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The many faces of AI and the role of mathematics

Gabriel Peyré
November 26, 2024

The many faces of AI and the role of mathematics

General audience introduction talk on deep learning.

Gabriel Peyré

November 26, 2024
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  1. The many faces of AI and the role of mathematics

    Gabriel Peyré É C O L E N O R M A L E S U P É R I E U R E
  2. AlexNet 2012 Adam 2014 Transformers 2017 ResNet 2016 X0R problem

    1969 ChatGPT 2022 Di ff usion 2021 SVMs 1995 BackProp 1986 ConvNets 1998 Birth of AI 1956 ADALINE 1959 Perceptron 1957 Turing test 1950 Arti fi cial Neuron 1943 Neocognitron 1980 Universality 1989 LSTM 1997 The revolutions of (deep) learning
  3. From Supervised to Generative Learning <latexit 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=) <latexit 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    =) <latexit 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k-means, DBScan <latexit 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PCA, t-SNE, UMAP Unsupervised learning Clustering Dimension reduction <latexit 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? <latexit 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=) <latexit 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y = +1 <latexit 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y = 1 Supervised learning <latexit 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? <latexit 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=) <latexit 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x <latexit 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y Regression Classification
  4. From Supervised to Generative Learning <latexit 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=) <latexit 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    =) <latexit 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k-means, DBScan <latexit 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PCA, t-SNE, UMAP Unsupervised learning Clustering Dimension reduction <latexit 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? <latexit 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=) <latexit 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y = +1 <latexit 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y = 1 Supervised learning <latexit 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? <latexit 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=) <latexit 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y Regression Classification Self supervised learning DALL·E 2 Add noise Denoise Masking Next token prediction
  5. Learning via optimizing <latexit sha1_base64="qogpSA+iGtyTRT9qW+kaVUzGk7A=">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</latexit> Dataset: (xi, yi)n i=1 .

    <latexit 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xi 2 Rd <latexit 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yi 2 R <latexit sha1_base64="CrCoGm457sfMJbC4UQwEtG/nICM=">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</latexit> Prediction: <latexit 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y ⇡ f✓(x) <latexit 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Empirical risk minimization: <latexit sha1_base64="lDqsLzbPrMD8XiEdi+sGTF4wS0s=">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</latexit> min ✓ 1 n n X i=1 `(f✓(xi), yi) <latexit 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Least square: `(y, y0) = (y y0)2 <latexit 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(xi, yi) <latexit 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f✓ <latexit 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f✓0 <latexit 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y
  6. Learning via optimizing <latexit sha1_base64="qogpSA+iGtyTRT9qW+kaVUzGk7A=">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</latexit> Dataset: (xi, yi)n i=1 .

    <latexit 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xi 2 Rd <latexit 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yi 2 R <latexit sha1_base64="CrCoGm457sfMJbC4UQwEtG/nICM=">AABE7HictVzbchu5EYU3t41z8yaPeZmN1ilvynFkxblUtlK1tijLWss2bVKyd03bxcuQpj3i0BySvnD1F6m8pFLJU34j35EPSFXylF9IX4ABhsRMY5SNpyRhMDjdjR6g0d3AuDdNxtl8e/sf59772te/8c1vvf/t89/57ve+/4MLH/zwOEsXs3581E+TdPao183iZDyJj+bjeRI/ms7i7kkviR/2Xu7i84fLeJaN00l7/nYaPznpjibj4bjfnUNVpzmLB+M+Fn8XPbuwtX1lm/5Fm4WrurCl9L9m+sGH/1QdNVCp6quFOlGxmqg5lBPVVRlcj9VVta2mUPdEraBuBqUxPY/VqToP2AW0iqFFF2pfwu8R3D3WtRO4R5oZofvAJYGfGSAjdREwKbSbQRm5RfR8QZSxtoz2imiibG/hb0/TOoHauXoOtRLOtAzFYV/maqh+S30YQ5+mVIO962sqC9IKSh45vZoDhSnUYXkAz2dQ7hPS6DkiTEZ9R9126fm/qCXW4n1ft12of5OUF+GKVEv3Ps0pdNWS6Ef0NhfwjOVJgPMIKMS6j1h6Tbo+od5PoP0K6u/CdUolo5MeXCuqPa1E7sLlQ+6KyH24fMh9EXkIlw95KCKbcPmQTY1E7Ix07se34PLhWyLn+3D5kPdF5AO4fMgHIvIYLh/yWER+AZcP+YWIvAmXD3lTRN6Gy4e8LSLbcPmQbRF5BJcPeSQi9+DyIfc0snymzuBKic5YmJXXoVzkgZYigZrronw3yDr6sDcC5nS/BCvP6gb89WMbATqNS7B7AeNuWIKVR94+2Eg/VrZFt2g18WFvidgDGAF+7IGI/Uy9KMF+FjDTXpZg5bl2CO38WNn63oE7P/aOiL0LJT9WXqPuQY0fey9gxZiWYJsi9r56VYINsfqzEqxs91tgV/xYeZ1qQ3s/NsSaLkqwsj09Bg/Gj5VXq4dQ68c+FLGP1JsS7CMR+zlYdz/284AV9l0J1qyx52kFGZE/EsOMraLWzWcllqZArSvwT/K1JSHfuAf1EmaUY0aEORER+zliPxBxmCMOg+XKcjuakb8rc2nliFYgopevTViai+0HeXssJQGIRo5orCGqPFJ816YvS/IuTI2EnOcrF5ZC+pTm9htLsR4P1ZbXIO4VEDy2n9PIv0zREkZQqKkqas/zNZ6REd1XIV5T9GZ6aXjIuHluFVzUGxHV86B6IuqtB/VWRC08qIWIWnpQSxFlZ76L6wSMAKt/fBcruuMRwD5y+RWBV3AdVp1bMEcjGD9N8AIfUM09+Nui2Fu6qiTDaB7XScxyPClY4hmUVmoL6m1U2KD4OqEZFoNk3PKejvHxDnMbKz3n2Aqf5it5lGdMwumMSZ5RTge9xYjmUz06t6nmlLw7LtXD38rnvSnVw++Rxk/Ji+dSPfxcSz8/g+xtjW2fAduC2TTV2rflujQ4/8I0TPk8rbpocfGtnugxg/Te1KR/oN/MwRneyy6VWD+2XI9G5vQvK/SvDg2r58zRcz0q6D2x12tKUe2eTHTca8t1ZUhpFZ1oOexd3TeDbQb6zZhyPRpN8Lh2KeZeOeW6o3ea98aW69E4Vpz3PCVP3pTr0RjRPevDluvRwGxLV8f5tlzXsqMGOHa25bpWfUJZYMwB8ZjnGusVzchPWmhqY/IPqrM1rs+/uY5hzuZpHiNUU7K+bTmdXr6WVUtk/IUYrNq8phzoXywcH6xIY6V2xPiKZZgX1vdNOnaNR80fghYjmP28ByDlzBOQ0OQk0HonQPGqGHUVe2ZwOyIOR8lwDdXRtXPRW7R8OWtUrHtGtVJcZntr9dghe53R2JuST3hImpX0cFj6hssoSho6LGhIpldHd+/0fC1qf1vETdcQ03yk9WlHiHfSquNUn9Zbjo4v6l2eOVy852PHL2abh9raYMyTki1CWap4uu1MHsmtw3X1srI5bn4W0RtFe7UkqzGmHalMjEJNtpi98RXdW9pHtCeHPJhGH95jpKlMFe+aYRYd8+kRWVTX3kq8UV8mQ8fljKyuscfV6JGDHnnQ9WOcXVgx7kKpDTHDEdy1A6Kc87muUtL4TP083x1N6Q1WR/RJwUIaGmxv4oKFrIqynxeovAY0jgaO0sNprNMx+M4GJTnq98ljY9ei5b9IO7dmf7tLY7x8NJdnYgbEdYe4RjRreFeX79Y5sAQr75Md8l+re4n86nBEGypxfepwZr1MaMc/pgh2Sp5xQrNNmh3F1m5+av2J4dRUZu8cd7NTspAR2b8I1qeUxmREP+7ZAbODzhYhIRsZYnfGuXfj83XG4hizftxY8akGO95ismUL4m/ourMro7HIEQOvA6drY9vo5JB8wZi4zrR1t3O7evVBpD0n4Y4SpmjHyiXi/zH9Nj9mnGxtjAjUML6BTNs63/tIKWZBHXVpla+2QaatK+VHuQxPtdR2/bMyfVSQrEERF8qDq/UAOPfpnnnhKJmR3NlGG15Hq7K5SHm6pkfs7ZCieLb7I70Co9yXaZXcojnXoVEyglEwz6MI01bKIq/zreZVpB5GO/u/ULe6LmoNKUbKZnBZQ1J+P6ZozZUygVHN4/clzSa/1mdrrar5TGgsnjhz+Uuo/RB+G7nNfRidXsEq3KAxwBTsndUI10QbLcJ43SjwMiPT0LL3lp8dk6aVW3OW+Jqtm42xl7WpNGnUvNFZC1M+C40XDo0XgTps016j1aKpN5bomRhbtPVuZSi/OtzaNSgvRMqyR2ZQ4wAp3VgqjOpApCrH+Ab1TqS1LdLqwmx1dwPcOR+C9M/19dn9Zb66R+om+TZ98sA4fhnQLB2Tz2VqqyM1poCcr2n76s7+DtUg9x5ZUKTM5zhxxvCuU5+u01zSn+qVLSU7by2CObf0WrcxNrZD5V9uIE9oTmQ0Lw3iGrWItfyuHNGaRbri+BwRZf675FOx31EdM7ut7TuJCv6EjTd5VlleHClMSP9S5u1gI3o9cOLXiGLChfaue0Cr/htGCowxmQS/Z5nRG8JVjncS2KPtkf3ctFO8izdxJLpCUq/U7wNsDEe9dqy7Y8v02PTtZ9AStW7fuq+FzC8J5ijxO8uOXpdWtRPto67W7s9Gq6tXueJ9lR4Wa3ytPhbUxo0sbJRXxHTUJ8FcWKJ6XBgTwqVeL+rIX0/yOjLz7lQoZdPaUC5mGtjGPKd4SToHigifd3fJ6819LPSjt0GvR1iXGtdIlDAbl+r8gGtpMSsVrUVIbr20JiXOelS2Xlge7qph7ThbypisYKKk3A23dvvQKUQrcjaGKfQVn+wtixNdmp/Ahb8j5YsSDceQHGIL/NzralftfQWnIl7pMmc2I6pBmzBYi8G7up/FFtU6euVQd+mHcAjnMQZdS9KPaUWtKztTliV3qYfTf03WYKZiUXrbsn4fXC5yTzY51enPmCyc3JuxMt/k1O2L4RDSkyKXcD68vyH1YqjMt031+mCoyz0ocqjDw5xnCHvntnV9Xi6nan1tcgnlweuA2XkxONwBLI9ZbLsQCzVz3shXzwGtw7CCulkt/td+GD6WU31eodwy+ubsRcBb53axzsyiX1x/zlhuIaO5nGM4zzTvnfWa/PzY/4tqvanU6c1XTx/9UjsGDK+V4nyoLB3j3VFk5Q2lgvsDPhlS9R/193PyVwmvchplctShZPYryqmZFjI18+Wlr3fmWYhMlk6ZTEVqNp5o0cnYXXWgbsLPbu4B1j0lyt9U8l/E+r+jHUDtkKyHyaZzBqFDdTFlQexu2oDu7TnaMonxTC+f8W1DDe6JH1Itnve9S+3xzG+70LfyL0l4rt9RqRoUIpP1XT47r3rQg+IOHOeCzPe+EZ2p52wWn0A7Cdhj5HNUHCmZr59XhBhQXLgu6YoQZrRUUe55KffoTFJcQrtX6FufRvhU7/TjvgOez+/m2aVI/YLqunp1wJVakqrpkeoxZQZ6pP9tiNB+pS7D38u67Je0uSFpRu+gKNEb51n1SbBT77iwXzNepDyYydQtdbuUonq7e1idiW2UcuET79X4UQV+5EjZorf1kuLumarOHS4qaC60TO5+7kSZvCfrAaPZbj4+quPnZQWvZUD/b5eibzuS7oMsPcq2R7SfNyN6idbNHknP5yqr87a3KqQ1X20yTXuy0o4Dc0ayek8g0eOufPbzOUgpVxOX0HHnOp/IlE6LjL2U5Pk5DTgN0Q3ordzXkJ5KVBaiJIuAL5GXAbIsA+gMBWmGIoWRKIm2D88ubF1d/78+NgvHO1eu/vrKtfs7W5/e0P8PyPvqx+on6hKsfb9Rn8L4b6oj4DRVf1R/UX9tTBp/aPyp8Wdu+t45jfmRKvxr/O2/kfBEPA==</latexit> Prediction: <latexit 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y ⇡ f✓(x) <latexit 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Empirical risk minimization: <latexit sha1_base64="lDqsLzbPrMD8XiEdi+sGTF4wS0s=">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</latexit> min ✓ 1 n n X i=1 `(f✓(xi), yi) <latexit 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Least square: `(y, y0) = (y y0)2 <latexit 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Logistic: `(y, y0) = log(1 + e yy0 ) <latexit 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yi 2 { 1, 1} <latexit 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Classification: <latexit 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y ⇡ sign(f✓(x)) <latexit 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f✓ <latexit 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f✓0 <latexit 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x <latexit sha1_base64="WI2HEUK7U7Ib1a7UKFHMW+OpF90=">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</latexit> y yi = +1 yi = 1
  7. Learning via optimizing <latexit sha1_base64="qogpSA+iGtyTRT9qW+kaVUzGk7A=">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</latexit> Dataset: (xi, yi)n i=1 .

    <latexit 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xi 2 Rd <latexit 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yi 2 R <latexit sha1_base64="CrCoGm457sfMJbC4UQwEtG/nICM=">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</latexit> Prediction: <latexit 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y ⇡ f✓(x) <latexit 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Empirical risk minimization: <latexit sha1_base64="lDqsLzbPrMD8XiEdi+sGTF4wS0s=">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</latexit> min ✓ 1 n n X i=1 `(f✓(xi), yi) <latexit 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Least square: `(y, y0) = (y y0)2 <latexit 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Logistic: `(y, y0) = log(1 + e yy0 ) <latexit 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yi 2 { 1, 1} <latexit 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Classification: <latexit 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y ⇡ sign(f✓(x)) <latexit 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f✓ <latexit 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f✓0 <latexit 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y <latexit 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Overfitting, regularization, . . . yi = +1 yi = 1
  8. Learning using gradient descent Small ⌧` Large ⌧` Optimal ⌧`

    = ⌧? ` ✓`+1 = ✓` ⌧` rE(✓`) Gradient descent: <latexit sha1_base64="bbJMdjCFXVxfhmGqAzIDZjVUKG4=">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</latexit> min ✓ E(✓) , 1 n n X i=1 `(f✓(xi), yi)
  9. Learning using gradient descent Small ⌧` Large ⌧` Optimal ⌧`

    = ⌧? ` ✓`+1 = ✓` ⌧` rE(✓`) Gradient descent: <latexit sha1_base64="bbJMdjCFXVxfhmGqAzIDZjVUKG4=">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</latexit> min ✓ E(✓) , 1 n n X i=1 `(f✓(xi), yi) Herbert Robbins Sutton Monro <latexit 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Stochastic gradient descent: <latexit 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✓`+1 = ✓` ⌧ ` rE`(✓`) <latexit 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E`(✓) , `(f✓(xi), yi) <latexit 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i rand
  10. Learning using gradient descent Small ⌧` Large ⌧` Optimal ⌧`

    = ⌧? ` ✓`+1 = ✓` ⌧` rE(✓`) Gradient descent: <latexit sha1_base64="bbJMdjCFXVxfhmGqAzIDZjVUKG4=">AABFMnictVxtbxu5EWaub9f0Ldd+7Je9+lI4hZvaafoCHA64xHIcX3yJE8lO7qLEWEkreZOVVtFKihOd/lXRr/0dLfqlKK5ogQL9C50ZkkuuxN3humkWtrlcPjPDWXI4M+SmM07ibLq9/edL733jm9/69nfe/+7l733/Bz/80ZUPfnySpbNJNzrupkk6edIJsyiJR9HxNJ4m0ZPxJAqHnSR63Hm5i88fz6NJFqej1vTNOHo2DAejuB93wylUnV45ag/j0Wl7ehZNw6C9F21C8VrQnk7icDRIoldBuz8Ju4ud5WK0DNrZbHi6iD/ZWT4fBe0oSTb7Crp5fhpf23oDv06vbGxf36Z/wXphRxU2hPp3lH7w4deiLXoiFV0xE0MRiZGYQjkRocjgeip2xLYYQ90zsYC6CZRieh6JpbgM2Bm0iqBFCLUv4fcA7p6q2hHcI82M0F3gksDPBJCBuAqYFNpNoIzcAno+I8pYW0Z7QTRRtjfwt6NoDaF2Ks6glsPplr447MtU9MXvqQ8x9GlMNdi7rqIyI62g5IHVqylQGEMdlnvwfALlLiG1ngPCZNR31G1Iz/9JLbEW77uq7Uz8i6S8Clcgmqr3aU4hFHOiH9DbnMEzKU8CnAdAIVJ9xNJr0vWQej+C9guovw/XkkpaJx24FlS7rETuwuVC7rLIfbhcyH0WeQiXC3nIIo/gciGPFBKxE9K5G9+Ey4VvspwfwuVCPmSRj+ByIR+xyBO4XMgTFvklXC7klyzyDlwu5B0WeQ8uF/Iei2zB5UK2WOQxXC7kMYvcg8uF3FPI8pk6gSslOjEzK29BucgDLUUCNbdY+W6TdXRhb3vM6W4Jlp/VDfjrxjY8dBqVYPc8xl2/BMuPvH2wkW4sb4vu0mriwt5lsQcwAtzYAxb7mXhRgv3MY6a9LMHyc+0Q2rmxvPX9HO7c2M9Z7H0oubH8GvUAatzYBx4rxrgEe8RiH4pXJVgfqz8pwfJ2vwl2xY3l16kWtHdjfazprATL29MT8GDcWH61egy1buxjFvtEnJdgn7DYL8C6u7FfeKywb0uweo29TCvIgPyRCGZsFbUwn5VYGgO1kOGf5GtLQr5xB+o5zCDHDAgzZBH7OWLfE3GYIw695cpyO5qRv8tzaeaIpieik69NWJqy7Xt5eywlHohGjmisIKo8UnzXui9z8i50DYec5isXlnz6lOb2G0uRGg/VllcjHhQQcmyf0cjfomgJIyjUVBW1s3yNl8iA7qsQryl6073UPHjcNLcKNuqcRXUcqA6LeuNAvWFRMwdqxqLmDtScRZmZb+PaHiPA6B/fxYLu5AiQPnL5FYBXcAtWnbswRwMYP0fgBT6imgfwt0mxN3dVSYbRPK6TmOV4VrDEEygtxAbUm6iwQfF1QjMsAslkywcqxsc7zG0s1JyTVniZr+RBnjHxpxOTPIOcDnqLAc2nenTuUc2SvDtZqoe/m897XaqH3yONL8mLl6V6+KmSfnoB2VsK27oAtgmzaay0b8p1acj8i6Shy5dp1UWLi291qMYM0juvSf9AvZmDC7yXXSpJ/ZhyPRqZ1b+s0L86NIyeM0vP9aig9yS9Xl0KavdkpOJeU64rQ0qr6EjJYe7qvhls01NvRpfr0TgCj2uXYu6FVa47esd5b0y5Ho0TIfOeS/LkdbkejQHdS32Ycj0amG0JVZxvynUtO2pAxs6mXNeqjygLjDkgOeZljfGKJuQnzRS1mPyD6myN7fOvr2OYs3mexwjVlIxvW06nk69l1RJpfyECqzatKQf6FzPLByvSWIgbbHwlZZgW1vd1OmaNR80fghYDmP1yD4DLmScgoc5JoPVOgOIOG3UVe6ZxN1gcjpL+Cqqtaqest2j4yqxRse6Uarm4zPTW6LFN9jqjsTcmn/CQNMvp4bD0DZdR5DR0WNAQT6+O7t6q+VrU/jaLG68gxvlI69KOkNxJq45TXVpvWjq+qnZ5pnDJPR8zfjHb3FfWBmOelGwRylLF026n80h2Ha6rW8LkuOWzgN4o2qs5WY2YdqQyNgrV2WLpjS/o3tA+pj055CFpdOE9BorKWMhdM8yiYz49IItq21uON+pLZ+hkOSOrq+1xNXpgoQcOdP0YZxdWjPtQakHMcAx3LY8o53Kuq5Q0PhG/zHdHU3qD1RF9UrCQmoa0N1HBQlZF2WcFKq8BjaNBRun+NFbpaHx7jRIf9bvkMbFr0fJfpZ1bvb8d0hgvH83lmZgecb1BXAOaNXJXV96tcpASLJxPbpD/Wt1L5FeHI9pQjutzi7PUy4h2/COKYMfkGSc027jZUWxt56dWn2hOR0LvneNudkoWMiD7F8D6lNKYDOjHPjugd9ClRUjIRvrYnTj3bly+TsyOMePHxUKeajDjLSJbNiP+mq49uzIaizJikOvAcmVsa50cki8YEdeJsu5mblevPog05yTsUSIpmrGySfyv0W/9o8fJxtqIQA3jG8iUrXO9j5RiFtRRSKt8tQ3SbW0pP8pleK6kNuufkemjgmQNirhQHlyte8C5S/eSF46SCcmdrbWR62hVNhcpj1f0iL3tUxQv7f5ArcAo9xatkhs059o0SgYwCqZ5FKHbclnkVb7VvIrU/Whn/xfqRtdFrSHFQJgMrtQQl9+PKFqzpUxgVMvx+5Jmk1vrk5VW1XxGNBaH1lz+Cmo/hN9abn3vR6dTsAq3aQxICubOaETWBGst/HjdLvDSI1PTMveGnxmTupVdc5H4Wlo3E2PPa1M5olFzrrIWunwRGi8sGi88ddiivUajRV2vLdEpG1u01G6lL7863Fo1KM9YyrxHplGxh5R2LOVHtcdS5WN8jXrL0tpmaYUwW+3dAHvO+yDdc311dn+Vr+6BuEO+TZc8MBm/9GiWxuRz6drqSE1SQM43lX21Z3+bapB7hywoUpbnOHHGyF2nLl3LXNKfq5UtJTtvLII+t/RatdE2tk3lX68hhzQnMpqXGnGTWkRKfluOYMUiXbd8joAy/yH5VNLvqI6Z7dbmnQQFf8LEm3JWGV4yUhiR/rnM28Fa9Hpgxa8BxYQz5V13gFb9N4wUJEZnEtyeZUZvCFc5uZMgPdoO2c91OyV38UaWRNdJ6oX4xMPGyKjXjHV7bOke6779Alqi1s1bd7Xg+SXeHDl+F9nRC2lVGyofdbFyfzFaoVrlivdVepit8DX6mFEbO7IwUV4R0xYfe3OREtXjIjE+XOr1oo789SSvI7PcnfKlrFtrysVMg7QxZxQvcedAEeHy7jad3tw1ph+dNXodwtrUZA1HCbNxqcoP2JYWs1LBSoRk13NrUmKtR2XrheFhrxrGjktLGZEVTASXu5Gt7T60C9EKn42RFLpCnuwtixNtmh/Dhb8D4YoSNUefHGIT/NxbYlfsvYNTEa9UWWY2A6pBm9BbicFD1c9ii2odvbKo2/R9OPjziEHXnPQxrah1ZZeUeclt6v70X5M1mIiIld60rN8Hmwvfk3VOdfoTk4XjexML/U1O3b5oDj49KXLx5yP3N7he9IX+tqleHzR1vgdFDnV46PMMfu/ctK7Py+ZUra91Lr485Dqgd140DncAy2MW087HQk2sN/LuOaB16FdQ16vF/9oPzcdwqs/Ll1tG35y98Hjrsl2kMrPoF9efM4abz2gu5+jPM817Z7wmNz/p/wW13lRq9ebd00e/1IwBzWshZD6Ul07i7VFk5PWlgvsDLhlS8R/xp0v8VwmvchplctShpPcryqnpFjw1/eWlq3f6mY9Mhk6ZTEVqJp5o0snYXXEg7sDPbu4B1j0lKr+plH8R6/6Otge1fbIeOpsuMwhtqosoC2J203p0b87RlkmMZ3rlGd8W1OCe+CHV4nnf+9Qez/y2Cn0r/5JEzvXPRSp6hchkdZfPzKsO9KC4AydzQfp734DO1MtsljyBNvTYY5TnqGSkpL9+XhCiR3HhqqQLQujRUkW546TcoTNJUQntTqFvXRrhY7XTj/sOeD4/zLNLgfgV1YVqdcCVmpPqyCHVU8oMdEj/2xCh/UZswd8tVXZLerQmaUbvoCjRufWs+iTY0jkuzNeMVykPpjN1c9Uupaje7B5WZ2IbpVzkifdq/KACP7CkbNLbeklx90RU5w5nFTRnSiZ7P3ckdN5T6gGj2TAfH9Xx87yC19yj//dK0fcsSfdBlg5l2wPaz5sQvUTpZo+kl+cqq/O2dyuk1V9tSprmZKUZB/qMZPWeQKLGXfnsl+cguVxNVELHnuvyRCZ3WiR2UuLn59jjNETo0Vu+rz495ajMWElmHl8izz1kmXvQ6TPS9FkKA1YSZR9Or2zsrP5fH+uFkxvXd357/ebDmxuf3lb/D8j74qfiZ2IT1r7fiU9h/B+JY+D0R/FX8bX4R+MPjb80/tb4u2z63iWF+Yko/Gv8+7+go1/A</latexit> min ✓ E(✓) , 1 n n X i=1 `(f✓(xi), yi) Herbert Robbins Sutton Monro <latexit 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Stochastic gradient descent: <latexit 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✓`+1 = ✓` ⌧ ` rE`(✓`) <latexit 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E`(✓) , `(f✓(xi), yi) <latexit 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i rand Adam [2016] momentum (acceleration) sign gradient (normalization) θℓ+1 ≈ θℓ − τℓ sign(∇ℰ(θℓ )) Diederik Kingma
  11. The Complexity of Gradient Computation Hypothesis: elementary operations (a ⇥

    b, log(a), p a . . . ) and their derivatives cost O(1). <latexit sha1_base64="XBU0O16nV93z/lhQgn5HZRGW7gQ=">AAA9x3ictVttc9u4EUaub5f0Ldd+7EyHrZ3OXSf12G6m7dyNZy6xncQXX+JEspO7U5LRCy0zkUSFpGwnOn/oT+ov6M/o9Fv7rf3Uv9DdBUCAEsgF3NQc2yCE59nFEljsAlRvOkryYn3971c++M53v/f9H3x49doPf/Tjn/z0+kc/O8rTWdaPD/vpKM2e9bp5PEom8WGRFKP42TSLu+PeKH7ae72Nnz89jbM8SSft4u00fj7uDifJcdLvFlD18vqDThGfF73jeSsuZtNPL6LVzm4cfRp1njx5MYg6WTI8KbpZlp5hzWrUT8fTWdEF9iiZRKsPVqN0GmfEla+9vL6yvrZOP9FyYUMVVoT6OUg/+uWm6IiBSEVfzMRYxGIiCiiPRFfkcH0jNsS6mELdczGHugxKCX0eiwtxDbAzaBVDiy7Uvoa/Q7j7RtVO4B45c0L3QcoIfjNARuIGYFJol0EZpUX0+YyYsbaOe06cqNtb+N9TXGOoLcQJ1HI43dIXh30pxLH4E/UhgT5NqQZ711csM7IKah5ZvSqAYQp1WB7A5xmU+4TUdo4Ik1Pf0bZd+vxf1BJr8b6v2s7Ev0nLG3BFoqV6n5YMXXFK/BE9zRl8JvUZgeQhMMSqj1g6I1uPqfcTaD+H+odwXVBJ26QH15xqLxqR23C5kNss8h5cLuQ9FrkPlwu5zyIP4HIhDxQSsRnZ3I1vweXCt1jJj+FyIR+zyCdwuZBPWOQRXC7kEYv8Gi4X8msWeRcuF/Iui3wAlwv5gEW24XIh2yzyEC4X8pBF7sLlQu4qZP1MzeBKiSdhZuVtKFdloKcYQc1tVr875B1d2Dsec7pfg+Vn9Q78d2N3PGwa12B3PcbdcQ2WH3n3wEe6sbwvuk+riQt7n8XuwQhwY/dY7BfiVQ32C4+Z9roGy8+1fWjnxvLe90u4c2O/ZLEPoeTG8mvUI6hxYx95rBjTGuwBi30s3tRgfbx+VoPl/X4L/Ioby69TbWjvxvp401kNlvenRxDBuLH8avUUat3Ypyz2mTivwT5jsV+Bd3djv/JYYd/VYPUae41WkCHFIzHM2Ca2bjkrsTQFti4jf1SuLSOKjXtQz2GGJWZImDGLuFci7nki9kvEvrdeeelHc4p3eSmtEtHyRPTKtQlLBdt+ULbH0sgDsVMidhYQTREpPmvdl1OKLnQNhyzKlQtLPn1KS/+NpViNh2bPqxGPKgg5tk9o5N+kbAkzKLRUE9tJucZLZET3TYgzyt50L7UMHleUXsFGnbOongPVY1FvHai3LGrmQM1Y1KkDdcqizMy3cR2PEWDsj89iTndyBMgYuf6KICq4DavOfZijEYyfA4gCn1DNI/jfotybu5o0w2we10nc5Xhe8cQZlOZiBepNVrhD+fWIZlgMmsmWj1SOj3e4tzFXc0564YtyJY/KHRN/noT0GZY8GC1GNJ/CeB5QzQVFd7IUhr9fzntdCsPvksUvKIqXpTB8obQvLqF7W2Hbl8C2YDZNlfVNOZRD7r9IDl2+Rqsuelx8qmM1ZpDvPJB/Tz2ZvUs8l20qSfuYchhHbvUvr/QvhMPYObfsHMaC0ZOMenUpCu7JROW9phyqQ0qr6ETpYe5Cnwy2Gagno8thHAcQcW1Tzj23yqGjd1r2xpTDOI6E3Pe8oEhel8M4hnQv7WHKYRy429JVeb4ph3p2tIDMnU051KtPaBcY94DkmJc1JirKKE6aKbaE4oPm3Ro75l9ex3DP5kWZIzQzmdi2nqdXrmXNGul4IQavVgTqgfHFzIrBqhxzscnmV1KHorK+L/OYNR4tvw9WjGD2yzMAbs98BBrqPQn03iNg3GCzrmrPNG6TxeEoOV5AdVRtwUaLRq7cNarWvaRaLi8zvTV27JC/zmnsTSkm3CfLcnbYr33CdYychfYrFuL5Qmz3Ts3XqvXXWdx0ATEtR1qfToTkSVpznuqyesuy8Q11ylPAJc98zPjF3eZj5W0w50nJF6EuTTLtdnofya7DdfWmMHvc8rOInij6q1PyGgmdSOVsFqp3i2U0Pqd7w31IZ3IoQ3L04TlGimUq5KkZ7qLjfnpEHtX2t5xstJfeoZPlnLyu9sfN6KGFHjrQ4TnONqwYD6HUhpzhEO7aHlnOtdJWKVk8E78rT0dTeoLNGf2o4iE1h/Q3ccVDNmXZJxWWM0DjaJBZuj/HIo/Gd5aY+KzfpY/JXaue/wad3Orz7S6N8frRXL8TMyCpmyQ1olkjT3Xl3aIEqcHc+ckmxa/NvUR5IRLRh3JSX1iSpV0mdOIfUwY7pch4RLONmx3V1vb+1OInWtKB0GfneJqdkoeMyP9FsD6lNCYj+rXfHdAn6NIjjMhH+vidpIxuXLFOwo4xE8clQr7VYMZbTL5sRvI1rz27chqLMmOQ68DFwtjWNtmnWDAmqZny7mZuN68+iDTvSdijRDKasfIxyf+E/upfPU5WlkYEWhifQK58net5pJSzoI26tMo3+yDd1tZytdThhdLarH9Gp9WKZjuUcaE+uFoPQHKf7qUsHCUZ6Z0vtZHraNNuLjJPF+yIvT2mLF76/aFagVHvm7RKrtCc69AoGcIoKMosQrfldpEX5TbLqrL7cef/F3Zj66rVkDESZgdXWojb348pW7O1HMGoluP3Nc0mt9WzhVbNciY0FsfWXP4Wan8Ff7Xe+t6Pp1fxCndoDEgGc2csImuipRZ+su5UZOmRqbnMvZFnxqRuZddcJr+W3s3k2KfBLAc0as7VroUuX4bjlcXxytOGbTprNFbU9doTvWRzi7Y6rfSVFyKtHcA8Y5n5iEyjEg8t7VzKj3XAsvI5vka9Y7nWWa4uzFb7NMCe8z5I91xfnN3flqt7JO5SbNOnCEzmLwOapQnFXLq2OVOTDCj5lvKv9uzvUA1K75EHRWb5HifOGHnq1KfrotT0N2plS8nPG4+g31s6U220j+1Q+fdLyDHNiZzmpUbcohax0t/WI1rwSGtWzBHRzn+XYioZdzTnzHZr80yiSjxh8k05q4wsmSlMyP7cztveUva6Z+WvEeWEMxVd94Ar/Akjg8TonQR3ZJnTE8JVTp4kyIi2R/5z2U/JU7yJpdEaaT0XWx4+Rma9ZqzbY0v3WPftt9ASrW6euqsFL2/kLZGTd5kTvS6tamMVo84X7i/H1VWrXPW+yQ6zBbnGHjNqY2cWJsurYjriM28pUqMwKRLjIyWsFyH6h2keorM8nfJl1q01c3WnQfqYE8qXuPdAEeGK7j52RnOfMP3oLfH1CGuzyRqOCXfjUrU/YHta3JW6urQOydqrjavRyFqJ6lYKzW6vFsZ/Sw8Zk/cbCW7PRra2de9UshR+F0Yy9IV8o7cuP7Q5P4ML/0bClR1qiT57hy2Ib2+LbbH7Ht6GeKPKckczohr0BYOF3Lur+llt0WyjNxa7ze8jwV9GArbmtE9oJQ3VXTLzmtvs/vxn5AUyEbPam5bhfbCl8D1ZlhTSn4Q8G9+bROjv4oT2RUvw6UlVir8cea7B9eJY6O80hfVBs/M9qEoIkaHfY/B75qZ1uCxbUrO9lqX4ypCrgD5x0Tg8+avPVUw7Hw+VWU/k/UtA73DcwK5Xi/+1H1qOkRQuy1daTt81e+Xx1GW7WO3IYjwcPmeMNJ/RXC/RX2Za9s5ES255Mu6Lgp5UavXm/fNjPGrGgJY1F3IflNdO4u1RZPT1ZcFzAZcOqfiP+OsV/tsIb0qOOj1CmPQ5RT2bbsGz6W9cunqnP/PRyfDU6VRlM3lEi96I3RZ74i78bpcRYOjbofK7lPI/Yt3fnx1A7TF5D72LLncOOlQX0+6HOUUb0L3aY3x5fWVj8VvIy4WjzbWNP6zdery58vkd9Q3lD8UvxK8hL9kQfxSfi/vQ30PQ6S/ib+If4p9be1vp1unWuWz6wRWF+bmo/Gz9+b9va/Dv</latexit> Setup: E : Rd ! R computable in K operations. <latexit sha1_base64="BLbI91EqNObAlbI/2RJe4xs+gGo=">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</latexit> Question: What is the complexity of computing rE : Rd ! Rd? Finite di↵erences: <latexit sha1_base64="IJjbmZX1RxHFJ6LjH+Uz9oKFiHc=">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</latexit> rE(✓) ⇡ 1 " (E(✓ + " 1) E(✓), . . . E(✓ + " d) E(✓)) <latexit sha1_base64="elfmtYFgpa8JKGP5IFiZSMc3eME=">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</latexit> K(d + 1) operations, intractable for large d.
  12. The Complexity of Gradient Computation Hypothesis: elementary operations (a ⇥

    b, log(a), p a . . . ) and their derivatives cost O(1). <latexit sha1_base64="XBU0O16nV93z/lhQgn5HZRGW7gQ=">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</latexit> Setup: E : Rd ! R computable in K operations. <latexit sha1_base64="BLbI91EqNObAlbI/2RJe4xs+gGo=">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</latexit> Question: What is the complexity of computing rE : Rd ! Rd? Finite di↵erences: <latexit sha1_base64="IJjbmZX1RxHFJ6LjH+Uz9oKFiHc=">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</latexit> rE(✓) ⇡ 1 " (E(✓ + " 1) E(✓), . . . E(✓ + " d) E(✓)) <latexit sha1_base64="elfmtYFgpa8JKGP5IFiZSMc3eME=">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</latexit> K(d + 1) operations, intractable for large d. This algorithm is reverse mode automatic di↵erentiation [Seppo Linnainmaa, 1970] Theorem: there is an algorithm to compute rE in O(K) operations. Seppo Linnainmaa
  13. Linear models (1 layer) <latexit sha1_base64="fpFKG9ItOKdtii1xLRoQrxil3sU=">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</latexit> f✓(x) = hx, ✓i

    = P k xk✓k x f(x) = 0 x y y = f(x) <latexit sha1_base64="+XMTGU2h84i8kefw5ntE/6bvDrk=">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</latexit> y = hx, ✓i <latexit 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✓d <latexit 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✓1 <latexit 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. . . <latexit 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Regression <latexit sha1_base64="OlDgU+OKIQz+ynHUitJ8TvkH7r8=">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</latexit> Classification:
  14. Linear models (1 layer) <latexit sha1_base64="fpFKG9ItOKdtii1xLRoQrxil3sU=">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</latexit> f✓(x) = hx, ✓i

    = P k xk✓k x f(x) = 0 x y y = f(x) <latexit sha1_base64="+XMTGU2h84i8kefw5ntE/6bvDrk=">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</latexit> y = hx, ✓i <latexit 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✓d <latexit 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✓1 <latexit 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. . . <latexit 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Regression <latexit 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Classification: <latexit 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min ✓ , 1 n n X i=1 `(hxi, ✓i i, yi) <latexit sha1_base64="PvseIPRi4Nf4IH/9H8AceNGnDTM=">AABE9nictVzbchTJES3WtzW+sbbf/NJrLQ52A2Mh40t4wxELGiG0CBiYkWAXATGX1tDQMz3MDcGsfsXhF4fDfvIf+Dv8AY6wn/wLzktVV/VMdWe1jOmQVF1dJzMruyorM6ua7jhNprPNzX+ce+9rX//GN7/1/rfPf+e73/v+Dy588MPDaTaf9OKDXpZmk0fdzjROk1F8MEtmafxoPIk7w24aP+y+3MbnDxfxZJpko/bszTh+MuwMRslx0uvMoOrZhR9vZ6NFfBJl41kyTN5S7e+eXdjYvLJJ/6L1wlVd2FD6XzP74MN/qiPVV5nqqbkaqliN1AzKqeqoKVyP1VW1qcZQ90QtoW4CpYSex+pUnQfsHFrF0KIDtS/h9wDuHuvaEdwjzSmhe8AlhZ8JICN1ETAZtJtAGblF9HxOlLG2jPaSaKJsb+BvV9MaQu1MPYdaCWdahuKwLzN1rH5LfUigT2Oqwd71NJU5aQUlj5xezYDCGOqw3IfnEyj3CGn0HBFmSn1H3Xbo+b+oJdbifU+3nat/k5QX4YpUS/c+yyl01ILoR/Q25/CM5UmB8wAoxLqPWHpNuh5S70fQfgn1d+E6pZLRSReuJdWeViK34fIht0XkLlw+5K6I3IfLh9wXkU24fMimRiJ2Qjr341tw+fAtkfN9uHzI+yLyAVw+5AMReQiXD3koIr+Ey4f8UkTehMuHvCkib8PlQ94WkW24fMi2iDyAy4c8EJE7cPmQOxpZPlMncGVEJxFm5XUoF3mgpUih5roo3w2yjj7sjYA53SvByrO6AX/92EaATuMS7E7AuDsuwcojbxdspB8r26JbtJr4sLdE7B6MAD92T8R+rl6UYD8PmGkvS7DyXNuHdn6sbH3vwJ0fe0fE3oWSHyuvUfegxo+9F7BijEuwTRF7X70qwYZY/UkJVrb7LbArfqy8TrWhvR8bYk3nJVjZnh6CB+PHyqvVQ6j1Yx+K2EfqpAT7SMR+Adbdj/0iYIV9W4I1a+x5WkEG5I/EMGOrqHXyWYmlMVDrCPzTfG1JyTfuQr2EGeSYAWGGImI3R+wGIvZzxH6wXNPcjk7J35W5tHJEKxDRzdcmLM3E9v28PZbSAEQjRzRWEFUeKb5r05cFeRemRkLO8pULSyF9ynL7jaVYj4dqy2sQ9woIHtvPaeRfpmgJIyjUVBW15/kaz8iI7qsQryl6M700PGTcLLcKLupERHU9qK6IeuNBvRFRcw9qLqIWHtRCRNmZ7+KOAkaA1T++iyXd8QhgH7n8isAruA6rzi2YoxGMnyZ4gQ+o5h78bVHsLV1VkmE0j+skZjmeFCzxBEpLtQH1NipsUHyd0gyLQTJueU/H+HiHuY2lnnNshU/zlTzKMybhdBKSZ5DTQW8xovlUj85tqjkl745L9fC38nlvSvXwO6TxU/LiuVQPP9PSz84ge1tj22fAtmA2jbX2bbkuDc6/MA1TPk+rLlpcfKtDPWaQ3klN+nv6zeyd4b1sU4n1Y8v1aEyd/k0L/atDw+p56ui5HhX0ntjrNaWodk9GOu615boyZLSKjrQc9q7um8E2ff1mTLkejSZ4XNsUcy+dct3RO857Y8v1aBwqznuekidvyvVoDOie9WHL9WhgtqWj43xbrmvZUQMcO9tyXas+oiww5oB4zHON9Yom5CfNNbWE/IPqbI3r86+vY5izeZrHCNWUrG9bTqebr2XVEhl/IQarNqspB/oXc8cHK9JYqi0xvmIZZoX1fZ2OXeNR8/ugxQhmP+8BSDnzFCQ0OQm03ilQvCpGXcWeGdyWiMNRcryCOtK1M9FbtHw5a1Sse0a1Ulxme2v1eET2ekpjb0w+4T5pVtLDfukbLqMoaWi/oCGZXh3dvdXztaj9TRE3XkGM85HWox0h3kmrjlN9Wm85Or6od3lmcPGejx2/mG0+1tYGY56MbBHKUsXTbWfySG4drquXlc1x87OI3ijaqwVZjYR2pKZiFGqyxeyNL+ne0j6gPTnkwTR68B4jTWWseNcMs+iYT4/Iorr2VuKN+jIZOi5Pyeoae1yNHjjogQddP8bZhhXjLpTaEDMcwF07IMo5n+sqI41P1M/z3dGM3mB1RJ8WLKShwfYmLljIqij7eYHKa0DjaOAoPZzGKh2DP1qjJEf9Pnls7Fq0/Bdp59bsb3dojJeP5vJMTJ+4bhHXiGYN7+ry3SoHlmDpfbJF/mt1L5FfHY5oQyWuTx3OrJcR7fjHFMGOyTNOabZJs6PY2s1PrT4xnJrK7J3jbnZGFjIi+xfB+pTRmIzoxz07YHbQ2SKkZCND7E6Sezc+XycRx5j14xLFpxrseIvJls2Jv6Hrzq4pjUWOGHgdOF0Z20Yn++QLxsR1oq27ndvVqw8i7TkJd5QwRTtWLhH/j+m3+THjZGNtRKCG8Q1Mta3zvY+MYhbUUYdW+WobZNq6Un6Uy/BUS23XPyvTRwXJGhRxoTy4WveBc4/umReOkgnJPV1rw+toVTYXKY9X9Ii9PaYonu3+QK/AKPdlWiU3aM4d0SgZwCiY5VGEaStlkVf5VvMqUg+jPf2/ULe6LmoNKUbKZnBZQ1J+P6ZozZUyhVHN4/clzSa/1icrrar5jGgsDp25/BXUfgi/jdzmPoxOt2AVbtAYYAr2zmqEa6K1FmG8bhR4mZFpaNl7y8+OSdPKrTlLfM3WzcbYi9pUmjRqTnTWwpTPQuOFQ+NFoA7btNdotWjqjSV6JsYWbb1bGcqvDrd2DcpzkbLskRlUEiClG0uFUe2LVOUY36DeirQ2RVodmK3uboA750OQ/rm+Oru/ylf3SN0k36ZHHhjHL32apQn5XKa2OlJjCsj5mrav7uw/ohrk3iULipT5HCfOGN516tF1mkv6M72yZWTnrUUw55Ze6zbGxh5R+ZdryCHNiSnNS4O4Ri1iLb8rR7Rika44PkdEmf8O+VTsd1THzG5r+06igj9h402eVZYXRwoj0r+Uedtbi173nPg1ophwrr3rLtCq/4aRAmNMJsHvWU7pDeEqxzsJ7NF2yX6u2ynexRs5El0hqZfq9wE2hqNeO9bdsWV6bPr2CbRErdu37msh80uDOUr8zrKj16FVbah91OXK/dlodfQqV7yv0sN8ha/Vx5zauJGFjfKKmCP1aTAXlqgeF8aEcKnXizry15O8jsy8OxVK2bQ2lIuZBrYxzyleks6BIsLn3V3yenMfC/3ortHrEtalxjUSJczGZTo/4FpazEpFKxGSWy+tSamzHpWtF5aHu2pYO86WMiYrmCopd8Ot3T4cFaIVORvDFHqKT/aWxYkuzU/hwt+R8kWJhmNIDrEFfu51ta123sGpiFe6zJnNiGrQJvRXYvCO7mexRbWOXjnUXfohHMJ5JKBrSfqEVtS6sjNlWXKXejj912QNJioWpbct6/fB5SL3ZJ1Tnf4kZOHk3iTKfJNTty+GQ0hPilzC+fD+htSLY2W+barXB0Nd7kGRQx0e5jxD2Du3revzcjlV62udSygPXgfMzovB4Q5gecxi24VYqInzRt49B7QOxxXUzWrxv/bD8LGc6vMK5Talb85eBLx1bhfrzCz6xfXnjOUWMprLOYbzzPLeWa/Jz4/9v6jWm8qc3rx7+uiX2jFgeC0V50Nl6RjvjiIrbygV3B/wyZCp/6i/n5O/SniV0yiTow4ls19RTs20kKmZLy99vTPPQmSydMpkKlKz8USLTsZuqz11E362cw+w7ilR/qaS/yLW/x1tH2qPyXqYbDpnEI6oLqYsiN1N69O9PUdbJjGe6eUzvm2owT3xfarF8753qT2e+W0X+lb+JQnP9TsqU/1CZLK6y2fnVRd6UNyB41yQ+d43ojP1nM3iE2jDgD1GPkfFkZL5+nlJiD7FhauSLglhRksV5a6XcpfOJMUltLuFvvVohI/1Tj/uO+D5/E6eXYrUL6iuo1cHXKklqZoeqR5TZqBL+t+ECO1X6jL8vazLfkmba5JO6R0UJTpxnlWfBDv1jgv7NeNFyoOZTN1Ct8soqre7h9WZ2EYpFz7xXo0fVOAHjpQtelsvKe6eqOrc4byC5lzL5O7njpTJe7IeMJrt5OOjOn5eVPBaBPT/din6tiPpLsjSpWx7RPt5E6KXat3skPR8rrI6b3urQlrz1SbTtCcr7TgwZySr9wRSPe7KZz+fg5RyNXEJHXeu84lM6bRI4qUkz89xwGmITkBv5b6G9FSiMhclmQd8ibwIkGURQOdYkOZYpDAQJdH24dmFjaur/9fHeuFw68rVX1+5dn9r47Mb+v8BeV/9RP1UXYK17zfqMxj/TXUAnN6qP6q/qL82Thp/aPyp8Wdu+t45jfmRKvxr/O2/xB1IRg==</latexit> Convex optimization:
  15. Linear models (1 layer) <latexit sha1_base64="fpFKG9ItOKdtii1xLRoQrxil3sU=">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</latexit> f✓(x) = hx, ✓i

    = P k xk✓k <latexit sha1_base64="5ovuZPU8dBWGm61iCoXPGu2QOzU=">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</latexit> Deep learning methods: learn '(x)! <latexit 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Kernel methods: replace x by '(x) 2 RD <latexit 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(D d, even D = 1!) x f(x) = 0 x y y = f(x) <latexit 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y = hx, ✓i <latexit 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✓d <latexit 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✓1 <latexit 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. . . <latexit 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Regression <latexit 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Classification: <latexit 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min ✓ , 1 n n X i=1 `(hxi, ✓i i, yi) <latexit 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Convex optimization:
  16. Multi-layer Perceptron <latexit 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x <latexit 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x1 <latexit 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    x2 <latexit 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xD 1 <latexit 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y = xD <latexit 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. . . <latexit 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Wk 2 Rdk+1 ⇥dk <latexit 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bk 2 Rdk+1 <latexit 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f✓(x0) = xD <latexit 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xk+1 , (Wkxk + bk) <latexit 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✓ = {(Wk, bk)}D 1 k=0 Frank Rosenblatt
  17. Multi-layer Perceptron <latexit 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x <latexit 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x1 <latexit 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    x2 <latexit 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xD 1 <latexit 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y = xD <latexit 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. . . <latexit 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Wk 2 Rdk+1 ⇥dk <latexit 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bk 2 Rdk+1 <latexit 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f✓(x0) = xD <latexit 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xk+1 , (Wkxk + bk) <latexit 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✓ = {(Wk, bk)}D 1 k=0 <latexit 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Sigmoid <latexit 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ReLu <latexit 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Non-linearity: must be non-polynomial to increase expressivity. Frank Rosenblatt
  18. Multi-layer Perceptron <latexit 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x <latexit 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x1 <latexit 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    x2 <latexit 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xD 1 <latexit 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y = xD <latexit 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. . . <latexit 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Wk 2 Rdk+1 ⇥dk <latexit 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bk 2 Rdk+1 <latexit 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f✓(x0) = xD <latexit 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xk+1 , (Wkxk + bk) <latexit 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✓ = {(Wk, bk)}D 1 k=0 <latexit 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Weight matrix: needs extra constraints (e.g. convolution & sub-sampling) <latexit 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s <latexit 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Sigmoid <latexit 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ReLu <latexit 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Non-linearity: must be non-polynomial to increase expressivity. Frank Rosenblatt
  19. Two Layers Perceptron: Universality <latexit 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Wx <latexit 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x

    <latexit 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w1 <latexit sha1_base64="dTEPGRjwpObKto22Pdj89jMy0R4=">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</latexit> a1 p = 6 neurons p = 30 neurons p = 100 neurons <latexit sha1_base64="v5o5dJsoyPefg3wihB9aJpDhTXc=">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</latexit> Input y = f(x) <latexit sha1_base64="ttgglA5lmQEZUY9EO2WF1erSv+8=">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</latexit> ! sum of “ridge” functions (hx, wi + b) <latexit 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f✓(x) , p X s=1 as (hx, ws i + bs) <latexit 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wp <latexit 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ap George Cybenko Andrew Barron
  20. Two Layers Perceptron: Universality <latexit 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Wx <latexit 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x

    <latexit 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w1 <latexit sha1_base64="dTEPGRjwpObKto22Pdj89jMy0R4=">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</latexit> a1 p = 6 neurons p = 30 neurons p = 100 neurons <latexit sha1_base64="v5o5dJsoyPefg3wihB9aJpDhTXc=">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</latexit> Input y = f(x) <latexit sha1_base64="ttgglA5lmQEZUY9EO2WF1erSv+8=">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</latexit> ! sum of “ridge” functions (hx, wi + b) <latexit 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f✓(x) , p X s=1 as (hx, ws i + bs) <latexit 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wp <latexit 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ap George Cybenko Andrew Barron
  21. Convolutional Networks (ConvNets) <latexit 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x1 <latexit

    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y = xD <latexit 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. . . <latexit 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! Leverage translation invariance of images. <latexit sha1_base64="Nz+CDjAByD39QLOSiLMnQuuLgrE=">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</latexit> ! Sub-sampling: breaks invariance but increase receptive fields. <latexit 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Pool <latexit 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x <latexit 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x1 Yann Lecun
  22. Convolutional Networks (ConvNets) <latexit 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x1 <latexit

    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y = xD <latexit 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. . . <latexit 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! Leverage translation invariance of images. <latexit sha1_base64="Nz+CDjAByD39QLOSiLMnQuuLgrE=">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</latexit> ! Sub-sampling: breaks invariance but increase receptive fields. <latexit 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AlexNet, 2011 <latexit 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Pool <latexit 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x <latexit 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x1 Yann Lecun Ilya Sutskever Alex Krizhevsky Geoffrey Hinton
  23. Conv Conv Conv Conv Conv Conv ReLu ReLu ReLu ReLu

    ReLu ReLu Pool Pool Pool Convolutional NN (example) Fully connected
  24. ResNet Architectures [He et al’ 16] ResNet-34 <latexit sha1_base64="hGNaHRogJoszxpvRv/VDNWjykms=">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