^{T}Pr(\mathbf{z}_{t-1}|\mathbf{z}_{t},\boldsymbol\phi_t)}{\prod_{t=2}^{T}q(\mathbf{z}_{t-1}|\mathbf{z}_{t},\mathbf{x})}\right]+\log\left[\frac{Pr(\mathbf{z}_{T})}{q(\mathbf{z}_{T}|\mathbf{x})} \right]\nonumber\\ + &&\hspace{-3cm}\approx\log\left[Pr(\mathbf{x}|\mathbf{z}_{1},\boldsymbol\phi_1)\right]+\sum_{t=2}^{T}\log\left[\frac{Pr(\mathbf{z}_{t-1}|\mathbf{z}_{t},\boldsymbol\phi_t)}{q(\mathbf{z}_{t-1}|\mathbf{z}_{t},\mathbf{x})}\right], + \end{eqnarray} + +\begin{eqnarray}\label{eq:diffusion_loss_2} + \mbox{ELBO}\bigl[\boldsymbol\phi_{1\ldots T}\bigr]&&\\ + &&\hspace{-1.8cm} =\int q(\mathbf{z}_{1\ldots T}|\mathbf{x})\log\left[ + \frac{Pr(\mathbf{x},\mathbf{z}_{1\ldots T}|\boldsymbol\phi_{1\ldots T}) }{q(\mathbf{z}_{1\ldots T}|\mathbf{x})}\right]d\mathbf{z}_{1\ldots T}\nonumber\\ + &&\hspace{-1.8cm} \approx\int q(\mathbf{z}_{1\ldots T}|\mathbf{x}) + \left(\log\left[Pr(\mathbf{x}|\mathbf{z}_{1},\boldsymbol\phi_1)\right]+\sum_{t=2}^{T}\log\left[\frac{Pr(\mathbf{z}_{t-1}|\mathbf{z}_{t},\boldsymbol\phi_t)}{q(\mathbf{z}_{t-1}|\mathbf{z}_{t},\mathbf{x})}\right]\right)d\mathbf{z}_{1\ldots T}\nonumber\\ + &&\hspace{-1.8cm} =\mathbb{E}_{q(\mathbf{z}_{1}|\mathbf{x})}\Bigl[\log\left[Pr(\mathbf{x}|\mathbf{z}_{1},\boldsymbol\phi_1)\right]\Bigr]\! -\! \sum_{t=2}^T \mathbb{E}_{q(\mathbf{z}_{t}|\mathbf{x})}\biggl[\mbox{D}_{KL}\Bigl[q(\mathbf{z}_{t-1}|\mathbf{z}_{t},\mathbf{x})\bigl|\bigr|Pr(\mathbf{z}_{t-1}|\mathbf{z}_{t},\boldsymbol\phi_t)\Bigr]\biggr],\nonumber + \end{eqnarray} + +\begin{eqnarray} + Pr(\mathbf{x}|\mathbf{z}_{1},\boldsymbol\phi_{1}) =\mbox{Norm}_{\mathbf{x}}\Bigl[\mbox{\bf f}_{1}[\mathbf{z}_{1},\boldsymbol\phi_{1}],\sigma_{1}^{2}\mathbf{I}\Bigr], + \end{eqnarray} + +\begin{eqnarray}\label{eq:diffusion_KL_normal_terms} + Pr(\mathbf{z}_{t-1}|\mathbf{z}_{t},\boldsymbol\phi_{t}) &=& \mbox{Norm}_{\mathbf{z}_{t-1}}\Bigl[\mbox{\bf f}_{t}[\mathbf{z}_{t},\boldsymbol\phi_{t}],\sigma_{t}^{2}\mathbf{I}\Bigr]\\ + q(\mathbf{z}_{t-1}|\mathbf{z}_{t},\mathbf{x}) &=& \mbox{Norm}_{\mathbf{z}_{t-1}}\left[\frac{(1-\alpha_{t-1})}{1-\alpha_{t}}\sqrt{1-\beta_{t}}\mathbf{z}_{t}+\frac{\sqrt{\alpha_{t-1}}\beta_{t}}{1-\alpha_{t}}\mathbf{x},\frac{\beta_{t}(1-\alpha_{t-1})}{1-\alpha_{t}}\mathbf{I} \right].\nonumber + \end{eqnarray} + +\begin{eqnarray}\label{eq:diffusion_kl_1} + D_{KL}\Bigl[q(\mathbf{z}_{t-1}|\mathbf{z}_{t},\mathbf{x})\bigl|\bigr|Pr(\mathbf{z}_{t-1}|\mathbf{z}_{t},\boldsymbol\phi_t)\Bigr] &=&\\ + &&\hspace{-3cm}\frac{1}{2\sigma_{t}^{2}}\left\lVert \frac{(1-\alpha_{t-1})}{1-\alpha_{t}}\sqrt{1-\beta_{t}}\mathbf{z}_{t}+\frac{\sqrt{\alpha_{t-1}}\beta_{t}}{1-\alpha_{t}}\mathbf{x}-\mbox{\bf f}_{t}[\mathbf{z}_{t},\boldsymbol\phi_{t}] \right\rVert^2+C.\nonumber + \end{eqnarray} + +\begin{eqnarray}\label{eq:diffusion_loss_function_1} + L[\boldsymbol\phi_{1\ldots T}] &=& \sum_{i=1}^{I}\biggl( + \begingroup\color{red} + \overbrace{\color{black}\rule[0cm]{0mm}{0.6cm}-\log\Bigl[\mbox{Norm}_{\mathbf{x}_i}\left[\mbox{\bf f}_{1}[\mathbf{z}_{i1},\boldsymbol\phi_{1}],\sigma_{1}^{2}\mathbf{I}\right]}^{\mbox{reconstruction term}} + \endgroup + \Bigr]\\ + &&\vspace{-1.5cm} + + \sum_{t=2}^{T}\frac{1}{2\sigma_{t}^{2}}\biggl\lVert + \begingroup\color{red} + \underbrace{\color{black}\mystrut{3.0ex}\frac{1-\alpha_{t-1}}{1-\alpha_{t}}\sqrt{1-\beta_{t}}\mathbf{z}_{it}+\frac{\sqrt{\alpha_{t-1}}\beta_{t}}{1-\alpha_{t}}\mathbf{x}_i}_{\mbox{target, mean of~$q(\mathbf{z}_{t-1}|\mathbf{z}_{t},\mathbf{x})$}} + \endgroup + - + \begingroup\color{red} + \underbrace{\color{black}\mystrut{3.0ex}\mbox{\bf f}_{t}[\mathbf{z}_{it},\boldsymbol\phi_{t}]}_{\mbox{predicted~$\mathbf{z}_{t-1}$}} + \endgroup + \biggr\rVert^2\biggr),\nonumber + \end{eqnarray} + + + +\begin{eqnarray}\label{eq:diffusion_diff_kernel2} + \mathbf{z}_{t} = \sqrt{\alpha_{t}}\cdot \mathbf{x} + \sqrt{1-\alpha_{t}}\cdot\boldsymbol\epsilon. + \end{eqnarray} + +\begin{eqnarray}\label{eq:diffusion_x_reparam} + \mathbf{x} = \frac{1}{\sqrt{\alpha_{t}}}\cdot \mathbf{z}_t - \frac{\sqrt{1-\alpha_{t}}}{\sqrt{\alpha_{t}}}\cdot\boldsymbol\epsilon. + \end{eqnarray} + +\begin{eqnarray} + \frac{(1-\alpha_{t-1})}{1-\alpha_{t}}\sqrt{1-\beta_{t}}\mathbf{z}_{t}+\frac{\sqrt{\alpha_{t-1}}\beta_{t}}{1-\alpha_{t}}\mathbf{x} \\ + &&\hspace{-3cm} + =\frac{(1-\alpha_{t-1})}{1-\alpha_{t}}\sqrt{1-\beta_{t}}\mathbf{z}_{t}+\frac{\sqrt{\alpha_{t-1}}\beta_{t}}{1-\alpha_{t}}\left(\frac{1}{\sqrt{\alpha_{t}}} \mathbf{z}_t - \frac{\sqrt{1-\alpha_{t}}}{\sqrt{\alpha_{t}}}\boldsymbol\epsilon\right)\nonumber\\ + &&\hspace{-3cm}= \frac{(1-\alpha_{t-1})}{1-\alpha_{t}}\sqrt{1-\beta_{t}}\mathbf{z}_{t}+\frac{\beta_{t}}{1-\alpha_{t}}\left(\frac{1}{\sqrt{1-\beta_{t}}} \mathbf{z}_t - \frac{\sqrt{1-\alpha_{t}}}{\sqrt{1-\beta_{t}}}\boldsymbol\epsilon\right),\nonumber + \end{eqnarray} + +\begin{eqnarray} + \frac{(1-\alpha_{t-1})}{1-\alpha_{t}}\sqrt{1-\beta_{t}}\mathbf{z}_{t}+\frac{\sqrt{\alpha_{t-1}}\beta_{t}}{1-\alpha_{t}}\mathbf{x}\\ + &&\hspace{-3cm}=\left(\frac{(1-\alpha_{t-1})\sqrt{1-\beta_{t}}}{1-\alpha_{t}}+\frac{\beta_{t}}{(1-\alpha_{t})\sqrt{1-\beta_{t}}}\right) \mathbf{z}_t - \frac{\beta_{t}}{\sqrt{1-\alpha_{t}}\sqrt{1-\beta_{t}}}\boldsymbol\epsilon\nonumber \\ + &&\hspace{-3cm}=\left(\frac{(1-\alpha_{t-1})(1-\beta_{t})}{(1-\alpha_{t})\sqrt{1-\beta_{t}}}+\frac{\beta_{t}}{(1-\alpha_{t})\sqrt{1-\beta_{t}}}\right) \mathbf{z}_t - \frac{\beta_{t}}{\sqrt{1-\alpha_{t}}\sqrt{1-\beta_{t}}}\boldsymbol\epsilon\nonumber \\ + &&\hspace{-3cm}=\frac{(1-\alpha_{t-1})(1-\beta_{t})+\beta_{t}}{(1-\alpha_{t})\sqrt{1-\beta_{t}}} \mathbf{z}_t - \frac{\beta_{t}}{\sqrt{1-\alpha_{t}}\sqrt{1-\beta_{t}}}\boldsymbol\epsilon\nonumber \\ + &&\hspace{-3cm}=\frac{1-\alpha_{t}}{(1-\alpha_{t})\sqrt{1-\beta_{t}}} \mathbf{z}_t - \frac{\beta_{t}}{\sqrt{1-\alpha_{t}}\sqrt{1-\beta_{t}}}\boldsymbol\epsilon\nonumber \\ + &&\hspace{-3cm}=\frac{1}{\sqrt{1-\beta_{t}}} \mathbf{z}_t - \frac{\beta_{t}}{\sqrt{1-\alpha_{t}}\sqrt{1-\beta_{t}}}\boldsymbol\epsilon,\nonumber + \end{eqnarray} + +\begin{eqnarray}\label{eq:diffusion_loss_function_2} + L[\boldsymbol\phi_{1\ldots T}] &=& \sum_{i=1}^{I}\biggl(-\log\Bigl[\mbox{Norm}_{\mathbf{x}_i}\left[\mbox{\bf f}_{1}[\mathbf{z}_{i1},\boldsymbol\phi_{1}],\sigma_{1}^{2}\mathbf{I}\right]\Bigr]\\ + &&\vspace{-2.0cm} + + \sum_{t=2}^{T}\frac{1}{2\sigma_{t}^{2}}\left\lVert + \left(\frac{1}{\sqrt{1-\beta_{t}}} \mathbf{z}_{it} - \frac{\beta_{t}}{\sqrt{1-\alpha_{t}}\sqrt{1-\beta_{t}}}\boldsymbol\epsilon_{it}\right)-\mbox{\bf f}_{t}[\mathbf{z}_{it},\boldsymbol\phi_{t}] \right\rVert^2\biggr).\nonumber + \end{eqnarray} + + +\begin{eqnarray}\label{eq:diffusion_f_reparam} + \mbox{\bf f}_{t}[\mathbf{z}_{t},\boldsymbol\phi_{t}] = \frac{1}{\sqrt{1-\beta_{t}}} \mathbf{z}_t - \frac{\beta_{t}}{\sqrt{1-\alpha_{t}}\sqrt{1-\beta_{t}}}\mbox{\bf g}_t[\mathbf{z}_{t},\boldsymbol\phi_t]. + \end{eqnarray} + +\begin{eqnarray}\label{eq:diffusion_loss_function_3} + L[\boldsymbol\phi_{1\ldots T}] =\\ + &&\hspace{-1.5cm}\sum_{i=1}^{I}-\log\Bigl[\mbox{Norm}_{\mathbf{x}_i}\left[\mbox{\bf f}_{1}[\mathbf{z}_{i1},\boldsymbol\phi_{1}],\sigma_{1}^{2}\mathbf{I}\right]\Bigr] + \sum_{t=2}^{T}\frac{\beta_t^2}{(1-\alpha_{t})(1-\beta_{t})2\sigma_{t}^{2}}\Big\lVert\mbox{\bf g}_{t}[\mathbf{z}_{it},\boldsymbol\phi_{t}]-\boldsymbol\epsilon_{it}\Big\rVert^2.\nonumber + \end{eqnarray} + +\begin{eqnarray} + L[\boldsymbol\phi_{1\ldots T}] = \sum_{i=1}^{I}\frac{1}{2\sigma_1^2}\Big\lVert\mathbf{x}_i-\mbox{\bf f}_{1}[\mathbf{z}_{i1},\boldsymbol\phi_{1}]\Bigr\rVert^2 + \sum_{t=2}^{T}\frac{\beta_t^2}{(1-\alpha_{t})(1-\beta_{t})2\sigma_{t}^{2}}\Big\lVert\mbox{\bf g}_{t}[\mathbf{z}_{it},\boldsymbol\phi_{t}]-\boldsymbol\epsilon_{it}\Big\rVert^2+C_i. + \end{eqnarray} + +\begin{eqnarray}\label{eq:diffusion_ls_term} + \frac{1}{2\sigma_1^2}\Big\lVert\mathbf{x}_i-\mbox{\bf f}_{1}[\mathbf{z}_{i1},\boldsymbol\phi_{1}]\Bigr\rVert^2 = \frac{1}{2\sigma_1^2}\biggl\lVert \frac{\beta_{1}}{\sqrt{1-\alpha_{1}}\sqrt{1-\beta_{1}}}\mbox{\bf g}_1[\mathbf{z}_{i1},\boldsymbol\phi_1]- \frac{\beta_{1}}{\sqrt{1-\alpha_{1}}\sqrt{1-\beta_{1}}}\boldsymbol\epsilon_{i1}\bigg\rVert^2. + \end{eqnarray} + +\begin{eqnarray} + L[\boldsymbol\phi_{1\ldots T}] \!\!&\!\!=\!\!&\!\! \sum_{i=1}^{I}\sum_{t=1}^{T}\frac{\beta_t^2}{(1-\alpha_{t})(1-\beta_{t})2\sigma_{t}^{2}}\Big\lVert\mbox{\bf g}_{t}[\mathbf{z}_{it},\boldsymbol\phi_{t}]-\boldsymbol\epsilon_{it}\Big\rVert^2, + \end{eqnarray} + +\begin{eqnarray}\label{eq:diffusion_loss_function_3b} + L[\boldsymbol\phi_{1\ldots T}] &=& \sum_{i=1}^{I}\sum_{t=1}^{T}\Big\lVert\mbox{\bf g}_{t}[\mathbf{z}_{it},\boldsymbol\phi_{t}]-\boldsymbol\epsilon_{it}\Big\rVert^2\\ + &=& \sum_{i=1}^{I}\sum_{t=1}^{T}\bigg\lVert\mbox{\bf g}_{t}\Bigl[\sqrt{\alpha_t}\cdot\mathbf{x}_i+\sqrt{1-\alpha_t}\cdot\boldsymbol\epsilon_{it},\boldsymbol\phi_{t}\Bigr]-\boldsymbol\epsilon_{it}\bigg\rVert^2\nonumber, + \end{eqnarray} + +\begin{eqnarray} + \mathbf{z}_{t-1} = \hat{\mathbf{z}}_{t-1} + \sigma_{t}^2\frac{\partial \log\bigl[\Pr(c|\mathbf{z}_{t})\bigr]}{\partial \mathbf{z}_{t}} + \sigma_{t}\boldsymbol\epsilon. + \end{eqnarray} + +\begin{eqnarray} + \mathbf{x}_{t} = \sqrt{1-\beta_{t}}\cdot\mathbf{x}_{t-1} + \sqrt{\beta_{t}}\cdot\boldsymbol\epsilon_{t}, + \end{eqnarray} + + + +\begin{eqnarray} + z = a\cdot \epsilon_{1}+b\cdot\epsilon_{2}, + \end{eqnarray} + +\begin{eqnarray} + \mathbb{E}[z] &=& 0\nonumber \\ + \mbox{Var}[z] &=& a^2+b^2, + \end{eqnarray} + +\begin{eqnarray} + \mathbf{z}_{3} &=&\sqrt{(1-\beta_{3})(1-\beta_{2})(1-\beta_{1}})\cdot\mathbf{x} + \sqrt{1-(1-\beta_{3})(1-\beta_{2})(1-\beta_{1})}\cdot\boldsymbol\epsilon', + \end{eqnarray} + +\begin{eqnarray} + \mbox{Norm}_{\mathbf{v}}\left[\mathbf{A}\mathbf{w},\mathbf{B}\right] \propto \mbox{Norm}_{\mathbf{w}}\Bigl[(\mathbf{A}^T\mathbf{B}^{-1}\mathbf{A})^{-1}\mathbf{A}^T\mathbf{B}^{-1}\mathbf{v},(\mathbf{A}^T\mathbf{B}^{-1}\mathbf{A})^{-1}\Bigr]. + \end{eqnarray} + +\begin{eqnarray} + \mbox{Norm}_{\mathbf{x}}[\mathbf{a},\mathbf{A}]\mbox{Norm}_{\mathbf{x}}[\mathbf{b},\mathbf{B}] &\!\! + \propto\!\! & + \mbox{Norm}_{\mathbf{x}}\Bigl[(\mathbf{A}^{-1}+\mathbf{B}^{-1})^{-1}(\mathbf{A}^{-1}\mathbf{a}+\mathbf{B}^{-1}\mathbf{b}),(\mathbf{A}^{-1}+\mathbf{B}^{-1})^{-1} \Bigr].\nonumber\\ + \end{eqnarray} + +\begin{eqnarray} + D_{KL}\Bigl[\mbox{Norm}_{\mathbf{w}}[\mathbf{a},\mathbf{A}]\bigl|\bigr|\mbox{Norm}_{\mathbf{w}}[\mathbf{b},\mathbf{B}]\Bigr]\!\! &\!=\!&\!\! + \frac{1}{2}\left(\mbox{tr}\left[\mathbf{B}^{-1}\mathbf{A}\right]-d+(\mathbf{a}-\mathbf{b})^{T}\mathbf{B}^{-1}(\mathbf{a}-\mathbf{b})+\log\left[\frac{ + |\mathbf{B}|}{|\mathbf{A}|}\right]\right).\nonumber\\ + \end{eqnarray} + +\begin{eqnarray} + \sqrt{\frac{\alpha_{t}}{\alpha_{t-1}}}=\sqrt{1-\beta_{t}}. + \end{eqnarray} + +\begin{eqnarray} + \frac{(1-\alpha_{t-1})(1-\beta_{t})+\beta_{t}}{(1-\alpha_{t})\sqrt{1-\beta_{t}}} =\frac{1}{\sqrt{1-\beta_{t}}}. + \end{eqnarray} + +\chapter{Reinforcement learning} + +\begin{eqnarray} + G_{t} = \sum_{k=0}^{\infty}\gamma^{k}r_{t+k+1}. + \end{eqnarray} + +\begin{eqnarray} + v[s_t|\pi] = \mathbb{E}\Bigl[G_{t}|s_{t},\pi\Bigr]. + \end{eqnarray} + +\begin{eqnarray} + q[s_t,a_t|\pi] = \mathbb{E}\Bigl[G_{t}|s_{t},a_{t},\pi\Bigr]. + \end{eqnarray} + +\begin{eqnarray} + v^{*}[s_{t}] = \max_{\pi}\biggl[\mathbb{E}\Bigl[G_{t}|s_{t},\pi\Bigr]\biggr]. + \end{eqnarray} + +\begin{eqnarray} + q^*[s_t,a_t] = \max_{\pi}\left[\mathbb{E}\Bigl[G_{t}|s_{t},a_{t},\pi \Bigr]\right]. + \end{eqnarray} + +\begin{eqnarray}\label{eq:reinforce_optimal_policy} + \pi[a_{t}|s_{t}] \leftarrow \mathop{\rm argmax}_{a_{t}}\Bigl[q^*[s_t,a_t]\Bigr]. + \end{eqnarray} + +\begin{eqnarray}\label{eq:reinforce_state_ito_action} + v[s_{t}] = \sum_{a_t} \pi[a_t|s_t]q[s_t,a_t]. + \end{eqnarray} + +\begin{eqnarray}\label{eq:reinforce_action_ito_state} + q[s_{t},a_t] = r[s_t,a_t] + \gamma \cdot \sum_{s_{t+1}} Pr(s_{t+1}|s_{t},a_{t})v[s_{t+1}]. + \end{eqnarray} + +\begin{eqnarray}\label{eq:reinforce_bellman_value} + v[s_{t}] = \sum_{a_t} \pi[a_t|s_t] + \left(r[s_t,a_t] + \gamma \cdot \sum_{s_{t+1}} Pr(s_{t+1}|s_{t},a_{t})v[s_{t+1}] + \right). + \end{eqnarray} + +\begin{eqnarray}\label{eq:reinforce_bellman_action} + q[s_{t},a_t] = r[s_t,a_t] + \gamma \cdot \sum_{s_{t+1}} Pr(s_{t+1}|s_{t},a_{t})\left(\sum_{a_{t+1}} \pi[a_{t+1}|s_{t+1}]q[s_{t+1},a_{t+1}]\right). + \end{eqnarray} + + +\begin{eqnarray}\label{eq:reinforce_dp1} + v[s_t]\leftarrow \sum_{a_t} \pi[a_t|s_t]\left(r[s_t,a_t]+\gamma \cdot\sum_{s_{t+1}}Pr(s_{t+1}|s_{t},a_t) v[s_{t+1}]\right), + \end{eqnarray} + +\begin{eqnarray}\label{eq:reinforce_dp2} + \pi[a_t|s_t] \leftarrow \mathop{\rm argmax}_{a_t}\biggl[r[s_t,a_t]+\gamma \cdot \sum_{s_{t+1}} Pr(s_{t+1}|s_t,a_t)v[s_{t+1}]\biggr]. + \end{eqnarray} + + +\begin{eqnarray}\label{eq:reinforce:MC_policy_update} + \pi[a|s] \leftarrow \mathop{\rm argmax}_{a}\Bigl[q[s,a]\Bigr]. + \end{eqnarray} + +\begin{eqnarray}\label{eq:reinforce_sarsa} + q[s_t,a_t] \leftarrow q[s_t,a_t] + \alpha \Bigl(r[s_t,a_t]+\gamma \cdot q[s_{t+1},a_{t+1}] - q[s_t,a_t]\Bigr), + \end{eqnarray} + +\begin{eqnarray}\label{eq:reinforce_qlearning} + q[s_t,a_t] \leftarrow q[s_t,a_t] + \alpha \Bigl(r[s_t,a_t]+\gamma \cdot \max_{a}\bigl[q[s_{t+1},a]\bigr] - q[s_t,a_t]\Bigr), + \end{eqnarray} + +\begin{eqnarray} + L[\boldsymbol\phi] = \biggl(r[\mathbf{s}_t,a_t]+\gamma \cdot \max_{a}\Bigl[q[\mathbf{s}_{t+1},a,\boldsymbol\phi]\Bigr] - q[\mathbf{s}_t,a_t,\boldsymbol\phi]\biggr)^2, + \end{eqnarray} + +\begin{eqnarray} + \boldsymbol\phi \leftarrow \boldsymbol\phi + \alpha \biggl(r[\mathbf{s}_t,a_t]+\gamma \cdot \max_{a}\Bigl[q[\mathbf{s}_{t+1},a,\boldsymbol\phi]\Bigr] - q[\mathbf{s}_t,a_t,\boldsymbol\phi]\biggr)\frac{\partial q[\mathbf{s}_{t},a_{t},\boldsymbol\phi]}{\partial \boldsymbol\phi}. + \end{eqnarray} + +\begin{eqnarray} + \boldsymbol\phi \leftarrow \boldsymbol\phi + \alpha \biggl(r[\mathbf{s}_t,a_t]+\gamma \cdot \max_{a}\Bigl[q[\mathbf{s}_{t+1},a,\boldsymbol\phi^{-}]\Bigr] - q[\mathbf{s}_t,a_t,\boldsymbol\phi]\biggr)\frac{\partial q[\mathbf{s}_{t},a_{t},\boldsymbol\phi]}{\partial \boldsymbol\phi}. + \end{eqnarray} + +\begin{eqnarray}\label{eq:reinforce_qlearning2} + q[s_t,a_t] \leftarrow q[s_t,a_t] + \alpha \Bigl(r[s_t,a_t]+\gamma \cdot \max_{a}\bigl[q[s_{t+1},a]\bigr] - q[s_t,a_t]\Bigr) + \end{eqnarray} + +\begin{eqnarray}\label{eq:reinforce_qlearning3} + q_1[s_t,a_t] &\leftarrow& q_1[s_t,a_t] + \alpha \Bigl(r[s_t,a_t]+\gamma \cdot q_2\left[s_{t+1},\mathop{\rm argmax}_{a}\Bigl[q_1[s_{t+1},a]\Bigr]\right] - q_1[s_t,a_t]\Bigr)\nonumber\\ + q_2[s_t,a_t] &\leftarrow& q_2[s_t,a_t] + \alpha \Bigl(r[s_t,a_t]+\gamma \cdot q_1\left[s_{t+1},\mathop{\rm argmax}_{a}\Bigl[q_2[s_{t+1},a]\Bigr]\right] - q_2[s_t,a_t]\Bigr).\nonumber \\ + \end{eqnarray} + +\begin{eqnarray} + \boldsymbol\phi_1\!\!\!&\!\!\leftarrow\!\!\!&\!\!\boldsymbol\phi_1\!\!+\!\alpha \biggl(\!r[\mathbf{s}_t,a_t]\!+\!\gamma \!\cdot\! q\biggl[\mathbf{s}_{t+1},\mathop{\rm argmax}_{a}\Bigl[q[\mathbf{s}_{t+1},a,\boldsymbol\phi_1]\Bigr],\boldsymbol\phi_2\biggr] \!\!-\! q[\mathbf{s}_t,a_t,\boldsymbol\phi_1]\!\biggr)\frac{\partial q[\mathbf{s}_{t},a_{t},\boldsymbol\phi_1]}{\partial \boldsymbol\phi_1}\nonumber \\ + \boldsymbol\phi_2\!\!\!&\!\!\leftarrow\!\!\!&\!\!\boldsymbol\phi_2\!\!+\!\alpha \biggl(\!r[\mathbf{s}_t,a_t]\!+\!\gamma \!\cdot\! q\biggl[\mathbf{s}_{t+1},\mathop{\rm argmax}_{a}\Bigl[q[\mathbf{s}_{t+1},a,\boldsymbol\phi_2]\Bigr],\boldsymbol\phi_1\!\biggr] \!\!-\! q[\mathbf{s}_t,a_t,\boldsymbol\phi_2]\!\biggr)\frac{\partial q[\mathbf{s}_{t},a_{t},\boldsymbol\phi_2]}{\partial \boldsymbol\phi_2}.\nonumber\\ + \end{eqnarray} + +\begin{eqnarray}\label{eq:reinforce_path_prob} + Pr(\boldsymbol\tau|\boldsymbol\theta) &=& Pr(\mathbf{s}_{1})\prod_{t=1}^{T} \pi[a_t|\mathbf{s}_{t},\boldsymbol\theta] Pr(\mathbf{s}_{t+1}|\mathbf{s}_{t},a_{t}). + \end{eqnarray} + +\begin{eqnarray}\label{eq:reinforce_policy_loss} + \boldsymbol\theta = \mathop{\rm argmax}_{\boldsymbol\theta}\biggl[\mathbb{E}_{\boldsymbol\tau}\Bigl[r[\boldsymbol\tau] \Bigr]\biggr] + = \mathop{\rm argmax}_{\boldsymbol\theta}\biggl[\int Pr(\boldsymbol\tau|\boldsymbol\theta) r[\boldsymbol\tau] d\boldsymbol\tau\biggr], + \end{eqnarray} + +\begin{eqnarray} + \boldsymbol\theta &\leftarrow& \boldsymbol\theta + \alpha\cdot \frac{\partial}{\partial \boldsymbol\theta}\int Pr(\boldsymbol\tau|\boldsymbol\theta) r[\boldsymbol\tau] d\boldsymbol\tau \nonumber \\ + &=& \boldsymbol\theta + \alpha \cdot\int \frac{\partial Pr(\boldsymbol\tau|\boldsymbol\theta)}{\partial \boldsymbol\theta} r[\boldsymbol\tau] d\boldsymbol\tau. + \end{eqnarray} + +\begin{eqnarray}\label{eq:reinforce_pg_explain} + \boldsymbol\theta &\leftarrow& \boldsymbol\theta + \alpha \cdot\int \frac{\partial Pr(\boldsymbol\tau|\boldsymbol\theta)}{\partial \boldsymbol\theta} r[\boldsymbol\tau] d\boldsymbol\tau \nonumber \\ + &=& \boldsymbol\theta + \alpha \cdot\int Pr(\boldsymbol\tau|\boldsymbol\theta) \frac{1}{Pr(\boldsymbol\tau|\boldsymbol\theta)} \frac{\partial Pr(\boldsymbol\tau|\boldsymbol\theta)}{\partial \boldsymbol\theta} r[\boldsymbol\tau] d\boldsymbol\tau\nonumber \\ + &\approx & \boldsymbol\theta + \alpha \cdot\frac{1}{I}\sum_{i=1}^{I} \frac{1}{Pr(\boldsymbol\tau_i|\boldsymbol\theta)} \frac{\partial Pr(\boldsymbol\tau_i|\boldsymbol\theta)}{\partial \boldsymbol\theta} r[\boldsymbol\tau_i]. + \end{eqnarray} + +\begin{eqnarray} + \frac{\partial \log[\mbox{f}[z]]}{\partial z} = \frac{1}{f[z]} \frac{\partial \mbox{f}[z]}{\partial z}, + \end{eqnarray} + +\begin{eqnarray}\label{eq:reinforce_update_2} + \boldsymbol\theta \leftarrow \boldsymbol\theta + \alpha \cdot\frac{1}{I}\sum_{i=1}^{I} \frac{\partial \log\bigl[Pr(\boldsymbol\tau_i|\boldsymbol\theta)\bigr]}{\partial \boldsymbol\theta} r[\boldsymbol\tau_i]. + \end{eqnarray} + +\begin{eqnarray} + \log[Pr(\boldsymbol\tau|\boldsymbol\theta)] &=& \log\Bigl[Pr(\mathbf{s}_{1})\prod_{t=1}^{T} \pi[a_{t}|\mathbf{s}_{t},\boldsymbol\theta] Pr(\mathbf{s}_{t+1}|\mathbf{s}_{t},a_{t})\Bigr] \\ + &=& \log\bigl[Pr(\mathbf{s}_{1})\bigr]+\sum_{t=1}^{T} \log\bigl[\pi[a_{t}|\mathbf{s}_{t},\boldsymbol\theta]\bigr] +\sum_{t=1}^{T} \log\bigl[ Pr(\mathbf{s}_{t+1}|\mathbf{s}_{t},a_{t})\bigr]\nonumber, + \end{eqnarray} + +\begin{eqnarray} + \boldsymbol\theta \!&\!\leftarrow\!&\! \boldsymbol\theta + \alpha \cdot\frac{1}{I}\sum_{i=1}^{I} \sum_{t=1}^{T} \frac{\partial\log\bigl[\pi[a_{it}|\mathbf{s}_{it},\boldsymbol\theta]\bigr]}{\partial \boldsymbol\theta} r[\boldsymbol\tau_i],\nonumber \\ + \end{eqnarray} + +\begin{eqnarray} + r[\boldsymbol\tau_i] = \sum_{t=1}^{T} r_{i,t+1} = \sum_{k=1}^{t}r_{i,k+1} + \sum_{k=t}^{T}r_{i,k+1}, + \end{eqnarray} + +\begin{eqnarray} + \boldsymbol\theta \!&\!\leftarrow\!&\! \boldsymbol\theta + \alpha \cdot\frac{1}{I}\sum_{i=1}^{I} \sum_{t=1}^{T} \frac{\partial\log\bigl[\pi[a_{it}|\mathbf{s}_{it},\boldsymbol\theta]\bigr]}{\partial \boldsymbol\theta} \sum_{k=t}^{T} r_{i,k+1}. + \end{eqnarray} + +\begin{eqnarray} + r[\boldsymbol\tau_{it}] = \sum_{k=t+1}^{T} \gamma^{k-t-1}r_{i,k+1}, + \end{eqnarray} + +\begin{eqnarray} + \boldsymbol\theta \leftarrow \boldsymbol\theta + \alpha \cdot \gamma^{t}\frac{\partial\log\bigl[\pi_{a_{it}}[\mathbf{s}_{it},\boldsymbol\theta]\bigr]}{\partial \boldsymbol\theta} r[\boldsymbol\tau_{it}]\hspace{2cm}\forall\hspace{1mm} i,t, + \end{eqnarray} + +\begin{eqnarray}\label{eq:reinforce_baseline_update} + \boldsymbol\theta \leftarrow \boldsymbol\theta + \alpha \cdot\frac{1}{I}\sum_{i=1}^{I} \sum_{t=1}^{T} \frac{\partial\log\bigl[\pi_{a_{it}}[\mathbf{s}_{it},\boldsymbol\theta]\bigr]}{\partial \boldsymbol\theta} \left( r[\boldsymbol\tau_{it}]- b\right). + \end{eqnarray} + +\begin{eqnarray} + \mathbb{E}_{\boldsymbol\tau} \left[\sum_{t=1}^{T} \frac{\partial\log\bigl[\pi_{a_{it}}[\mathbf{s}_{it},\boldsymbol\theta]\bigr]}{\partial \boldsymbol\theta}\cdot b\right] = 0, + \end{eqnarray} + +\begin{eqnarray} + b = \sum_{i}\frac{\sum_{t=1}^{T} \left({\partial\log\bigl[\pi_{a_{it}}[\mathbf{s}_{it},\boldsymbol\theta]\bigr]}/{\partial \boldsymbol\theta}\right)^2 r[\boldsymbol\tau_{it}]}{\sum_{t=1}^{T} \left({\partial\log\bigl[\pi_{a_{it}}[\mathbf{s}_{it},\boldsymbol\theta]\bigr]}/{\partial \boldsymbol\theta}\right)^2}. + \end{eqnarray} + +\begin{eqnarray} + b = \frac{1}{I}\sum_{i} r[\boldsymbol\tau_{i}]. + \end{eqnarray} + +\begin{eqnarray}\label{eq:reinforce_baseline_update2} + \boldsymbol\theta \leftarrow \boldsymbol\theta + \alpha \cdot\frac{1}{I}\sum_{i=1}^{I} \sum_{t=1}^{T} \frac{\partial\log\bigl[\pi_{a_{it}}[\mathbf{s}_{it},\boldsymbol\theta]\bigr]}{\partial \boldsymbol\theta} \left( r[\boldsymbol\tau_{it}]- b[\mathbf{s}_{it}]\right). + \end{eqnarray} + +\begin{eqnarray} + L[\boldsymbol\phi] = \sum_{i=1}^{I}\sum_{t=1}^{T}\left(v[\mathbf{s}_{it},\boldsymbol\phi] - \sum_{j=t}^{T} r_{i,j+1}\right)^2. + \end{eqnarray} + + + +\begin{eqnarray} + r[\boldsymbol\tau_{it}] \approx r_{i,t+1} + \gamma \cdot v[\mathbf{s}_{i,t+1},\boldsymbol\phi]. + \end{eqnarray} + +\begin{eqnarray} + \boldsymbol\theta \leftarrow \boldsymbol\theta + \alpha \cdot\frac{1}{I}\sum_{i=1}^{I} \sum_{t=1}^{T} \frac{\partial\log\bigl[Pr(a_{it}|\mathbf{s}_{it},\boldsymbol\theta)]\bigr]}{\partial \boldsymbol\theta} \Bigl( r_{i,t+1} + \gamma\cdot v[\mathbf{s}_{i,t+1},\boldsymbol\phi] - v[\mathbf{s}_{i,t},\boldsymbol\phi]\Bigr). + \end{eqnarray} + +\begin{eqnarray} + L[\boldsymbol\phi] = \sum_{i=1}^{I}\sum_{t=1}^{T}\left(r_{i,t+1} + \gamma \cdot v[\mathbf{s}_{i,t+1},\boldsymbol\phi]- v[\mathbf{s}_{i,t},\boldsymbol\phi] \right)^2. + \end{eqnarray} + +\begin{eqnarray} + \pi'[a_t|s_t] \leftarrow \mathop{\rm argmax}_{a_t}\biggl[r[s_t,a_t]+\gamma \cdot \sum_{s_{t+1}} Pr(s_{t+1}|s_t,a_t)v[s_{t+1}|\pi]\biggr]. + \end{eqnarray} + +\begin{eqnarray} + v[s_t|\pi] &\leq& q\Bigl[s_t,\pi'[a_{t}|s_{t}]\Bigl|\pi\Bigr]\nonumber\\ + &=& \mathbb{E}_{\pi'}\Bigl[r_{t+1} + \gamma\cdot v[s_{t+1}|\pi]\Bigr]. + \end{eqnarray} + +\begin{eqnarray} + \pi[a|s] = \frac{\exp\bigl[q[s,a]/\tau\bigr]}{\sum_{a'}\exp\bigl[q[s,a']/\tau\bigr]}. + \end{eqnarray} + +\begin{eqnarray} + \mbox{f}\bigl[q[s,a]\bigr] =r[s,a]+\gamma \cdot \max_{a}\bigl[q[s',a]\bigr]. + \end{eqnarray} + +\begin{eqnarray} + \biggl|\biggr|\mbox{f}\bigl[q_1[s,a]\bigr]-\mbox{f}\bigl[q_2[s,a]\bigr]\biggl|\biggr|_{\infty} < + \biggl|\biggr|q_1[s,a]- q_2[s,a]\biggl|\biggr|_\infty\quad\quad\quad\quad \forall\; q_1,q_2. + \end{eqnarray} + + +\begin{eqnarray} + \mathbb{E}_{\boldsymbol\tau}\left[\frac{\partial}{\partial \boldsymbol\theta} \log\bigl[Pr(\boldsymbol\tau|\boldsymbol\theta)\bigr] b \right]= 0, + \end{eqnarray} + + + +\begin{eqnarray} + a' = a - c(b-\mu_b). + \end{eqnarray} + +\begin{eqnarray} + \mathbb{E}_{\boldsymbol\tau}\Bigl[g[\boldsymbol\theta](r[\boldsymbol\tau_t]-b)\Bigr], + \end{eqnarray} + +\begin{eqnarray} + g[\theta] = \sum_{t=1}^{T} \frac{\partial\log\bigl[Pr(a_{t}|\mathbf{s}_{t},\boldsymbol\theta)]\bigr]}{\partial \boldsymbol\theta}, + \end{eqnarray} + +\begin{eqnarray} + r[\boldsymbol\tau_t] = \sum_{k=t}^{T}r_k. + \end{eqnarray} + +\begin{eqnarray} + b = \frac{\mathbb{E}[g[\boldsymbol\tau]^2]r[\boldsymbol\tau]}{\mathbb{E}[g[\boldsymbol\tau]^2]}. + \end{eqnarray} + + +\chapter{Why does deep learning work?} + + +\chapter{Ethics} + + +\end{document} \ No newline at end of file

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<111278851+aleksandrskoselevs@users.noreply.github.com> Date: Fri, 11 Oct 2024 10:54:05 +0200 Subject: [PATCH 29/49] Update 13_4_Graph_Attention_Networks.ipynb `phi` is defined in the book as a column vector --- Notebooks/Chap13/13_4_Graph_Attention_Networks.ipynb | 4 ++-- 1 file changed, 2 insertions(+), 2 deletions(-) diff --git a/Notebooks/Chap13/13_4_Graph_Attention_Networks.ipynb b/Notebooks/Chap13/13_4_Graph_Attention_Networks.ipynb index 7c33404..a67e829 100644 --- a/Notebooks/Chap13/13_4_Graph_Attention_Networks.ipynb +++ b/Notebooks/Chap13/13_4_Graph_Attention_Networks.ipynb @@ -109,7 +109,7 @@ "# Choose random values for the parameters\n", "omega = np.random.normal(size=(D,D))\n", "beta = np.random.normal(size=(D,1))\n", - "phi = np.random.normal(size=(1,2*D))" + "phi = np.random.normal(size=(2*D,1))" ], "metadata": { "id": "79TSK7oLMobe" @@ -210,4 +210,4 @@ } } ] -} \ No newline at end of file +} From be86733a9359c20a13d5ac6609d51bc9252c8152 Mon Sep 17 00:00:00 2001 From: aleksandrskoselevs <111278851+aleksandrskoselevs@users.noreply.github.com> Date: Tue, 22 Oct 2024 12:11:15 +0200 Subject: [PATCH 30/49] Update 15_2_Wasserstein_Distance.ipynb Scaling of the distance matrix was not mentioned in the book. --- Notebooks/Chap15/15_2_Wasserstein_Distance.ipynb | 4 +++- 1 file changed, 3 insertions(+), 1 deletion(-) diff --git a/Notebooks/Chap15/15_2_Wasserstein_Distance.ipynb b/Notebooks/Chap15/15_2_Wasserstein_Distance.ipynb index 332b36f..9f70e26 100644 --- a/Notebooks/Chap15/15_2_Wasserstein_Distance.ipynb +++ b/Notebooks/Chap15/15_2_Wasserstein_Distance.ipynb @@ -86,6 +86,8 @@ "cell_type": "code", "source": [ "# TODO Define the distance matrix from figure 15.8d\n", + "# Each element (i, j) in the distance matrix represents the absolute difference between positions i and j,\n", + "# scaled by the maximum possible distance to ensure the values lie within [0, 1].\n", "# Replace this line\n", "dist_mat = np.zeros((10,10))\n", "\n", @@ -243,4 +245,4 @@ } } ] -} \ No newline at end of file +} From 7ba844f2b5646fa957603d46504e32bed6676267 Mon Sep 17 00:00:00 2001 From: udlbook <110402648+udlbook@users.noreply.github.com> Date: Thu, 24 Oct 2024 16:04:27 -0400 Subject: [PATCH 31/49] Created using Colab --- Notebooks/Chap07/7_2_Backpropagation.ipynb | 7 +++---- 1 file changed, 3 insertions(+), 4 deletions(-) diff --git a/Notebooks/Chap07/7_2_Backpropagation.ipynb b/Notebooks/Chap07/7_2_Backpropagation.ipynb index e977edd..dac7e89 100644 --- a/Notebooks/Chap07/7_2_Backpropagation.ipynb +++ b/Notebooks/Chap07/7_2_Backpropagation.ipynb @@ -4,7 +4,6 @@ "metadata": { "colab": { "provenance": [], - "authorship_tag": "ABX9TyM2kkHLr00J4Jeypw41sTkQ", "include_colab_link": true }, "kernelspec": { @@ -230,8 +229,8 @@ "# We'll need the indicator function\n", "def indicator_function(x):\n", " x_in = np.array(x)\n", - " x_in[x_in>=0] = 1\n", - " x_in[x_in<0] = 0\n", + " x_in[x_in>0] = 1\n", + " x_in[x_in<=0] = 0\n", " return x_in\n", "\n", "# Main backward pass routine\n", @@ -353,4 +352,4 @@ "outputs": [] } ] -} +} \ No newline at end of file From 5eb264540d7a4fb03795ddc5de805602ca10a3f8 Mon Sep 17 00:00:00 2001 From: udlbook <110402648+udlbook@users.noreply.github.com> Date: Thu, 24 Oct 2024 16:40:27 -0400 Subject: [PATCH 32/49] Created using Colab --- .../Chap10/10_5_Convolution_For_MNIST.ipynb | 19 +++++++------------ 1 file changed, 7 insertions(+), 12 deletions(-) diff --git a/Notebooks/Chap10/10_5_Convolution_For_MNIST.ipynb b/Notebooks/Chap10/10_5_Convolution_For_MNIST.ipynb index 33322a0..fb5b32b 100644 --- a/Notebooks/Chap10/10_5_Convolution_For_MNIST.ipynb +++ b/Notebooks/Chap10/10_5_Convolution_For_MNIST.ipynb @@ -4,7 +4,7 @@ "metadata": { "colab": { "provenance": [], - "authorship_tag": "ABX9TyNELb86uz5qbhEKH81UqFKT", + "authorship_tag": "ABX9TyORZF8xy4X1yf4oRhRq8Rtm", "include_colab_link": true }, "kernelspec": { @@ -72,8 +72,12 @@ "# even before you make changes.\n", "batch_size_train = 64\n", "batch_size_test = 1000\n", + "\n", + "# TODO Change this directory to point towards an existing directory\n", + "myDir = '/files/'\n", + "\n", "train_loader = torch.utils.data.DataLoader(\n", - " torchvision.datasets.MNIST('/files/', train=True, download=True,\n", + " torchvision.datasets.MNIST(myDir, train=True, download=True,\n", " transform=torchvision.transforms.Compose([\n", " torchvision.transforms.ToTensor(),\n", " torchvision.transforms.Normalize(\n", @@ -82,7 +86,7 @@ " batch_size=batch_size_train, shuffle=True)\n", "\n", "test_loader = torch.utils.data.DataLoader(\n", - " torchvision.datasets.MNIST('/files/', train=False, download=True,\n", + " torchvision.datasets.MNIST(myDir, train=False, download=True,\n", " transform=torchvision.transforms.Compose([\n", " torchvision.transforms.ToTensor(),\n", " torchvision.transforms.Normalize(\n", @@ -96,15 +100,6 @@ "execution_count": null, "outputs": [] }, - { - "cell_type": "code", - "source": [], - "metadata": { - "id": "YGwbxJDEm88i" - }, - "execution_count": null, - "outputs": [] - }, { "cell_type": "code", "source": [ From dc6b346bdadf6f08c0971992b30b987686b77efd Mon Sep 17 00:00:00 2001 From: udlbook <110402648+udlbook@users.noreply.github.com> Date: Thu, 24 Oct 2024 16:43:14 -0400 Subject: [PATCH 33/49] Created using Colab From 3801b8d52d1ddaf2c4a4eb0d6c337331d790409d Mon Sep 17 00:00:00 2001 From: udlbook <110402648+udlbook@users.noreply.github.com> Date: Thu, 24 Oct 2024 16:45:43 -0400 Subject: [PATCH 34/49] Created using Colab --- Notebooks/Chap03/3_3_Shallow_Network_Regions.ipynb | 5 ++--- 1 file changed, 2 insertions(+), 3 deletions(-) diff --git a/Notebooks/Chap03/3_3_Shallow_Network_Regions.ipynb b/Notebooks/Chap03/3_3_Shallow_Network_Regions.ipynb index 6f2584d..4ca8d8e 100644 --- a/Notebooks/Chap03/3_3_Shallow_Network_Regions.ipynb +++ b/Notebooks/Chap03/3_3_Shallow_Network_Regions.ipynb @@ -4,7 +4,6 @@ "metadata": { "colab": { "provenance": [], - "authorship_tag": "ABX9TyNioITtfAcfxEfM3UOfQyb9", "include_colab_link": true }, "kernelspec": { @@ -221,7 +220,7 @@ { "cell_type": "code", "source": [ - "# Now let's plot the graph from figure 3.9a (takes ~1min)\n", + "# Now let's plot the graph from figure 3.9b (takes ~1min)\n", "dims = np.array([1,5,10,50,100])\n", "regions = np.zeros((dims.shape[0], 200))\n", "params = np.zeros((dims.shape[0], 200))\n", @@ -256,4 +255,4 @@ "outputs": [] } ] -} +} \ No newline at end of file From 1adb96e00619a4fd935fb660257e3c471b6f38a9 Mon Sep 17 00:00:00 2001 From: aleksandrskoselevs <111278851+aleksandrskoselevs@users.noreply.github.com> Date: Wed, 30 Oct 2024 09:19:22 +0100 Subject: [PATCH 35/49] Update 15_2_Wasserstein_Distance.ipynb --- Notebooks/Chap15/15_2_Wasserstein_Distance.ipynb | 3 +-- 1 file changed, 1 insertion(+), 2 deletions(-) diff --git a/Notebooks/Chap15/15_2_Wasserstein_Distance.ipynb b/Notebooks/Chap15/15_2_Wasserstein_Distance.ipynb index 9f70e26..9384399 100644 --- a/Notebooks/Chap15/15_2_Wasserstein_Distance.ipynb +++ 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