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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=">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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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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=">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=">AABFHnictVxfc9u4EUeu/67pv1z72BdefekkHV9qp+mfmevNXGI5ji9O4kSyk7so8VASJTOhREWUFCc6fZdOv0jfOn3pdNqn9gN0pn3qV+juAiBACeSCbhqObRDEb3exBBa7CzCdcRJn062tf1x472tf/8Y3v/X+ty9+57vf+/4PLn3ww+MsnU260VE3TdLJk06YRUk8io6m8TSJnownUTjsJNHjzssdfP54Hk2yOB21pm/G0bNhOBjF/bgbTqHq5NJv28N4dNKenkbTMGj3J2F3sb1cjJZBO5sNTxbxp9vL56OgHSXJlb5qduXsJL66+QZ+nVza2Lq2Rf+C9cK2KmwI9e8w/eDDf4q26IlUdMVMDEUkRmIK5USEIoPrqdgWW2IMdc/EAuomUIrpeSSW4iJgZ9AqghYh1L6E3wO4e6pqR3CPNDNCd4FLAj8TQAbiMmBSaDeBMnIL6PmMKGNtGe0F0UTZ3sDfjqI1hNqpOIVaDqdb+uKwL1PRF7+hPsTQpzHVYO+6isqMtIKSB1avpkBhDHVY7sHzCZS7hNR6DgiTUd9RtyE9/xe1xFq876q2M/FvkvIyXIFoqt6nOYVQzIl+QG9zBs+kPAlwHgCFSPURS69J10Pq/QjaL6D+PlxLKmmddOBaUO2yErkDlwu5wyL34HIh91jkAVwu5AGLPITLhTxUSMROSOdufBMuF77Jcn4Ilwv5kEU+gsuFfMQij+FyIY9Z5JdwuZBfssjbcLmQt1nkXbhcyLsssgWXC9likUdwuZBHLHIXLhdyVyHLZ+oErpToxMysvAnlIg+0FAnU3GTlu0XW0YW95TGnuyVYflY34K8b2/DQaVSC3fUYd/0SLD/y9sBGurG8LbpDq4kLe4fF7sMIcGP3Wezn4kUJ9nOPmfayBMvPtQNo58by1vce3Lmx91jsfSi5sfwa9QBq3NgHHivGuAR7yGIfilclWB+rPynB8na/CXbFjeXXqRa0d2N9rOmsBMvb02PwYNxYfrV6DLVu7GMW+0SclWCfsNgvwLq7sV94rLBvS7B6jb1IK8iA/JEIZmwVtTCflVgaA7WQ4Z/ka0tCvnEH6jnMIMcMCDNkEXs5Ys8TcZAjDrzlynI7mpG/y3Np5oimJ6KTr01YmrLte3l7LCUeiEaOaKwgqjxSfNe6L3PyLnQNh5zmKxeWfPqU5vYbS5EaD9WWVyMeFBBybJ/SyN+kaAkjKNRUFbXTfI2XyIDuqxCvKXrTvdQ8eNw0two26oxFdRyoDot640C9YVEzB2rGouYO1JxFmZlv49oeI8DoH9/Fgu7kCJA+cvkVgFdwE1adOzBHAxg/h+AFPqKaB/C3SbE3d1VJhtE8rpOY5XhWsMQTKC3EBtSbqLBB8XVCMywCyWTLByrGxzvMbSzUnJNWeJmv5EGeMfGnE5M8g5wOeosBzad6dO5SzZK8O1mqh7+Tz3tdqoffJY0vyYuXpXr4qZJ+eg7ZWwrbOge2CbNprLRvynVpyPyLpKHLF2nVRYuLb3WoxgzSO6tJf1+9mf1zvJcdKkn9mHI9GpnVv6zQvzo0jJ4zS8/1qKD3JL1eXQpq92Sk4l5TritDSqvoSMlh7uq+GWzTU29Gl+vROASPa4di7oVVrjt6x3lvTLkejWMh855L8uR1uR6NAd1LfZhyPRqYbQlVnG/KdS07akDGzqZc16qPKAuMOSA55mWN8Yom5CfNFLWY/IPqbI3t86+vY5izeZ7HCNWUjG9bTqeTr2XVEml/IQKrNq0pB/oXM8sHK9JYiOtsfCVlmBbW93U6Zo1HzR+AFgOY/XIPgMuZJyChzkmg9U6A4jYbdRV7pnHXWRyOkv4Kqq1qp6y3aPjKrFGx7oRqubjM9NbosU32OqOxNyaf8IA0y+nhoPQNl1HkNHRQ0BBPr47u3qr5WtT+FosbryDG+Ujr0o6Q3EmrjlNdWm9aOr6sdnmmcMk9HzN+MdvcV9YGY56UbBHKUsXTbqfzSHYdrqubwuS45bOA3ijaqzlZjZh2pDI2CtXZYumNL+je0D6iPTnkIWl04T0GispYyF0zzKJjPj0gi2rbW4436ktn6GQ5I6ur7XE1emChBw50/RhnB1aM+1BqQcxwBHctjyjnYq6rlDQ+ER/nu6MpvcHqiD4pWEhNQ9qbqGAhq6Ls0wKV14DG0SCjdH8aq3Q0vr1GiY/6XfKY2LVo+S/Tzq3e3w5pjJeP5vJMTI+4XieuAc0auasr71Y5SAkWzifXyX+t7iXyq8MRbSjH9bnFWeplRDv+EUWwY/KME5pt3OwotrbzU6tPNKdDoffOcTc7JQsZkP0LYH1KaUwG9GOfHdA76NIiJGQjfexOnHs3Ll8nZseY8eNiIU81mPEWkS2bEX9N155dGY1FGTHIdWC5Mra1Tg7IF4yI60RZdzO3q1cfRJpzEvYokRTNWLlC/K/Sb/2jx8nG2ohADeMbyJStc72PlGIW1FFIq3y1DdJtbSk/ymV4rqQ265+R6aOCZA2KuFAeXK17wLlL95IXjpIJyZ2ttZHraFU2FymPV/SIve1TFC/t/kCtwCj3Jq2SGzTn2jRKBjAKpnkUodtyWeRVvtW8itT9aGf/F+pG10WtIcVAmAyu1BCX348oWrOlTGBUy/H7kmaTW+uTlVbVfEY0FofWXP4Kaj+E31pufe9Hp1OwCrdoDEgK5s5oRNYEay38eN0q8NIjU9My94afGZO6lV1znvhaWjcTY89rUzmkUXOmsha6fB4aLywaLzx12KK9RqNFXa8t0QkbW7TUbqUvvzrcWjUoz1jKvEemUbGHlHYs5Ue1x1LlY3yNesvS2mJphTBb7d0Ae877IN1zfXV2f5Wv7oG4Tb5NlzwwGb/0aJbG5HPp2upITVJAzjeUfbVnf5tqkHuHLChSluc4ccbIXacuXctc0p+qlS0lO28sgj639Fq10Ta2TeVfrCGHNCcympcacYNaREp+W45gxSJds3yOgDL/IflU0u+ojpnt1uadBAV/wsSbclYZXjJSGJH+uczb/lr0um/FrwHFhDPlXXeAVv03jBQkRmcS3J5lRm8IVzm5kyA92g7Zz3U7JXfxRpZE10jqhfjUw8bIqNeMdXts6R7rvv0MWqLWzVt3teD5Jd4cOX7n2dELaVUbKh91sXJ/PlqhWuWK91V6mK3wNfqYURs7sjBRXhHTFp94c5ES1eMiMT5c6vWijvz1JK8js9yd8qWsW2vKxUyDtDGnFC9x50AR4fLurji9uatMPzpr9DqEtanJGo4SZuNSlR+wLS1mpYKVCMmu59akxFqPytYLw8NeNYwdl5YyIiuYCC53I1vbfWgXohU+GyMpdIU82VsWJ9o0P4ELfwfCFSVqjj45xCb4uTfFjth9B6ciXqmyzGwGVIM2obcSg4eqn8UW1Tp6ZVG36ftw8OcRg6456WNaUevKLinzktvU/em/JmswERErvWlZvw82F74n65zq9CcmC8f3Jhb6m5y6fdEcfHpS5OLPR+5vcL3oC/1tU70+aOp8D4oc6vDQ5xn83rlpXZ+XzalaX+tcfHnIdUDvvGgc7gCWxyymnY+Fmlhv5N1zQOvQr6CuV4v/tR+aj+FUn5cvt4y+OXvh8dZlu0hlZtEvrj9nDDef0VzO0Z9nmvfOeE1uftL/C2q9qdTqzbunj36pGQOa10LIfCgvncTbo8jI60sF9wdcMqTiP+IPF/ivEl7lNMrkqENJ71eUU9MteGr6y0tX7/QzH5kMnTKZitRMPNGkk7E7Yl/chp+d3AOse0pUflMp/yLW/R1tD2r7ZD10Nl1mENpUF1EWxOym9ejenKMtkxjP9Mozvi2owT3xA6rF8773qT2e+W0V+lb+JYmc6/dEKnqFyGR1l8/Mqw70oLgDJ3NB+nvfgM7Uy2yWPIE29NhjlOeoZKSkv35eEKJHceGqpAtC6NFSRbnjpNyhM0lRCe1OoW9dGuFjtdOP+w54Pj/Ms0uB+DnVhWp1wJWak+rQIdVTygx0SP9bEKH9UmzC301Vdkt6uCZpRu+gKNGZ9az6JNjSOS7M14yXKQ+mM3Vz1S6lqN7sHlZnYhulXOSJ92r8oAI/sKRs0tt6SXH3RFTnDmcVNGdKJns/dyR03lPqAaPZMB8f1fHzvILX3KP/d0vRdy1J90CWDmXbA9rPmxC9ROlml6SX5yqr87Z3KqTVX21KmuZkpRkH+oxk9Z5AosZd+eyX5yC5XE1UQsee6/JEJndaJHZS4ufn2OM0ROjRW76vPj3lqMxYSWYeXyLPPWSZe9DpM9L0WQoDVhJlH04ubWyv/l8f64Xj69e2f3XtxsMbG5/dUv8PyPvix+In4gqsfb8Wn8H4PxRHwOn34k/ir+Jvjd81/tj4c+Mvsul7FxTmR6Lwr/H3/wLFElgG</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=">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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 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=">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</latexit> Setup: E : Rd ! R computable in K operations. <latexit sha1_base64="BLbI91EqNObAlbI/2RJe4xs+gGo=">AAA92nictVtfc9u4EUeu/y7pv1z72JkOWyedu06asd1M27kbTy+xncQXX2JHspO7U5KhJFpmQokKSdlOdH7pW6ev/T596ZfoN2j70of2A3R3ARCgBHJBNzXGNgjht7tYAPsHoPrTJM6L1dW/XXrvG9/81re/8/7lK9/93vd/8MOrH/zoME9n2SA6GKRJmj3th3mUxJPooIiLJHo6zaJw3E+iJ/1Xm/j5k5Moy+N00i3eTKNn43A0iY/iQVhA04urz3tFdFb0j+b7syjHpo/PgyfHYRHEeVAcR8EgHU+T6Cwu3gTpET3NingyCq71JmE/CYPedhR8HPQeP34+DHpZPDouwixLT2XLtd+/uLqyenOVfoLlypqqrAj1s5d+8NN10RNDkYqBmImxiMREFFBPRChyKF+JNbEqptD2TMyhLYNaTJ9H4lxcAewMekXQI4TWV/B3BE9fqdYJPCPNnNAD4JLAbwbIQFwHTAr9Mqgjt4A+nxFlbK2jPSeaKNsb+N9XtMbQWohjaOVwuqcvDsdSiCPxOxpDDGOaUguObqCozEgrKHlgjaoAClNow/oQPs+gPiCk1nNAmJzGjroN6fO/U09sxeeB6jsT/yApr0MJREeNPi0phOKE6Ac0mzP4TMqTAOcRUIjUGLF2Sroe0+gn0H8O7Q+hnFNN66QPZU6t543ITSgu5CaLvAfFhbzHInehuJC7LHIPigu5p5CIzUjnbnwHigvfYTnvQ3Eh91nkYygu5GMWeQjFhTxkkV9CcSG/ZJF3obiQd1nkAygu5AMW2YXiQnZZ5AEUF/KARW5DcSG3FbJ+p2ZQUqITM7vyNtSrPNBSJNBym5XvDllHF/aOx54e1GD5Xb0F/93YLQ+dRjXYbY91d1SD5VfePbCRbixvi+6TN3Fh77PYHVgBbuwOi/1MvKzBfuax017VYPm9tgv93Fje+n4OT27s5yz2IdTcWN5HPYIWN/aRh8eY1mD3WOy+eF2D9bH6WQ2Wt/sdsCtuLO+nutDfjfWxprMaLG9PDyGCcWN5b/UEWt3YJyz2qTirwT5lsV+AdXdjv/DwsG9rsNrHXiEPMqJ4JIId20QtLHcl1qZALWT4J6VvSSg27kM7hxmVmBFhxiziXom454nYLRG73nLlpR3NKd7luXRKRMcT0S99E9YKtv+w7I+1xAOxVSK2FhBNESnOtR7LCUUXuoVDFqXnwprPmNLSfmMtUuuh2fJqxKMKQq7tY1r5NyhbwgwKNdVE7bj08RIZ0HMT4pSyNz1KzYPHFaVVsFFnLKrvQPVZ1BsH6g2LmjlQMxZ14kCdsCiz821cz2MFGP3jXMzpSa4AGSPXlwCigtvgde7DHg1g/exBFPiYWh7B/w7l3lxpkgyzefSTeMrxrGKJM6jNxQq0m6xwi/LrhHZYBJLJno9Ujo9PeLYxV3tOWuHz0pMH5YmJP52Y5BmVdDBaDGg/taPzgFrOKbqTtXb4++W+17V2+G3S+DlF8bLWDl8o6YsLyN5V2O4FsB3YTVOlfVNvS0Oev0gaun6FvC5aXJzVsVozSO+sJf0dNTM7F5iXTapJ/Zh6Oxq5Nb68Mr42NIyec0vP7ahg9CSjXl0LWo9kovJeU28rQ0pedKLkME9tZwb7DNXM6Ho7GnsQcW1Szj236m1X77Qcjam3o3Eo5LnnOUXyut6OxoiepT5MvR0NPG0JVZ5v6m0tO2pA5s6m3taqT+gUGM+A5JqXLSYqyihOmilqMcUHzac1dsy/7MfwzOZ5mSM0UzKxbT2dfunLmiXS8UIEVq1oKQfGFzMrBqvSmIt1Nr+SMhQV/75Mx/h41PwuaDGA3S/vALgz8wQk1GcSaL0ToLjGZl3VkWncOovDVXK0gOqp1oKNFg1feWpUbXtBrVxeZkZr9Ngje53T2ptSTLhLmuX0sFs7w3UUOQ3tVjTE02uju7dqv1a1v8ripguIabnSBnQjJG/SmvNUl9Y7lo6vq1ueAoq88zHrF0+bj5S1wZwnJVuEsjTxtPvpcyS7Df3qDWHOuOVnAc0o2qsTshox3UjlbBaqT4tlND6nZ0P7gO7kkIekMYB5DBSVqZC3ZniKjufpAVlU295yvFFf+oRO1nOyutoeN6NHFnrkQLfPcTbBYzyEWhdyhgN46npkOVdKXaWk8Uz8qrwdTWkGmzP6pGIhNQ1pb6KKhWzKso8rVE4BjatBZun+NBbpaHxviRKf9bvkMblr1fJfp5tbfb8d0hqvX831JzFD4rpOXAPaNfJWVz4tcpASzJ2frFP82jxK5NeGI9pQjutzi7PUy4Ru/CPKYKcUGSe027jdUe1tn08tfqI57Ql9d4632SlZyIDsXwD+KaU1GdCv/e6AvkGXFiEhG+ljd+IyunHFOjG7xkwcFwv5VoNZbxHZshnx13Tt3ZXTWpQZg/QD5wtrW+tkl2LBiLhmyrqbvd3sfRBp3pOwV4mkaNbKh8T/I/qrf/U6WVlaEahhnIFc2TrXfKSUs6COQvLyzTZI97WlvFbK8FxJbfyfkelaRbItyrhQHvTWQ+A8oGfJC1dJRnLnS32kH206zUXK0wU94miPKIuXdn+kPDDKfYO85ArtuR6tkhGsgqLMInRf7hR5kW8zryp1P9r5/4W60XVVa0gxEOYEV2qIO9+PKFuzpUxgVcv1+4p2k1vr2UKvZj4TWotjay9/Da0/g79abv3sR6dfsQp3aA1ICubJaES2BEs9/HjdqfDSK1PTMs+Gn1mTupfdcpH8Wlo3k2OftKayR6vmTJ1a6PpFaLy0aLz01GGX7hqNFnW7tkQv2Nyiq24rffm14dZtQXnGUuYjMo2KPaS0cyk/qkOWKp/ja9RbltYqSyuE3WrfBth73gfp3uuLu/vr0rsH4i7FNgOKwGT+MqRdGlPMpVubMzVJATnfUvbV3v09akHufbKgSFm+x4k7Rt46Daicl5L+Qnm2lOy8sQj6vaVT1Ufb2B7Vf72EHNOeyGlfasQt6hEp+W05ggWLdNOKOQI6+Q8pppJxR3PObPc2cxJU4gmTb8pdZXjJTGFC+udO3naWstcdK38NKCecqei6D7TazzBSkBh9kuCOLHOaIfRy8iZBRrR9sp/Ldkre4k0siW6S1HOx4WFjZNZr1rq9tvSI9dh+CT1R62bWXT14fok3R47fRW70QvJqYxWjzheeL0YrVF6u+tykh9kCX6OPGfWxMwuT5VUxPfGJNxcpUTsuEuPDpd0o2sjfTvI2MsvbKV/KuremXD1pkDbmmPIl7j1QRLiiuw+d0dxHzDj6S/T6hLWpyRaOEp7Gpep8wLa0eCp1eckPydbLjd4osTxRnafQ1G1vYey3tJARWb9EcGc2srcte6+SpfCnMJLCQMg3euvyQ5vmJ1DwbyBc2aHm6HN22IH49rbYFNvv4G2I16ouTzQDakFbMFzIvUM1zmqPZh29tqjb9H04+POIQdec9DF50rayS8q85DZ1f/qnZAUyEbHSm57tx2Bz4UeyzKnNeGKybPxoYqG/i9N2LJqDz0iqXPz5yHsNbhRHQn+nqd0YNHV+BFUObXjo9xj85tz0bs/L5tSsr2UuvjykF9A3LhqHN3/1uYrp52OhMmtG3j0HtA5HDdS1t/hfx6H5GE7teflyy+m7Zi89Zl32i9SJLMbD7feM4eazmus5+vNMy9GZaMnNT8Z9QauZSq3RvHv6GI+aNaB5zYU8B+Wlk3h7FRl5fangvYBLhlT8S/zlEv9thNcljTo52lDS9xT11HQPnpr+xqVrdPozH5kMnTqZqtRMHtGhN2I3xY64C7+bZQTY9u1Q+V1K+R+x7u/PDqH1iKyHPkWXJwc9aovo9MPcog3pWZ0xvri6srb4LeTlyuH6zbXf3Ly1v77y6R31DeX3xU/EzyEvWRO/FZ+K+zDeA5Dpr+Kf4t/iPxu9jT9s/HHjT7Lre5cU5sei8rPx5/8CFHT5UQ==</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=">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=">AAA9q3ictVttc9u4EUaub5f0LXf92JkOWzudpM35bDfTdubGM5fYefHFlziR7ORySjKURCtMaFEhJedF55/RX9MP/dL+iP6D9lP/QncXAAFKIBdwU3NsgxCeZxdLYLELUP1JlpbT9fV/nvvoe9//wQ9/9PH5Cz/+yU9/9vOLn3x6WOazYpAcDPIsLx734zLJ0nFyME2nWfJ4UiTxcT9LHvVfbePnj06SokzzcXf6bpI8PY5H4/QoHcRTqHp+8fPVu5eHv9+4shrlk6SgyvJqlI6nRTyYxsASHeVFlMXFKIlWh6trzy+urK+t00+0XNhQhRWhfvbzT361KXpiKHIxEDNxLBIxFlMoZyIWJVzfig2xLiZQ91TMoa6AUkqfJ+JUXADsDFol0CKG2lfwdwR336raMdwjZ0noAUjJ4LcAZCQuASaHdgWUUVpEn8+IGWubuOfEibq9g/99xXUMtVPxAmo5nG7pi8O+TMWR+DP1IYU+TagGezdQLDOyCmoeWb2aAsME6rA8hM8LKA8Iqe0cEaakvqNtY/r8X9QSa/F+oNrOxL9Jy0twRaKjep9XDLE4If6InuYMPpP6ZCB5BAyJ6iOW3pCtj6n3Y2g/h/p7cJ1SSdukD9ecak9bkdtwuZDbLPI2XC7kbRa5B5cLucci9+FyIfcVErEF2dyN78DlwndYyQ/gciEfsMiHcLmQD1nkIVwu5CGLfAKXC/mERd6Cy4W8xSLvwuVC3mWRXbhcyC6LPIDLhTxgkTfhciFvKmTzTC3gyoknZWbldSjXZaCnyKDmOqvfDfKOLuwNjzk9aMDys3oH/ruxOx42TRqwNz3G3VEDlh95t8FHurG8L7pDq4kLe4fF7sIIcGN3WexX4mUD9iuPmfaqAcvPtT1o58by3vdruHNjv2ax96DkxvJr1H2ocWPve6wYkwbsPot9IF43YH28ftGA5f1+B/yKG8uvU11o78b6eNNZA5b3p4cQwbix/Gr1CGrd2Ecs9rF424B9zGK/Ae/uxn7jscK+b8DqNfYCrSAjikcSmLFtbHE1K7E0AbaYkZ9Va0tGsXEf6jnMqMKMCHPMIm5XiNueiL0KseetV1n50ZLiXV5Kp0J0PBH9am3C0pRtP6zaYynzQOxUiJ0FRFtEis9a9+WEogtdwyGn1cqFJZ8+5ZX/xlKixkO759WI+zWEHNsvaORfpWwJMyi0VBvbi2qNl8iI7tsQbyh7073UMnjctPIKNuoti+o7UH0W9c6BeseiZg7UjEWdOFAnLMrMfBvX8xgBxv74LOZ0J0eAjJGbrwiiguuw6tyBORrB+NmHKPAh1dyH/x3KvbmrTTPM5nGdxF2OpzVPXEBpLlag3mSFO5RfZzTDEtBMtryvcny8w72NuZpz0gufVit5VO2Y+POkpM+o4sFoMaL5FMZzl2pOKbqTpTD8nWre61IY/iZZ/JSieFkKw0+V9tMz6N5V2O4ZsB2YTRNlfVMO5ZD7L5JDly/QqoseF5/qsRozyPc2kH9XPZndMzyXbSpJ+5hyGEdp9a+s9S+Ew9i5tOwcxoLRk4x6dSkK7slY5b2mHKpDTqvoWOlh7kKfDLYZqiejy2Ec+xBxbVPOPbfKoaN3UvXGlMM4DoXc9zylSF6XwzhGdC/tYcphHLjbEqs835RDPTtaQObOphzq1ce0C4x7QHLMyxoTFRUUJ80UW0rxQftujR3zL69juGfzrMoR2plMbNvM06/WsnaNdLyQgFebBuqB8cXMisHqHHOxyeZXUodpbX1f5jFrPFp+D6wYweyXZwDcnnkGGuo9CfTeGTBusFlXvWcat8nicJQcLaB6qnbKRotGrtw1qtc9p1ouLzO9NXbskb8uaexNKCbcI8tydthrfMJNjJyF9moW4vlCbPdezde69ddZ3GQBMalG2oBOhORJWnue6rJ6x7LxJXXKM4VLnvmY8Yu7zUfK22DOk5MvQl3aZNrt9D6SXYfr6lVh9rjlZxE9UfRXJ+Q1UjqRKtksVO8Wy2h8TveG+4DO5FCG5BjAc4wUy0TIUzPcRcf99Ig8qu1vOdloL71DJ8sleV3tj9vRIws9cqDDc5xtWDHuQakLOcMB3HU9spwLla1ysnghPqtOR3N6gu0ZfVbzkJpD+puk5iHbsuwXNZY3gMbRILN0f45FHo3vLTHxWb9LH5O71j3/JTq51efbMY3x5tHcvBMzJKmbJDWiWSNPdeXdogSpwdz5ySbFr+29RHkhEtGHclKfWZKlXcZ04p9QBjuhyDij2cbNjnpre39q8RMtaV/os3M8zc7JQ0bk/yJYn3IakxH92u8O6BN06REy8pE+fietohtXrJOyY8zEcamQbzWY8ZaQL5uRfM1rz66SxqLMGOQ6cLowtrVN9igWTEhqoby7mdvtqw8izXsS9iiRjGasXCb5V+iv/tXjZGVpRKCF8QmUyte5nkdOOQvaKKZVvt0H6ba2lquVDs+U1mb9Mzqt1jTboYwL9cHVegiSB3QvZeEoKUjvcqmNXEfbdnORebJgR+ztEWXx0u+P1AqMel+lVXKF5lyPRskIRsG0yiJ0W24XeVFuu6w6ux93+X9hN7auWw0ZI2F2cKWFuP39hLI1W8sMRrUcv69oNrmtXiy0apczprF4bM3l76D21/BX663v/Xj6Na9wg8aAZDB3xiKyJlpq4SfrRk2WHpmay9wbeWZM6lZ2zVnya+ndTI59EsyyT6Pmrdq10OWzcLy0OF562rBLZ43Girpee6LnbG7RVaeVvvJCpHUDmGcsMx+RaVTqoaWdS/mxDllWPsfXqPcs1zrLFcNstU8D7Dnvg3TP9cXZ/V21ukfiFsU2A4rAZP4ypFmaUsyla9szNcmAkq8p/2rP/h7VoPQ+eVBklu9x4oyRp04Duk4rTX+rVrac/LzxCPq9pTeqjfaxPSr/YQl5THOipHmpEdeoRaL0t/WIFjzSmhVzRLTzH1NMJeOO9pzZbm2eSVSLJ0y+KWeVkSUzhTHZn9t5213KXnet/DWinHCmous+cIU/YWSQGL2T4I4sS3pCuMrJkwQZ0fbJfy77KXmKN7Y0WiOt52LLw8fIrNeMdXts6R7rvv0OWqLVzVN3teDlZd4SOXlnOdGLaVU7VjHqfOH+bFyxWuXq9212mC3INfaYURs7szBZXh3TE194S5EahUmRGB8pYb0I0T9M8xCd5emUL7NurZnrOw3Sx7ygfIl7DxQRrujusjOau8L0o7/E1yeszSZrOCbcjcvV/oDtaXFX6vzSOiRrz7euRpm1EjWtFJrdXi2M/5YeMiHvlwluz0a2tnXv1bIUfhdGMgyEfKO3KT+0Ob+AC/9GwpUdaok+e4cdiG+vi21x8wO8DfFaleWOZkQ16AuGC7l3rPpZb9Fuo9cWu83vI8FfRgq25rRPaSUN1V0y85rb7P78b8gLFCJhtTctw/tgS+F7siwppD8peTa+N6nQ38UJ7YuW4NOTuhR/OfJcg+vFkdDfaQrrg2bne1CXECJDv8fg98xN63BZtqR2ey1L8ZUhVwF94qJxePLXnKuYdj4eqrCeyIeXgN7hqIVdrxb/az+0HCMpXJavtJK+a/bS46nLdonakcV4OHzOGGk+o7lZor/MvOqdiZbc8mTcFwU9qdzqzYfnx3jUjAEtay7kPiivncTbo8jo68uC5wIuHXLxH/G3c/y3EV5XHE16hDDpc4pmNt2CZ9PfuHT1Tn/mo5PhadKpzmbyiA69EbstdsUt+N2uIsDQt0Pldynlf8S6vz87hNoj8h56F13uHPSoLqHdD3OKNqR7tcf4/OLKxuK3kJcLh5trG39cu/Zgc+XLG+obyh+LX4rfQF6yIf4kvhR3oL8HoNNfxF/F38U/tj7b6mw92erJph+dU5hfiNrPVvJfREDk/Q==</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=">AABFD3ictVxfc9u4EUeu/67pn8u1j33h1Ukn18mljpv+mbm5mUssx/FFSZRIdnIXJR5RomTFlKiIkuJEpw/R6Yfp9KXttE/9CP0AnWmf+hW6uwAIUAK5oJuGYxsE8dtdLIHF7gJMOImH6Wx7+x8X3vvGN7/17e+8/92L3/v+D374waUPf3SUJvNpNzrsJnEyfRp20igejqPD2XAWR08n06gzCuPoSXi6i8+fLKJpOkzGrdmbSfR81BmMh/1htzODquNLn1zuH7dnJ1fPPg4+C9q9ZDZZnq2WULPC+3Q+Oj4NzuAHao5PLx9f2tq+vk3/gs3CDVXYEupfI/nwo3+KtuiJRHTFXIxEJMZiBuVYdEQK1zNxQ2yLCdQ9F0uom0JpSM8jsRIXATuHVhG06EDtKfwewN0zVTuGe6SZEroLXGL4mQIyEFcAk0C7KZSRW0DP50QZa4toL4kmyvYG/oaK1ghqZ+IEajmcbumLw77MRF/8lvowhD5NqAZ711VU5qQVlDywejUDChOow3IPnk+h3CWk1nNAmJT6jrrt0PN/UUusxfuuajsX/yYpr8AViKbqfZJR6IgF0Q/obc7hmZQnBs4DoBCpPmLpNel6RL0fQ/sl1D+Aa0UlrZMQriXVrkqRu3C5kLssch8uF3KfRdbhciHrLLIBlwvZUEjETknnbnwTLhe+yXJ+BJcL+YhFPobLhXzMIo/gciGPWORXcLmQX7HIO3C5kHdY5D24XMh7LLIFlwvZYpGHcLmQhyxyDy4Xck8hi2fqFK6E6AyZWXkLynkeaCliqLnFynebrKMLe9tjTncLsPysrsFfN7bmodOoALvnMe76BVh+5O2DjXRjeVt0l1YTF/Yuiz2AEeDGHrDYL8TLAuwXHjPttADLz7U6tHNjeet7H+7c2Pss9gGU3Fh+jXoINW7sQ48VY1KAbbDYR+JVAdbH6k8LsLzdb4JdcWP5daoF7d1YH2s6L8Dy9vQIPBg3ll+tnkCtG/uExT4VZwXYpyz2S7DubuyXHivs2wKsXmMv0goyIH8kghlbRq2TzUosTYBah+EfZ2tLTL5xCPUcZpBhBoQZsYj9DLHviahniLq3XGlmR1Pyd3kuzQzR9ESE2dqEpRnbvpe1x1LsgahliNoaoswjxXet+7Ig70LXcMhZtnJhyadPSWa/sRSp8VBueTXiYQ4hx/YJjfxrFC1hBIWaKqN2kq3xEhnQfRniNUVvupeaB4+bZVbBRp2xqNCBClnUGwfqDYuaO1BzFrVwoBYsysx8G9f2GAFG//gulnQnR4D0kYuvALyCW7Dq3IU5GsD4aYAX+JhqHsLfJsXe3FUmGUbzuE5iluN5zhJPobQUW1BvosIaxdcxzbAIJJMtH6oYH+8wt7FUc05a4VW2kgdZxsSfzpDkGWR00FsMaD5Vo3OPalbk3clSNfzdbN7rUjX8Hml8RV68LFXDz5T0s3PI3lLY1jmwTZhNE6V9U65KQ+ZfJA1dvkirLlpcfKsjNWaQ3llF+gfqzRyc473sUknqx5Sr0Uit/qW5/lWhYfScWnquRgW9J+n16lJQuSdjFfeaclUZElpFx0oOc1f1zWCbnnozulyNRgM8rl2KuZdWueronWS9MeVqNI6EzHuuyJPX5Wo0BnQv9WHK1WhgtqWj4nxTrmrZUQMydjblqlZ9TFlgzAHJMS9rjFc0JT9prqgNyT8oz9bYPv/mOoY5mxdZjFBOyfi2xXTCbC0rl0j7CxFYtVlFOdC/mFs+WJ7GUuyw8ZWUYZZb3zfpmDUeNV8HLQYw++UeAJczj0FCnZNA6x0DxRts1JXvmcbtsDgcJf01VFvVzlhv0fCVWaN83THVcnGZ6a3RY5vsdUpjb0I+YZ00y+mhXviGiyhyGqrnNMTTq6K7t2q+5rW/zeIma4hJNtK6tCMkd9LK41SX1puWjq+oXZ4ZXHLPx4xfzDb3lbXBmCchW4SylPG02+k8kl2H6+o1YXLc8llAbxTt1YKsxpB2pFI2CtXZYumNL+ne0D6kPTnkIWl04T0GispEyF0zzKJjPj0gi2rbW4436ktn6GQ5Jaur7XE5emChBw509RhnF1aMB1BqQcxwCHctjyjnYqarhDQ+FZ9ku6MJvcHyiD7OWUhNQ9qbKGchy6LskxyV14DG0SCjdH8a63Q0vr1BiY/6XfKY2DVv+a/Qzq3e3+7QGC8ezcWZmB5x3SGuAc0auasr79Y5SAmWzic75L+W9xL5VeGINpTj+sLiLPUyph3/iCLYCXnGMc02bnbkW9v5qfUnmlND6L1z3M1OyEIGZP8CWJ8SGpMB/dhnB/QOurQIMdlIH7szzLwbl68zZMeY8eOGQp5qMOMtIls2J/6arj27UhqLMmKQ68BqbWxrndTJF4yI61RZdzO3y1cfRJpzEvYokRTNWLlK/D+m3/pHj5OtjRGBGsY3kCpb53ofCcUsqKMOrfLlNki3taW8nMnwQklt1j8j0+WcZDWKuFAeXK17wLlL95IXjpIpyZ1utJHraFk2FylP1vSIve1TFC/t/kCtwCj3NVolt2jOtWmUDGAUzLIoQrflssjrfMt55an70U7/L9SNrvNaQ4qBMBlcqSEuvx9RtGZLGcOoluP3lGaTW+vTtVblfMY0FkfWXP4aaj+C31pufe9HJ8xZhds0BiQFc2c0ImuCjRZ+vG7neOmRqWmZe8PPjEndyq45T3wtrZuJsReVqTRo1JyprIUun4fGS4vGS08dtmiv0WhR12tLdMzGFi21W+nLrwq3VgXKc5Yy75Fp1NBDSjuW8qPaY6nyMb5GvWVpbbO0OjBb7d0Ae877IN1zfX12f52t7oG4Q75NlzwwGb/0aJYOyefSteWRmqSAnG8q+2rP/jbVIPeQLChSluc4ccbIXacuXatM0p+plS0hO28sgj639Fq10Ta2TeVfbiBHNCdSmpcacZNaREp+W45gzSJdt3yOgDL/HfKppN9RHjPbrc07CXL+hIk35awyvGSkMCb9c5m3g43o9cCKXwOKCefKuw6BVvU3jBQkRmcS3J5lSm8IVzm5kyA92pDs56adkrt4Y0ui6yT1UnzmYWNk1GvGuj22dI91334OLVHr5q27WvD8Ym+OHL/z7Oh1aFUbKR91uXZ/Plodtcrl78v0MF/ja/QxpzZ2ZGGivDymLT715iIlqsZFYny4VOtFFfmrSV5FZrk75UtZt9aU85kGaWNOKF7izoEiwuXdXXV6cx8z/Qg36IWEtanJGo4SZuMSlR+wLS1mpYK1CMmu59ak2FqPitYLw8NeNYwdl5YyIisYCy53I1vbfWjnohU+GyMpdIU82VsUJ9o0P4ULfwfCFSVqjj45xCb4ubfErth7B6ciXqmyzGwGVIM2obcWg3dUP/MtynX0yqJu0/fh4M9jCLrmpB/SilpVdkmZl9ym7k//NVmDqYhY6U3L6n2wufA92eRUpT9DsnB8b4ZCf5NTtS+ag09P8lz8+cj9Da4XfaG/barWB02d70GeQxUe+jyD3zs3ravzsjmV62uTiy8PuQ7onReNwx3A4pjFtPOxUFPrjbx7Dmgd+iXU9Wrxv/ZD8zGcqvPy5ZbSN2cvPd66bBepzCz6xdXnjOHmM5qLOfrzTLLeGa/JzU/6f0GlN5VYvXn39NEvNWNA81oKmQ/lpZN4exQZeX2p4P6AS4ZE/Ef8+QL/VcKrjEaRHFUo6f2KYmq6BU9Nf3np6p1+5iOToVMkU56aiSeadDJ2VxyIO/Czm3mAVU+Jym8q5V/Eur+j7UFtn6yHzqbLDEKb6iLKgpjdtB7dm3O0RRLjmV55xrcFNbgnXqdaPO/7gNrjmd9Wrm/FX5LIuX5fJKKXi0zWd/nMvAqhB/kdOJkL0t/7BnSmXmaz5Am0kcceozxHJSMl/fXzkhA9igvXJV0SQo+WMsqhk3JIZ5KiAtphrm9dGuETtdOP+w54Pr+TZZcC8Quq66jVAVdqTqqGQ6pnlBkISf/bEKH9SlyDv9dU2S1pY0PSlN5BXqIz61n5SbCVc1yYrxmvUB5MZ+oWql1CUb3ZPSzPxNYKucgT7+X4QQl+YEnZpLd1SnH3VJTnDuclNOdKJns/dyx03lPqAaPZTjY+yuPnRQmvhUf/7xWi71mS7oMsIWXbA9rPmxK9WOlmj6SX5yrL87Z3S6TVX21KmuZkpRkH+oxk+Z5ArMZd8eyX5yC5XE1UQMee6/JEJndaZOikxM/PicdpiI5Hb/m++vSUozJnJZl7fIm88JBl4UGnz0jTZykMWEmUfTi+tHVj/f/62Cwc7Vy/8evrNx/tbH1+W/0/IO+Ln4ifiquw9v1GfA7jvyEOgdPvxR/FX8Xfar+r/aH2p9pfZNP3LijMj0XuX+3v/wV/OFEG</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=">AABFD3ictVxfc9u4EUeu/67pn8u1j33h1Ukn18mljpv+mbm5mUssx/FFSZRIdnIXJR5RomTFlKiIkuJEpw/R6Yfp9KXttE/9CP0AnWmf+hW6uwAIUAK5oJuGYxsE8dtdLIHF7gJMOImH6Wx7+x8X3vvGN7/17e+8/92L3/v+D374waUPf3SUJvNpNzrsJnEyfRp20igejqPD2XAWR08n06gzCuPoSXi6i8+fLKJpOkzGrdmbSfR81BmMh/1htzODquNLn1zuH7dnJ1fPPg4+C9q9ZDZZnq2WULPC+3Q+Oj4NzuAHao5PLx9f2tq+vk3/gs3CDVXYEupfI/nwo3+KtuiJRHTFXIxEJMZiBuVYdEQK1zNxQ2yLCdQ9F0uom0JpSM8jsRIXATuHVhG06EDtKfwewN0zVTuGe6SZEroLXGL4mQIyEFcAk0C7KZSRW0DP50QZa4toL4kmyvYG/oaK1ghqZ+IEajmcbumLw77MRF/8lvowhD5NqAZ711VU5qQVlDywejUDChOow3IPnk+h3CWk1nNAmJT6jrrt0PN/UUusxfuuajsX/yYpr8AViKbqfZJR6IgF0Q/obc7hmZQnBs4DoBCpPmLpNel6RL0fQ/sl1D+Aa0UlrZMQriXVrkqRu3C5kLssch8uF3KfRdbhciHrLLIBlwvZUEjETknnbnwTLhe+yXJ+BJcL+YhFPobLhXzMIo/gciGPWORXcLmQX7HIO3C5kHdY5D24XMh7LLIFlwvZYpGHcLmQhyxyDy4Xck8hi2fqFK6E6AyZWXkLynkeaCliqLnFynebrKMLe9tjTncLsPysrsFfN7bmodOoALvnMe76BVh+5O2DjXRjeVt0l1YTF/Yuiz2AEeDGHrDYL8TLAuwXHjPttADLz7U6tHNjeet7H+7c2Pss9gGU3Fh+jXoINW7sQ48VY1KAbbDYR+JVAdbH6k8LsLzdb4JdcWP5daoF7d1YH2s6L8Dy9vQIPBg3ll+tnkCtG/uExT4VZwXYpyz2S7DubuyXHivs2wKsXmMv0goyIH8kghlbRq2TzUosTYBah+EfZ2tLTL5xCPUcZpBhBoQZsYj9DLHviahniLq3XGlmR1Pyd3kuzQzR9ESE2dqEpRnbvpe1x1LsgahliNoaoswjxXet+7Ig70LXcMhZtnJhyadPSWa/sRSp8VBueTXiYQ4hx/YJjfxrFC1hBIWaKqN2kq3xEhnQfRniNUVvupeaB4+bZVbBRp2xqNCBClnUGwfqDYuaO1BzFrVwoBYsysx8G9f2GAFG//gulnQnR4D0kYuvALyCW7Dq3IU5GsD4aYAX+JhqHsLfJsXe3FUmGUbzuE5iluN5zhJPobQUW1BvosIaxdcxzbAIJJMtH6oYH+8wt7FUc05a4VW2kgdZxsSfzpDkGWR00FsMaD5Vo3OPalbk3clSNfzdbN7rUjX8Hml8RV68LFXDz5T0s3PI3lLY1jmwTZhNE6V9U65KQ+ZfJA1dvkirLlpcfKsjNWaQ3llF+gfqzRyc473sUknqx5Sr0Uit/qW5/lWhYfScWnquRgW9J+n16lJQuSdjFfeaclUZElpFx0oOc1f1zWCbnnozulyNRgM8rl2KuZdWueronWS9MeVqNI6EzHuuyJPX5Wo0BnQv9WHK1WhgtqWj4nxTrmrZUQMydjblqlZ9TFlgzAHJMS9rjFc0JT9prqgNyT8oz9bYPv/mOoY5mxdZjFBOyfi2xXTCbC0rl0j7CxFYtVlFOdC/mFs+WJ7GUuyw8ZWUYZZb3zfpmDUeNV8HLQYw++UeAJczj0FCnZNA6x0DxRts1JXvmcbtsDgcJf01VFvVzlhv0fCVWaN83THVcnGZ6a3RY5vsdUpjb0I+YZ00y+mhXviGiyhyGqrnNMTTq6K7t2q+5rW/zeIma4hJNtK6tCMkd9LK41SX1puWjq+oXZ4ZXHLPx4xfzDb3lbXBmCchW4SylPG02+k8kl2H6+o1YXLc8llAbxTt1YKsxpB2pFI2CtXZYumNL+ne0D6kPTnkIWl04T0GispEyF0zzKJjPj0gi2rbW4436ktn6GQ5Jaur7XE5emChBw509RhnF1aMB1BqQcxwCHctjyjnYqarhDQ+FZ9ku6MJvcHyiD7OWUhNQ9qbKGchy6LskxyV14DG0SCjdH8a63Q0vr1BiY/6XfKY2DVv+a/Qzq3e3+7QGC8ezcWZmB5x3SGuAc0auasr79Y5SAmWzic75L+W9xL5VeGINpTj+sLiLPUyph3/iCLYCXnGMc02bnbkW9v5qfUnmlND6L1z3M1OyEIGZP8CWJ8SGpMB/dhnB/QOurQIMdlIH7szzLwbl68zZMeY8eOGQp5qMOMtIls2J/6arj27UhqLMmKQ68BqbWxrndTJF4yI61RZdzO3y1cfRJpzEvYokRTNWLlK/D+m3/pHj5OtjRGBGsY3kCpb53ofCcUsqKMOrfLlNki3taW8nMnwQklt1j8j0+WcZDWKuFAeXK17wLlL95IXjpIpyZ1utJHraFk2FylP1vSIve1TFC/t/kCtwCj3NVolt2jOtWmUDGAUzLIoQrflssjrfMt55an70U7/L9SNrvNaQ4qBMBlcqSEuvx9RtGZLGcOoluP3lGaTW+vTtVblfMY0FkfWXP4aaj+C31pufe9HJ8xZhds0BiQFc2c0ImuCjRZ+vG7neOmRqWmZe8PPjEndyq45T3wtrZuJsReVqTRo1JyprIUun4fGS4vGS08dtmiv0WhR12tLdMzGFi21W+nLrwq3VgXKc5Yy75Fp1NBDSjuW8qPaY6nyMb5GvWVpbbO0OjBb7d0Ae877IN1zfX12f52t7oG4Q75NlzwwGb/0aJYOyefSteWRmqSAnG8q+2rP/jbVIPeQLChSluc4ccbIXacuXatM0p+plS0hO28sgj639Fq10Ta2TeVfbiBHNCdSmpcacZNaREp+W45gzSJdt3yOgDL/HfKppN9RHjPbrc07CXL+hIk35awyvGSkMCb9c5m3g43o9cCKXwOKCefKuw6BVvU3jBQkRmcS3J5lSm8IVzm5kyA92pDs56adkrt4Y0ui6yT1UnzmYWNk1GvGuj22dI91334OLVHr5q27WvD8Ym+OHL/z7Oh1aFUbKR91uXZ/Plodtcrl78v0MF/ja/QxpzZ2ZGGivDymLT715iIlqsZFYny4VOtFFfmrSV5FZrk75UtZt9aU85kGaWNOKF7izoEiwuXdXXV6cx8z/Qg36IWEtanJGo4SZuMSlR+wLS1mpYK1CMmu59ak2FqPitYLw8NeNYwdl5YyIisYCy53I1vbfWjnohU+GyMpdIU82VsUJ9o0P4ULfwfCFSVqjj45xCb4ubfErth7B6ciXqmyzGwGVIM2obcWg3dUP/MtynX0yqJu0/fh4M9jCLrmpB/SilpVdkmZl9ym7k//NVmDqYhY6U3L6n2wufA92eRUpT9DsnB8b4ZCf5NTtS+ag09P8lz8+cj9Da4XfaG/barWB02d70GeQxUe+jyD3zs3ravzsjmV62uTiy8PuQ7onReNwx3A4pjFtPOxUFPrjbx7Dmgd+iXU9Wrxv/ZD8zGcqvPy5ZbSN2cvPd66bBepzCz6xdXnjOHmM5qLOfrzTLLeGa/JzU/6f0GlN5VYvXn39NEvNWNA81oKmQ/lpZN4exQZeX2p4P6AS4ZE/Ef8+QL/VcKrjEaRHFUo6f2KYmq6BU9Nf3np6p1+5iOToVMkU56aiSeadDJ2VxyIO/Czm3mAVU+Jym8q5V/Eur+j7UFtn6yHzqbLDEKb6iLKgpjdtB7dm3O0RRLjmV55xrcFNbgnXqdaPO/7gNrjmd9Wrm/FX5LIuX5fJKKXi0zWd/nMvAqhB/kdOJkL0t/7BnSmXmaz5Am0kcceozxHJSMl/fXzkhA9igvXJV0SQo+WMsqhk3JIZ5KiAtphrm9dGuETtdOP+w54Pr+TZZcC8Quq66jVAVdqTqqGQ6pnlBkISf/bEKH9SlyDv9dU2S1pY0PSlN5BXqIz61n5SbCVc1yYrxmvUB5MZ+oWql1CUb3ZPSzPxNYKucgT7+X4QQl+YEnZpLd1SnH3VJTnDuclNOdKJns/dyx03lPqAaPZTjY+yuPnRQmvhUf/7xWi71mS7oMsIWXbA9rPmxK9WOlmj6SX5yrL87Z3S6TVX21KmuZkpRkH+oxk+Z5ArMZd8eyX5yC5XE1UQMee6/JEJndaZOikxM/PicdpiI5Hb/m++vSUozJnJZl7fIm88JBl4UGnz0jTZykMWEmUfTi+tHVj/f/62Cwc7Vy/8evrNx/tbH1+W/0/IO+Ln4ifiquw9v1GfA7jvyEOgdPvxR/FX8Xfar+r/aH2p9pfZNP3LijMj0XuX+3v/wV/OFEG</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=">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</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=">AABB0XictVzdchPJFW42fwv5WTa5zM0khhSkCDEOlaRqa6vW2MZ4EWCQbNhdAaWfsRCMNEIjGRutq1K5zSPkNnmOPEfeILnKK+T8dE/3SD1zehxCl+2eVn/nnD7Tffqc0y26k2SYzdbX/3nho+9893vf/8HHFy/98Ec//sknlz/96WGWzqe9+KCXJun0WbeTxclwHB/MhrMkfjaZxp1RN4mfdt9s4edPj+NpNkzHrdnpJH4+6gzGw6NhrzODppeXP9kbT+az6Mrp50fXTq5fiV5eXlu/uU7/otXKLV1ZU/rffvppdKDaqq9S1VNzNVKxGqsZ1BPVURmUb9Qtta4m0PZcLaBtCrUhfR6rM3UJsHPoFUOPDrS+gd8DePpGt47hGWlmhO4BlwR+poCM1FXApNBvCnXkFtHnc6KMrWW0F0QTZTuFv11NawStM/UKWiWc6RmKw7HM1JH6I41hCGOaUAuOrqepzEkrKHnkjGoGFCbQhvU+fD6Feo+QRs8RYTIaO+q2Q5//i3piKz73dN+5+jdJeRVKpJp69GlOoaOOiX5Eb3MOn7E8CXAeAIVYjxFr70jXIxr9GPovoP0hlDOqGZ10oSyo9awSuQXFh9wSkbtQfMhdEdmA4kM2ROQ+FB9yXyMROyWd+/FNKD58U+T8GIoP+VhEPoHiQz4RkYdQfMhDEfk1FB/yaxF5F4oPeVdE3ofiQ94XkS0oPmRLRB5A8SEPROQOFB9yRyPLV+oUSkp0hsKq3IR6kQdaigRaNkX57pB19GHvBKzpXglWXtXb8NeP3Q7QaVyC3QmYd0clWHnm7YKN9GNlW3SPdhMf9p6I3YMZ4Mfuidgv1esS7JcBK+1NCVZeaw3o58fK1vcBPPmxD0TsQ6j5sfIe9Qha/NhHATvGpAS7L2Ifq7cl2BCrPy3Byna/CXbFj5X3qRb092NDrOm8BCvb00PwYPxYebd6Cq1+7FMR+0ydlGCfidivwLr7sV8F7LDvS7Bmj71EO8iA/JEYVmwVtU6+KrE2AWodgX+S7y0J+cZdaJcwgxwzIMxIROzmiN1ARCNHNILlynI7mpG/K3Np5ohmIKKb701Ym4n9+3l/rCUBiO0csb2EqPJI8V2bsRyTd2FaJOQs37mwFjKmNLffWIv1fKi2vAbxqIDguf2KZv4NipYwgkJNVVF7le/xjIzouQrxjqI3M0rDQ8bNcqvgok5EVNeD6oqoUw/qVETNPai5iDr2oI5FlF35Lq4dMAOs/vFdLOiJZwD7yOUlAq9gE3ade7BGI5g/++AFPqGWR/C3SbG3VKokw2ge90nMcjwvWOIp1BZqDdptVLhN8XVCKywGybjnIx3j4xPmNhZ6zbEVPst38ijPmITTGZI8g5wOeosRrad6dO5Tyxl5d1yrh7+Xr3tTq4ffIY2fkRfPtXr4mZZ+dg7ZWxrbOge2CatporVv63VpcP6FaZj6Jdp10eLiWx3pOYP0TmrS39NvZu8c72WLaqwfW69HI3PGlxXGV4eG1XPm6LkeFfSe2Os1taj2SMY67rX1ujKktIuOtRz2qe6bwT59/WZMvR6NffC4tijmXjj1urN3ko/G1uvROFSc9zwjT97U69EY0DPrw9br0cBsS0fH+bZe17KjBjh2tvW6Vn1MWWDMAfGc5xbrFU3JT5prakPyD6qzNa7Pv7qPYc7mRR4jVFOyvm05nW6+l1VLZPyFGKzarKYc6F/MHR+sSGOhNsT4imWYFfb3VTp2j0fNN0CLEax+PgOQcuYJSGhyEmi9E6B4S4y6iiMzuA0Rh7PkaAnV1q0z0Vu0fDlrVGx7Sa1SXGZHa/XYJnud0dybkE/YIM1KemiUvuEyipKGGgUNyfTq6O69Xq9F7a+LuMkSYpLPtB6dCPFJWnWc6tN609HxVX3KM4PCZz52/mK2+UhbG4x5UrJFKEsVT7efySO5bbiv3lA2x82fRfRG0V4dk9UY0olUJkahJlvM3viCni3tAzqTQx5MowfvMdJUJopPzTCLjvn0iCyqa28l3qgvk6HjekZW19jjavTAQQ886PoxzhbsGA+h1oKY4QCeWgFRzqVcVylpfKp+k5+OpvQGqyP6pGAhDQ22N3HBQlZF2a8KVN4BGmcDR+nhNJbpGHx7hZIc9fvksbFr0fJfpZNbc77doTlePpvLMzF94rpBXCNaNXyqy0/LHFiChfeTDfJfq0eJ/OpwRBsqcX3hcGa9jOnEP6YIdkKecUKrTVodxd5ufmr5E8NpX5mzczzNTslCRmT/ItifUpqTEf24dwfMCTpbhIRsZIjdGebejc/XGYpzzPpxQ8W3Gux8i8mWzYm/oeuurozmIkcMvA+cLc1to5MG+YIxcZ1q627XdvXug0h7T8KdJUzRzpVrxP86/TY/Zp6srcwI1DC+gUzbOt/7SClmQR11aJevtkGmryvllVyGF1pqu/9Zma4UJNumiAvlwd26D5x79My8cJZMSe5spQ/vo1XZXKQ8WdIjjvaIoni2+wO9A6PcN2iXXKM116ZZMoBZMMujCNNXyiIv863mVaQeRjv7v1C3ui5qDSlGymZwWUNSfj+maM2VMoFZzfP3Da0mv9anS72q+YxpLo6ctfwttP4Cfhu5zXMYnW7BKtyhOcAU7JPVCLdEKz3CeN0p8DIz09Cyz5afnZOml9tynviarZuNsY9rU9mnWXOisxamfh4arx0arwN12KKzRqtF024s0Usxtmjp08pQfnW4tWpQnouUZY/MoIYBUrqxVBjVvkhVjvEN6r1Ia12k1YHV6p4GuGs+BOlf68ur+9t8d4/UXfJteuSBcfzSp1U6JJ/LtFZHakwBOd/W9tVd/W1qQe5dsqBIme9x4orhU6celbNc0l/pnS0lO28tgrm39E73MTa2TfXfrSBHtCYyWpcGcZt6xFp+V45oySLddHyOiDL/HfKp2O+ojpnd3vadRAV/wsabvKosL44UxqR/KfO2txK97jnxa0Qx4Vx7112gVf8NIwXGmEyC37PM6A3hLscnCezRdsl+rtopPsUbOxLdJKkX6vMAG8NRr53r7twyIzZj+zX0RK3bt+7rIfNLgjlK/M5zotehXW2kfdTF0vP5aHX0Lld8rtLDfImv1cec+riRhY3yipi2+iyYC0tUjwtjQrjUG0Ud+etJXkdmPp0KpWx6G8rFTAPbmFcUL0n3QBHh8+6ueb2568I4uiv0uoR1qXGLRAmzcanOD7iWFrNSF1f2IW69WLkbJc5OVLZTGOrubmHtN1vImKxfoqScDfd2ZW8XohQ5C8MUeopv9JbFhy7Nz6Dg70j5okPDMSR32AT/dlNtqZ0PcBvira5zRjOiFrQF/aXYu6PHWexRraO3DnWXfgiHcB5D0LUk/ZB20rqyM2VZcpd6OP13ZAWmKhaltz3rj8HlIo9klVOd8QzJssmjGSrzXZy6YzEcQkZS5BLOh881pFEcKfOdpnpjMNTlERQ51OFh7jGEvXPbuz4vl1O1vla5hPLgXcCcuBgcnvyVxyq2X4iFmjpv5MNzQOtwVEHd7Bb/6zgMH8upPq9Qbhl91+x1wFvnfrHOyKI/XH/NWG4hs7mcYzjPNB+d9Zb8/Njvi2q9qdQZzYenj/6onQOG10JxHlSWjvHuLLLyhlLBcwGfDKn6j/rHBfnbCG9zGmVy1KFkzinKqZkeMjXzjUvf6MxnITJZOmUyFanZOKJJN2K31J66Cz9buQdY93Yof5eS/yLW//3ZPrQekfUwWXTOHLSpLabshz1F69OzvT9bJjHe5eW7vS1owbPwBrXiPd+H1B/v+rYKYyv/Bgmv9QcqVf1CRLJ8umfXVRdGUDx54xyQ+Z5vRHfpOYvFN89GAWeL5v7UskQL+kS+WdAtxXcdKXs0Vyf6rB5PDvCGfSfPD0Xqt9TW0XYe91yJ834p5/0lzhlpp8jhxPms+m5WGZcth0s/z50d634pxdn2PK86N7pdyoXvoFfjBxX4gSNlk7T/hiLhqarO5s0raM61TO4J61iZTCTrAePMTv6+qyPb4wpexwHjv1+Kvu9IuguydCn/HdEJ25ToJVo3OyQ933SszqTeq5BWf4/y5eW1W8v/l8Fq5XDj5q3f37z9eGPtizv6/zn4WP1c/VJdgzX+B/UFUNtXBwr/v4O/qr+pv282N083/7T5Z+760QWN+Zkq/Nv8y38BeaelZQ==</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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</latexit> image

    <latexit 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skip-connexion <latexit 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xt = xt 1 + v✓t (xt 1) Kaiming He
  25. ResNet Architectures [He et al’ 16] ResNet-34 <latexit sha1_base64="hGNaHRogJoszxpvRv/VDNWjykms=">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</latexit> image

    <latexit sha1_base64="T6b9HaNHHjCNOHzV6onUoURA+iA=">AABFInictVzdc9TIER8uXxfyxSWPedHFkONSnGMc8lF1lSrAC8aHAcOuDXcscNpd7VpYu1q0WhvY83+Tyr+R91ReUqmkKlV5TVXylH8hPd0zmtHuSD1yCCrbo9H8untaMz3dPSN60ySe5Rsbfz/33te+/o1vfuv9b5//zne/9/0fXPjghwezdJ71o/1+mqTZk144i5J4Eu3ncZ5ET6ZZFI57SfS4d7Qlnz8+jrJZnE46+Ztp9GwcjibxMO6HOVS9uHD9YjeLR4d5mGXpycXgXngUzYL8MAq+/DKeDONJnEfBIJrmhx99FCTxOM6DSTr5ZBCNokmUhXm0/uLC2sb6Bv4LVgtXVWFNqH976Qcf/lN0xUCkoi/mYiwiMRE5lBMRihlcT8VVsSGmUPdMLKAug1KMzyNxKs4Ddg6tImgRQu0R/B7B3VNVO4F7SXOG6D5wSeAnA2QgLgEmhXYZlCW3AJ/PkbKsraK9QJpStjfwt6dojaE2F4dQy+F0S1+c7EsuhuI32IcY+jTFGtm7vqIyR61IyQOrVzlQmEKdLA/geQblPiK1ngPEzLDvUrchPv8XtpS18r6v2s7Fv1HKS3AFoq16nxYUQnGM9AN8m3N4RvIkwHkEFCLVR1k6QV2PsfcTaL+A+vtwnWJJ66QH1wJrT2uRW3C5kFsschsuF3KbRe7C5ULussg9uFzIPYWU2Ax17sa34XLh2yznh3C5kA9Z5CO4XMhHLPIALhfygEV+AZcL+QWLvA2XC3mbRd6Fy4W8yyI7cLmQHRa5D5cLuc8ib8HlQt5SyOqZmsGVIp2YmZU3oFzmIS1FAjU3WPluonV0YW96zOl+BZaf1S3468a2PHQaVWBveYy7YQWWH3nbYCPdWN4W3cHVxIW9w2J3YAS4sTss9jPxsgL7mcdMO6rA8nNtF9q5sbz1vQd3buw9FnsfSm4sv0Y9gBo39oHHijGtwO6x2IfiVQXWx+pnFVje7rfBrrix/DrVgfZurI81nVdgeXt6AB6MG8uvVo+h1o19zGKfiNcV2Ccs9nOw7m7s5x4r7NsKrF5jz+MKMkJ/JIIZW0ctLGalLE2BWsjwT4q1JUHfuAf1HGZUYEaIGbOI7QKx7YnYLRC73nLNCjs6Q3+X59IuEG1PRK9Ym2QpZ9sPivaylHggWgWitYSo80jlu9Z9OUbvQtdwyLxYuWTJp09pYb9lKVLjod7yasSDEoLG9iGO/CsYLckISmqqjtphscYTMsD7OsQJRm+6l5oHj8sLq2CjXrOongPVY1FvHKg3LGruQM1Z1LEDdcyizMy3cV2PEWD0L9/FAu9oBJCPXH0F4BXcgFXnDszRAMbPHniBj7DmAfxtY+zNXXWSyWherpMyy/GsZIkzKC3EGtSbqLCF8XWCMywCyajlAxXjyzuZ21ioOUdW+LRYyYMiY+JPJ0Z5RgUd6S0GOJ+a0bmLNafo3VGpGf5OMe91qRn+Fmr8FL14KjXD50r6/AyydxS2cwZsG2bTVGnflJvSoPwL0dDl87jqSosr3+pYjRlJ73VD+jvqzeyc4b1sYYn0Y8rNaMys/s1K/WtCw+h5Zum5GRXpPZHXq0tB455MVNxryk1lSHEVnSg5zF3TNyPbDNSb0eVmNPbA49rCmHthlZuO3mnRG1NuRuNAUN7zFD15XW5GY4T3pA9TbkZDZltCFeebclPLLjVAsbMpN7XqE8wCyxwQjXmqMV5Rhn7SXFGL0T+oz9bYPv/qOiZzNs+LGKGekvFtq+n0irWsXiLtL0Rg1fKGckj/Ym75YGUaC7HJxlckQ15a31fpmDVean4XtBjA7Kc9AC5nnoCEOichrXcCFK+yUVe5Zxq3yeLkKBkuobqqNme9RcOXskbluhdYy8VlprdGj1201zMce1P0CXdRs5wedivfcBVFTkO7JQ3x9Jro7q2ar2Xtb7C46RJiWoy0Pu4I0U5afZzq0nrb0vEltcuTw0V7Pmb8ymzzUFkbGfOkaIukLHU87XY6j2TXyXX1ijA5bnoW4BuV9uoYrUaMO1IzNgrV2WLyxhd4b2jv456c5EE0+vAeA0VlKmjXTGbRZT49QItq21uOt9SXztBReYZWV9vjevTIQo8c6OYxzhasGPeh1IGYYR/uOh5RzvlCVylqPBOfFLujKb7B+og+KVlITYPsTVSykHVR9mGJygmg5WigKN2fxjIdje+uUOKjfpc8JnYtW/5LuHOr97dDHOPVo7k6EzNArpvINcBZQ7u6dLfMgSRYOJ9sov9a30vJrwlHaUM5rs8tzqSXCe74RxjBTtEzTnC2cbOj3NrOTy0/0Zz2hN47l7vZKVrIAO1fAOtTimMywB/77IDeQSeLkKCN9LE7ceHduHydmB1jxo+LBZ1qMOMtQls2R/6arj27ZjgWKWKgdeB0aWxrneyiLxgh10xZdzO361cfiTTnJOxRQhTNWLmM/D/G3/pHj5O1lREhNSzfwEzZOtf7SDFmkToKcZWvt0G6rS3lxUKG50pqs/4ZmS6WJGthxCXlkav1ADj38Z54yVGSodyzlTa0jtZlcyXl6ZIeZW+HGMWT3R+pFVjKfQVXyTWcc10cJSMYBXkRRei2XBZ5mW89rzJ1P9qz/wt1o+uy1iTFQJgMLmmIy+9HGK3ZUiYwqmn8HuFscms9W2pVz2eCY3FszeWvoPZD+K3l1vd+dHolq3ATxwBRMHdGI1QTrLTw43WzxEuPTE3L3Bt+ZkzqVnbNWeJrsm4mxj5uTGUPR81rlbXQ5bPQeGnReOmpww7uNRot6nptiV6wsUVH7Vb68mvCrdOA8pylzHtkGhV7SGnHUn5UByxVPsbXqLcsrQ2WVgiz1d4NsOe8D9I915dn91fF6h6I2+jb9NEDo/hlgLM0Rp9L19ZHakRBcr6m7Ks9+7tYI7n30IJKynSOU84Y2nXq43VaSPpTtbKlaOeNRdDnlk5UG21ju1j+xQpyjHNihvNSI65hi0jJb8sRLFmkdcvnCDDzH6JPRX5HfcxstzbvJCj5EybepFlleFGkMEH9c5m3nZXodceKXwOMCefKu+4BreZvWFIgjM4kuD3LGb4hucrRTgJ5tD20n6t2inbxJpZE6yj1QvzWw8ZQ1GvGuj22dI91334GLaXWzVt3teD5Jd4cOX5n2dELcVUbKx91sXR/NlqhWuXK93V6mC/xNfqYYxs7sjBRXhnTFZ96cyGJmnEhjA+XZr1oIn8zyZvITLtTvpR1a025nGkgG3OI8RJ3DlQiXN7dZac39zHTj94KvR5ibWpUw1GS2bhU5QdsSyuzUsFShGTXc2tSYq1HVeuF4WGvGsaOk6WM0AomgsvdUGu7D91StMJnY4hCX9DJ3qo40ab5KVzydyBcUaLm6JNDbIOfe0NsiVvv4FTEK1WmzGaANdImDJZi8FD1s9yiXkevLOo2fR8O/jxi0DUnfYwralPZiTIvuU3dn/4JWoNMRKz0pmXzPthc+J6scmrSnxgtHN+bWOhvcpr2RXPw6UmZiz8f2t/gejEU+tumZn3Q1PkelDk04aHPM/i9c9O6OS+bU72+Vrn48qB1QO+8aJzcAayOWUw7HwuVWW/k3XOQ1mFYQ12vFv9rPzQfw6k5L19uM/zm7KXHW6d2kcrMSr+4+Zwx3HxGczVHf55p0TvjNbn5kf8XNHpTqdWbd09f+qVmDGheC0H5UF46wtujyMjrS0XuD7hkSMV/xB/O8V8lvCpoVMnRhJLer6implvw1PSXl67e6Wc+Mhk6VTKVqZl4oo0nY7fEjrgNP1uFB9j0lCh9U0l/Jdb9He0AaodoPXQ2nTIIXayLMAtidtMGeG/O0VZJLM/00hnfDtTIPfFdrJXnfe9je3nmt1PqW/WXJDTX74lUDEqRyfIun5lXPehBeQeOckH6e98Az9RTNotOoI099hjpHBVFSvrr5wUiBhgXLku6QIQeLXWUe07KPTyTFFXQ7pX61scRPlU7/XLfQZ7PD4vsUiB+jnWhWh3kSs1JteeQ6ilmBnqo/w2I0H4prsDfK6rslnRvRdIZvoOyRK+tZ/UnwU6d48J8zXgJ82A6U3es2qUY1Zvdw/pMbKuSC514r8ePavAjS8o2vq0jjLszUZ87nNfQnCuZ7P3cidB5T9KDjGbDYnzUx8/HNbyOPfp/txJ915J0G2TpYbY9wP28DOklSje3UHo6V1mft71TI63+apNompOVZhzoM5L1ewKJGnfVs5/OQXK5mqiCjj3X6UQmd1okdlLi5+fU4zRE6NFbvq8+PeWozFlJ5h5fIh97yHLsQWfISDNkKYxYSZR9eHFh7ery//WxWjjYXL/6q/VrDzfXrt9U/w/I++LH4ifiMqx9vxbXYfzviX3g9HvxJ/FX8bfW71p/bP259Rdq+t45hfmRKP1r/eO/IpVZOw==</latexit> ! Makes the “infinite depth” limit non-degenerate. <latexit 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x0 <latexit 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xT <latexit 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! Enable v✓ = 0 initialization, i.e. identity map. <latexit 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x0 <latexit 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x1 <latexit 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x0 <latexit 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x1 <latexit 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skip-connexion <latexit sha1_base64="Rv0zJz4pzD6Ong7U911+V2gV4aM=">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</latexit> xt = xt 1 + v✓t (xt 1) Kaiming He
  26. conv5 x 7⇥7  3⇥3, 512 3⇥3, 512 ⇥2 

    3⇥3, 512 3⇥3, 512 ⇥3 4 1⇥1, 512 3⇥3, 512 1⇥1, 2048 5 ⇥3 4 1⇥1, 512 3⇥3, 512 1⇥1, 2048 5 ⇥3 4 1⇥1, 512 3⇥3, 512 1⇥1, 2048 5 ⇥3 1⇥1 average pool, 1000-d fc, softmax FLOPs 1.8⇥109 3.6⇥109 3.8⇥109 7.6⇥109 11.3⇥109 1. Architectures for ImageNet. Building blocks are shown in brackets (see also Fig. 5), with the numbers of blocks stacked. Down- ing is performed by conv3 1, conv4 1, and conv5 1 with a stride of 2. 0 10 20 30 40 50 20 30 40 50 60 iter. (1e4) error (%) plain-18 plain-34 0 10 20 30 40 50 20 30 40 50 60 iter. (1e4) error (%) ResNet-18 ResNet-34 18-layer 34-layer 18-layer 34-layer e 4. Training on ImageNet. Thin curves denote training error, and bold curves denote validation error of the center crops. Left: plain rks of 18 and 34 layers. Right: ResNets of 18 and 34 layers. In this plot, the residual networks have no extra parameter compared to plain counterparts. Standard Neural Networks Residual Neural Networks (ResNets) xn+1 = f(xn , θN n ) xn+1 = xn + 1 N f(xn , θN n ) The deeper, the better
  27. Infinite Depth and Neural-ODEs <latexit sha1_base64="c+kKnAxKjdK3kNTUmMbn0Klm8uI=">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</latexit> where <latexit sha1_base64="qhSLJgQq/yuorQiDFfjqGL3B6xk=">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</latexit> ResNet

    [He et al, 2016] <latexit 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✓(x0) , xT <latexit sha1_base64="qbpiB4MRNcTm1Jd0Cno/ONu/uAc=">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</latexit> xt+1 = xt + 1 T v✓t (xt) <latexit 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x0 <latexit 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x1 <latexit 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xT
  28. Infinite Depth and Neural-ODEs <latexit sha1_base64="c+kKnAxKjdK3kNTUmMbn0Klm8uI=">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</latexit> where <latexit sha1_base64="qhSLJgQq/yuorQiDFfjqGL3B6xk=">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</latexit> ResNet

    [He et al, 2016] <latexit 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✓(x0) , xT <latexit sha1_base64="qbpiB4MRNcTm1Jd0Cno/ONu/uAc=">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</latexit> xt+1 = xt + 1 T v✓t (xt) <latexit 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x0 <latexit 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Neural ODE [Chen et al, 2018] <latexit 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where <latexit 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T ! +1 <latexit 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dx(t) dt = v✓(t) (x(t)) <latexit 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✓(x(0)) , x(1)
  29. Infinite Depth and Neural-ODEs <latexit sha1_base64="c+kKnAxKjdK3kNTUmMbn0Klm8uI=">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</latexit> where <latexit sha1_base64="qhSLJgQq/yuorQiDFfjqGL3B6xk=">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</latexit> ResNet

    [He et al, 2016] <latexit 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✓(x0) , xT <latexit sha1_base64="qbpiB4MRNcTm1Jd0Cno/ONu/uAc=">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</latexit> xt+1 = xt + 1 T v✓t (xt) <latexit 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x0 <latexit 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Neural ODE [Chen et al, 2018] <latexit 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where <latexit 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T ! +1 <latexit 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dx(t) dt = v✓(t) (x(t)) <latexit 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✓(x(0)) , x(1) <latexit 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T ! +1 is a singular limit (✓ can “explodes” during training) <latexit 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Trajectories cannot cross: ✓ defines a di↵eomorphism.
  30. Architectures for text and sequences Shallow Recurrent Neural Networks Long

    Short Term Memory (LSTM) [1997] yt+1 = fθ (xt+1 , yt ) xt+1 yt+1 yt fθ xt+2 yt+2 fθ xt+1 fθ Jürgen Schmidhuber John Hopfield Usages : Training: next token prediction! Text to text (translation) Auto-regressive (generative) ~ skip connections (avoid vanishing gradients)
  31. Ilya Sutskever Architectures for text and sequences Shallow Recurrent Neural

    Networks Long Short Term Memory (LSTM) [1997] yt+1 = fθ (xt+1 , yt ) Deep Recurrent Networks [2013] Non-recurrent: Transformers [2017] xt+1 yt+1 yt fθ xt+2 yt+2 fθ xt+1 fθ Jürgen Schmidhuber John Hopfield Usages : Training: next token prediction! Text to text (translation) Auto-regressive (generative) ~ skip connections (avoid vanishing gradients)
  32. ˜ xi := ∑ j e⟨Qxi ,Kxj ⟩ ∑ ℓ

    e⟨Qxi ,Kxℓ ⟩ Vxj Transformers and attention mechanism … + <latexit 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x1 <latexit 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x2 <latexit 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{xi }i Points cloud Positional encoding Token encoding Tokenize Le groupe thématique SMAI- S I G M A ( S i g n a l - I m a g e - Géométrie-Modélisation- Approximation) est issu de l ’A s s o c i a t i o n F r a n ç a i s e d ’A p p r o x i m a t i o n ( A FA ) , association créée en 1989 et intégrée en tant que groupe au sein de la SMAI en 2000. Le groupe thématique SMAI- S I G M A ( S i g n a l - I m a g e - G é o m é t r i e - M o d é l i s a t i o n - Approximation) est issu de l ’ A s s o c i a t i o n F r a n ç a i s e d ’ A p p ro x i m a t i o n ( A FA ) , association créée en 1989 et intégrée en tant que groupe au sein de la SMAI en 2000. xi xj (Unmasked) Attention layer … next token probabilities Attention Norm MLP Classif N × … Ashish Vaswani
  33. ˜ xi := ∑ j e⟨Qxi ,Kxj ⟩ ∑ ℓ

    e⟨Qxi ,Kxℓ ⟩ Vxj Transformers and attention mechanism … + <latexit 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x1 <latexit 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x2 <latexit sha1_base64="Fgw+vWgPriclgLxpVoHDEicBDLw=">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</latexit> {xi }i Points cloud Positional encoding Token encoding Tokenize Le groupe thématique SMAI- S I G M A ( S i g n a l - I m a g e - Géométrie-Modélisation- Approximation) est issu de l ’A s s o c i a t i o n F r a n ç a i s e d ’A p p r o x i m a t i o n ( A FA ) , association créée en 1989 et intégrée en tant que groupe au sein de la SMAI en 2000. Le groupe thématique SMAI- S I G M A ( S i g n a l - I m a g e - G é o m é t r i e - M o d é l i s a t i o n - Approximation) est issu de l ’ A s s o c i a t i o n F r a n ç a i s e d ’ A p p ro x i m a t i o n ( A FA ) , association créée en 1989 et intégrée en tant que groupe au sein de la SMAI en 2000. xi xj (Unmasked) Attention layer Arbitrary number of tokens Arbitrary number of layers Expressivity Understanding … next token probabilities Attention Norm MLP Classif N × … Ashish Vaswani
  34. Generative AI: images vs. texts DALL·E 2 Représente un chercheur

    en mathématiques en train de présenter l’IA générative devant un large auditoire. Raconte de façon rigolotte l'histoire d'un chercheur CNRS qui présente l’IA générative devant un large auditoire. Le professeur Jacques Leclair, célèbre directeur de l’Institut de Mathématiques du CNRS, monte sur scène avec l’enthousiasme d’un enfant qui vient de découvrir que son yaourt préféré a un nouveau goût. Devant lui, un large auditoire aux regards curieux. “Aujourd’hui, mes amis,” commence-t-il avec un clin d’œil complice, “je vais vous prouver que les mathématiques peuvent être aussi sexy qu’un chaton jouant avec une pelote de laine… et je parle bien sûr de l’IA générative!”. D’un geste dramatique, il appuie sur sa télécommande, et l’écran derrière lui se remplit d’équations complexes qui dansent comme des notes de musique sur un karaoké.
  35. Pre-Training via Self-supervision DALL·E 2 Add noise Denoise Image: Architecture:

    U-Net/transformers pre-training: denoising (score matching). generation: iterative denoising (inverse diffusion)
  36. Pre-Training via Self-supervision DALL·E 2 Add noise Denoise Image: Architecture:

    U-Net/transformers pre-training: denoising (score matching). generation: iterative denoising (inverse diffusion) Masking Next token prediction Text: Architecture: transformers pre-training: next-token prediction (teacher forcing) generation: auto-regressive (logistic models on tokens) … …
  37. Image generation: Denoising diffusion models Goal: learn a map from

    lattent space to image space . X Y Earlier attempts: VAE, GANs, Normalizing flows, … X Y generation
  38. Image generation: Denoising diffusion models Goal: learn a map from

    lattent space to image space . X Y Earlier attempts: VAE, GANs, Normalizing flows, … X Y generation noising X Y Y de-noising Denoising models Synthesis by inverting the noising (diffusion) Learning the denoiser (score matching) [Valentin de Bortoli]
  39. Text Generation: LLM and in-context learning Very large transformer networks.

    (Pre)-trained for data generation Can be « tuned » easily (LORA, etc) Large Language Model ⟺ Foundation models Frontier models ⟺
  40. Text Generation: LLM and in-context learning Very large transformer networks.

    (Pre)-trained for data generation Can be « tuned » easily (LORA, etc) Large Language Model ⟺ Pour prouver que le produit scalaire matriciel peut s'écrire sous la forme , considérons les définitions suivantes : le produit scalaire entre deux matrices et de taille est donné par . La trace du produit est définie comme . En développant l'élément diagonal, nous avons , ce qui donne . Ainsi, . ⟨S, T⟩ tr(STt) S T n × n ⟨S, T⟩ = ∑n i=1 ∑n j=1 sij tij STt tr(STt) = ∑n i=1 (STt)ii (STt)ii = ∑n j=1 sij tij tr(STt) = ∑n i=1 ∑n j=1 sij tij = ⟨S, T⟩ ⟨S, T⟩ = tr(STt) Prouve que le produit scalaire matriciel peut s'écrire tr . ⟨S, T⟩ (STt) … but used to solve (unseen?) problems! Foundation models Frontier models ⟺
  41. Text Generation: LLM and in-context learning Very large transformer networks.

    (Pre)-trained for data generation Can be « tuned » easily (LORA, etc) Large Language Model ⟺ Pour prouver que le produit scalaire matriciel peut s'écrire sous la forme , considérons les définitions suivantes : le produit scalaire entre deux matrices et de taille est donné par . La trace du produit est définie comme . En développant l'élément diagonal, nous avons , ce qui donne . Ainsi, . ⟨S, T⟩ tr(STt) S T n × n ⟨S, T⟩ = ∑n i=1 ∑n j=1 sij tij STt tr(STt) = ∑n i=1 (STt)ii (STt)ii = ∑n j=1 sij tij tr(STt) = ∑n i=1 ∑n j=1 sij tij = ⟨S, T⟩ ⟨S, T⟩ = tr(STt) Prouve que le produit scalaire matriciel peut s'écrire tr . ⟨S, T⟩ (STt) … but used to solve (unseen?) problems! Genomics? What is the equivalent of next token prediction for science? How can these approaches be extended beyond images and text? Astrophysics ? Chemistry? Materials? … Maths ? Foundation models Frontier models ⟺
  42. Mathematics at the heart of LLMs « The fact that

    the program can come up with a non-obvious construction like this is very impressive, and well beyond what I thought was state of the art. » Prove that ∠KIL + ∠XPY = 180° Timothy Gowers AlphaProof and AlphaGeometry 2: silver medal level at the Olympiad. Toward gold medal? ? COQ Theoretically understanding the processes at work in transformers. Interoperability with formal proof languages. The evolution of the profession of mathematician. The industrial challenge of leveraging mathematics to train LLMs (reasoning, evaluation).
  43. Foundations Models in Genomics and Astrophysics AstroCLIP: A Cross-Modal Foundation

    Model for Galaxies scPRINT: Large Cell Model for scRNAseq data Establishing connections between models from different disciplines. What is the equivalent of “next token prediction” in different scientific domains?
  44. Foundation models in the humanities. Generate metadata: • Analyze large

    archival collections. • Historical, scientific, legislative texts… • Annotations, summaries, etc. From image to text: Handwritten text. Ancient texts. relatively reliable. very cheap.
  45. Flash Attention Beyond attention (better scaling laws)? High performance computing

    (flash attention, triton…) Numerics The future ? Machine learning (AI) is changing at breakneck speed. Impossible to make 5 year predictions... [Mensch et al 2023]
  46. Flash Attention Beyond attention (better scaling laws)? High performance computing

    (flash attention, triton…) Numerics The future ? Machine learning (AI) is changing at breakneck speed. Impossible to make 5 year predictions... Connect diffusion (continuous) and LLM (discrete) models Theory Understand the “in context learning” process. [Mensch et al 2023]
  47. Flash Attention Beyond attention (better scaling laws)? High performance computing

    (flash attention, triton…) Numerics The future ? Machine learning (AI) is changing at breakneck speed. Impossible to make 5 year predictions... Connect diffusion (continuous) and LLM (discrete) models Theory Understand the “in context learning” process. [Mensch et al 2023] Applications AI for science: beyond next token prediction Towards multimodal (video, 3D, etc.). scGPT
  48. Flash Attention Beyond attention (better scaling laws)? High performance computing

    (flash attention, triton…) Numerics The future ? Industry Business model (few qualified engineers + fine tuning)? Open weights vs. open source (training data?) Interface with daily life? Machine learning (AI) is changing at breakneck speed. Impossible to make 5 year predictions... Connect diffusion (continuous) and LLM (discrete) models Theory Understand the “in context learning” process. [Mensch et al 2023] Applications AI for science: beyond next token prediction Towards multimodal (video, 3D, etc.). scGPT