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The Mathematics of Neural Networks

The Mathematics of Neural Networks

Tutorial talk at the conference F2S "Science et Progrès" 2023

Gabriel Peyré

May 05, 2023
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  1. The Mathematics of Neural Networks Gabriel Peyré É C O

    L E N O R M A L E S U P É R I E U R E www.numerical-tours.com
  2. Overview <latexit 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1 layer <latexit 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2 layers <latexit

    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Convolutional <latexit 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ResNet <latexit 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Transformers Networks Architectures Optimization <latexit 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non-linear <latexit 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structured <latexit 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very deep <latexit 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very non-linear Machine Learning <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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x <latexit 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y yi = +1 yi = 1
  3. Supervised vs Unsupervised Learning <latexit sha1_base64="I8r1jo2BiNzWFgtYEKBvick83w0=">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</latexit> Un-supervised learning: <latexit sha1_base64="UwjFjyD/HgpbP/nZSDdMdYT8t5g=">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</latexit>

    Clustering <latexit sha1_base64="fHX1xpg2l1d48CtAmyQ/3vEzPDQ=">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</latexit> Generative modeling <latexit 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Dimensionality reduction <latexit 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=) <latexit 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=) <latexit 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VAE, GAN, di↵usion <latexit 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k-means, DBScan <latexit 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PCA, t-SNE, UMAP
  4. Supervised vs Unsupervised Learning <latexit sha1_base64="I8r1jo2BiNzWFgtYEKBvick83w0=">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</latexit> Un-supervised learning: <latexit sha1_base64="UwjFjyD/HgpbP/nZSDdMdYT8t5g=">AABE6nictVzbchTJES3WtzW+sfajX3qtxcE6tFjI+BKx4YgFjRBaBAhmJFgQEHNpDQ2t6WF6ZrjM6iccfnE47Cf/h7/DH+AI+8m/4LxUdVXPVHdWy5gOSdXVdTKzsquyMrOq6Y3TJJ9ubPzj3Aff+Oa3vv2dD797/nvf/8EPf3Thox8f5tls0o8P+lmaTR72unmcJqP4YJpM0/jheBJ3T3pp/KD3cgufP5jHkzzJRp3p23H85KQ7HCXHSb87hapHW+ksn8aTZDR8dmFt4/IG/YtWC1d0YU3pf/vZRx//Ux2pgcpUX83UiYrVSE2hnKquyuF6rK6oDTWGuidqAXUTKCX0PFan6jxgZ9AqhhZdqH0Jv4dw91jXjuAeaeaE7gOXFH4mgIzURcBk0G4CZeQW0fMZUcbaKtoLoomyvYW/PU3rBGqn6jnUSjjTMhSHfZmqY/U76kMCfRpTDfaur6nMSCsoeeT0agoUxlCH5QE8n0C5T0ij54gwOfUdddul5/+illiL933ddqb+TVJehCtSbd37rKDQVXOiH9HbnMEzlicFzkOgEOs+Yuk16fqEej+C9guovwPXKZWMTnpwLaj2tBa5BZcPuSUid+DyIXdE5B5cPuSeiNyHy4fc10jETkjnfnwbLh++LXK+B5cPeU9E3ofLh7wvIg/h8iEPReQjuHzIRyLyBlw+5A0ReQsuH/KWiOzA5UN2ROQBXD7kgYjchsuH3NbI6pk6gSsjOokwK69BucwDLUUKNddE+a6TdfRhrwfM6X4FVp7VLfjrx7YCdBpXYLcDxt1xBVYeeTtgI/1Y2RbdpNXEh70pYndhBPixuyL2S/WiAvtlwEx7WYGV59oetPNjZet7G+782Nsi9g6U/Fh5jboLNX7s3YAVY1yB3Rex99SrCmyI1Z9UYGW73wa74sfK61QH2vuxIdZ0VoGV7ekheDB+rLxaPYBaP/aBiH2o3lRgH4rYr8C6+7FfBayw7yqwZo09TyvIkPyRGGZsHbVuMSuxNAZqXYF/WqwtKfnGPaiXMMMCMyTMiYjYKRA7gYi9ArEXLFde2NGc/F2ZS7tAtAMRvWJtwtJUbD8o2mMpDUC0CkRrCVHnkeK7Nn2Zk3dhaiTktFi5sBTSp6yw31iK9Xiot7wGcbeE4LH9nEb+OkVLGEGhpuqoPS/WeEZGdF+HeE3Rm+ml4SHjpoVVcFFvRFTPg+qJqLce1FsRNfOgZiJq7kHNRZSd+S7uKGAEWP3ju1jQHY8A9pGrrwi8gmuw6tyEORrB+NkHL/A+1dyFv22KvaWrTjKM5nGdxCzHk5IlnkBpodag3kaFLYqvU5phMUjGLe/qGB/vMLex0HOOrfBpsZJHRcYknE5C8gwLOugtRjSfmtG5RTWn5N1xqRn+ZjHvTakZfps0fkpePJea4ada+ukZZO9obOcM2DbMprHWvi03pcH5F6Zhyudp1UWLi2/1RI8ZpPemIf1d/WZ2z/BetqjE+rHlZjRyp395qX9NaFg9546em1FB74m9XlOKGvdkpONeW24qQ0ar6EjLYe+avhlsM9BvxpSb0dgHj2uLYu6FU246esdFb2y5GY1DxXnPU/LkTbkZjSHdsz5suRkNzLZ0dZxvy00tO2qAY2dbbmrVR5QFxhwQj3musV7RhPykmaaWkH9Qn61xff7VdQxzNk+LGKGekvVtq+n0irWsXiLjL8Rg1aYN5UD/Yub4YGUaC7Upxlcsw7S0vq/SsWs8an4PtBjB7Oc9AClnnoKEJieB1jsFilfEqKvcM4PbFHE4So6XUEe6dip6i5YvZ43Kdc+oVorLbG+tHo/IXuc09sbkE+6RZiU97FW+4SqKkob2ShqS6TXR3Ts9X8va3xBx4yXEuBhpfdoR4p20+jjVp/W2o+OLepdnChfv+djxi9nmY21tMObJyBahLHU83XYmj+TW4bq6rmyOm59F9EbRXs3JaiS0I5WLUajJFrM3vqB7S/uA9uSQB9Pow3uMNJWx4l0zzKJjPj0ii+raW4k36stk6Lick9U19rgePXTQQw+6eYyzBSvGHSh1IGY4gLtOQJRzvtBVRhqfqM+K3dGM3mB9RJ+WLKShwfYmLlnIuij7eYnKa0DjaOAoPZzGMh2DP1qhJEf9Pnls7Fq2/Bdp59bsb3dpjFeP5upMzIC4bhLXiGYN7+ry3TIHlmDhfbJJ/mt9L5FfE45oQyWuTx3OrJcR7fjHFMGOyTNOabZJs6Pc2s1PLT8xnPaV2TvH3eyMLGRE9i+C9SmjMRnRj3t2wOygs0VIyUaG2J2k8G58vk4ijjHrxyWKTzXY8RaTLZsRf0PXnV05jUWOGHgdOF0a20Yne+QLxsR1oq27ndv1qw8i7TkJd5QwRTtWLhH/T+m3+THjZG1lRKCG8Q3k2tb53kdGMQvqqEurfL0NMm1dKT8pZHiqpbbrn5Xpk5JkLYq4UB5crQfAuU/3zAtHyYTkzlfa8Dpal81FyuMlPWJvjymKZ7s/1Cswyr1Oq+QazbkjGiVDGAXTIoowbaUs8jLfel5l6mG08/8LdavrstaQYqRsBpc1JOX3Y4rWXClTGNU8fl/SbPJrfbLUqp7PiMbiiTOXv4baj+G3kdvch9HplazCdRoDTMHeWY1wTbTSIozX9RIvMzINLXtv+dkxaVq5NWeJr9m62Rh73pjKPo2aNzprYcpnofHCofEiUIcd2mu0WjT1xhI9E2OLjt6tDOXXhFunAeWZSFn2yAwqCZDSjaXCqA5EqnKMb1DvRFobIq0uzFZ3N8Cd8yFI/1xfnt1fF6t7pG6Qb9MnD4zjlwHN0oR8LlNbH6kxBeR8VdtXd/YfUQ1y75EFRcp8jhNnDO869ek6LST9uV7ZMrLz1iKYc0uvdRtjY4+o/KsV5AnNiZzmpUFcpRaxlt+VI1qySJcdnyOizH+XfCr2O+pjZre1fSdRyZ+w8SbPKsuLI4UR6V/KvO2uRK+7TvwaUUw40951D2g1f8NIgTEmk+D3LHN6Q7jK8U4Ce7Q9sp+rdop38UaORJdJ6oX6fYCN4ajXjnV3bJkem779Alqi1u1b97WQ+aXBHCV+Z9nR69KqdqJ91MXS/dlodfUqV76v08Nsia/Vx4zauJGFjfLKmCP1eTAXlqgZF8aEcGnWiybyN5O8icy8OxVK2bQ2lMuZBrYxzyleks6BIsLn3V3yenOfCv3ordDrEdalxjUSJczGZTo/4FpazEpFSxGSWy+tSamzHlWtF5aHu2pYO86WMiYrmCopd8Ot3T4claIVORvDFPqKT/ZWxYkuzc/hwt+R8kWJhmNIDrENfu41taW238OpiFe6zJnNiGrQJgyWYvCu7me5Rb2OXjnUXfohHMJ5JKBrSfqEVtSmsjNlWXKXejj912QNJioWpbctm/fB5SL3ZJVTk/4kZOHk3iTKfJPTtC+GQ0hPylzC+fD+htSLY2W+bWrWB0Nd7kGZQxMe5jxD2Du3rZvzcjnV62uVSygPXgfMzovB4Q5gdcxi24VYqInzRt4/B7QOxzXUzWrxv/bD8LGcmvMK5ZbTN2cvAt46t4t1Zhb94uZzxnILGc3VHMN5ZkXvrNfk58f+X9ToTWVOb94/ffRL7RgwvBaK86GydIx3R5GVN5QK7g/4ZMjUf9Tfz8lfJbwqaFTJ0YSS2a+opmZayNTMl5e+3plnITJZOlUylanZeKJNJ2O31K66AT9bhQfY9JQof1PJfxHr/452ALXHZD1MNp0zCEdUF1MWxO6mDejenqOtkhjP9PIZ3w7U4J74HtXied871B7P/HZKfav+koTn+m2VqUEpMlne5bPzqgc9KO/AcS7IfO8b0Zl6zmbxCbSTgD1GPkfFkZL5+nlBiAHFhcuSLghhRksd5Z6Xco/OJMUVtHulvvVphI/1Tj/uO+D5/G6RXYrUL6muq1cHXKklqfY9Uj2mzECP9L8BEdqv1Tr8Xddlv6T7K5Lm9A7KEr1xntWfBDv1jgv7NeNFyoOZTN1ct8soqre7h/WZ2FYlFz7xXo8f1uCHjpRtelsvKe6eqPrc4ayG5kzL5O7njpTJe7IeMJrtFuOjPn6e1/CaB/T/ViX6liPpDsjSo2x7RPt5E6KXat1sk/R8rrI+b3uzRlrz1SbTtCcr7TgwZyTr9wRSPe6qZz+fg5RyNXEFHXeu84lM6bRI4qUkz89xwGmIbkBv5b6G9FSiMhMlmQV8iTwPkGUeQOdYkOZYpDAUJdH24dmFtSvL/9fHauFw8/KV31y+em9z7Yvr+v8B+VD9VP1MXYK177fqCxj/++oAOI3UH9Vf1F9baesPrT+1/sxNPzinMT9RpX+tv/0XqDdD3Q==</latexit>

    Clustering <latexit sha1_base64="fHX1xpg2l1d48CtAmyQ/3vEzPDQ=">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</latexit> Generative modeling <latexit 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Dimensionality reduction <latexit 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Supervised learning: <latexit 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Regression <latexit 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? <latexit 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=) <latexit 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Classification <latexit 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? <latexit 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=) <latexit 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y = +1 <latexit 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y = 1 <latexit 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=) <latexit 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=) <latexit 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VAE, GAN, di↵usion <latexit 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k-means, DBScan <latexit 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PCA, t-SNE, UMAP
  5. Supervised vs Unsupervised Learning <latexit sha1_base64="I8r1jo2BiNzWFgtYEKBvick83w0=">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</latexit> Un-supervised learning: <latexit sha1_base64="UwjFjyD/HgpbP/nZSDdMdYT8t5g=">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</latexit>

    Clustering <latexit sha1_base64="fHX1xpg2l1d48CtAmyQ/3vEzPDQ=">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</latexit> Generative modeling <latexit 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Dimensionality reduction <latexit 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Supervised learning: <latexit 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Regression <latexit 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? <latexit 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=) <latexit 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Classification <latexit 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? <latexit 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=) <latexit 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y = +1 <latexit 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y = 1 <latexit 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=) <latexit 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=) <latexit 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VAE, GAN, di↵usion <latexit 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k-means, DBScan <latexit 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PCA, t-SNE, UMAP <latexit 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Self-supervised learning: Demain, dès l'aube, à l'heure où blanchit la campagne, Je partirai. Vois- tu, je sais que tu m’attends. J'irai par la forêt, j'irai par la montagne. Je ne puis demeurer loin de toi plus longtemps. Je marcherai les yeux fixés sur mes pensées,… Demain, dès l'aube, à l'heure où blanchit la campagne, Je partirai. Vois- tu, je sais que tu m’attends. J'irai par la forêt, j'irai par la montagne. Je ne puis demeurer loin de toi plus longtemps. Je marcherai les yeux fixés sur mes pensées,… <latexit 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? ? ? <latexit 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? ? ? <latexit 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=) <latexit 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=)
  6. Empirical Risk Minimization <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 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y ⇡ f✓(x) <latexit 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Empirical risk minimization: <latexit 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min ✓ 1 n n X i=1 `(f✓(xi), yi) <latexit 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f✓ <latexit 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f✓0 <latexit 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y <latexit 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Regression: yi 2 R <latexit 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Least square: <latexit sha1_base64="GTs/6CYLabKgJFBv6ol/ZD+zFtI=">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</latexit> `(y, y0) = (y y0)2
  7. Empirical Risk Minimization <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 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y ⇡ f✓(x) <latexit 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Empirical risk minimization: <latexit 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min ✓ 1 n n X i=1 `(f✓(xi), yi) <latexit 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f✓ <latexit 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f✓0 <latexit 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y <latexit 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Regression: yi 2 R <latexit 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Least square: <latexit 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`(y, y0) = (y y0)2 <latexit 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yi 2 { 1, 1} <latexit 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Classification: <latexit 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y ⇡ sign(f✓(x)) yi = +1 yi = 1 <latexit 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Logistic: <latexit sha1_base64="YhFRGauKhVgWcYg0cUejUVbCDx8=">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</latexit> `(y, y0) = log(1 + e yy0 )
  8. Empirical Risk Minimization <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 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y ⇡ f✓(x) <latexit 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Empirical risk minimization: <latexit 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min ✓ 1 n n X i=1 `(f✓(xi), yi) <latexit 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f✓ <latexit 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f✓0 <latexit 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Overfitting, regularization, . . . <latexit 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Regression: yi 2 R <latexit 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Least square: <latexit 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`(y, y0) = (y y0)2 <latexit 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yi 2 { 1, 1} <latexit 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Classification: <latexit 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y ⇡ sign(f✓(x)) yi = +1 yi = 1 <latexit 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Logistic: <latexit sha1_base64="YhFRGauKhVgWcYg0cUejUVbCDx8=">AABFAnictVxZcxTJES7W1xpfrP3ol14LvGBrsSTjI2JjIxY0QmgRMDAjwS4DxBytYaA1PcwFYlZvDv8Yh18cDvvJEf4d/gGOsJ/8F5xHVVf1THVntYzpkFRdXV9mVnZVVmZWNZ1RMphMNzb+ce69r339G9/81vvfPv+d737v+z+48MEPDyfpbNyND7ppko4fddqTOBkM44PpYJrEj0bjuH3cSeKHnZfb+PzhPB5PBumwOT0ZxU+O2/3h4GjQbU+h6tmF6GIrTpLLJ+snH12JPo1aSdq/HG3+PH66+Pjk5KPTKxefXVjbuLpB/6LVwqYurCn9r55+8OE/VUv1VKq6aqaOVayGagrlRLXVBK7HalNtqBHUPVELqBtDaUDPY3WqzgN2Bq1iaNGG2pfwuw93j3XtEO6R5oTQXeCSwM8YkJG6BJgU2o2hjNwiej4jylhbRHtBNFG2E/jb0bSOoXaqnkOthDMtQ3HYl6k6Ur+lPgygTyOqwd51NZUZaQUlj5xeTYHCCOqw3IPnYyh3CWn0HBFmQn1H3bbp+b+oJdbifVe3nal/k5SX4IpUQ/c+zSi01ZzoR/Q2Z/CM5UmAcx8oxLqPWHpNuj6m3g+h/QLq78J1SiWjkw5cC6o9LUVuw+VDbovIXbh8yF0RuQ+XD7kvIutw+ZB1jUTsmHTuxzfg8uEbIuf7cPmQ90XkA7h8yAci8hAuH/JQRH4Jlw/5pYi8CZcPeVNE3obLh7wtIptw+ZBNEXkAlw95ICJ34PIhdzSyeKaO4UqJzkCYldehnOeBliKBmuuifDfIOvqwNwLmdLcAK8/qGvz1Y2sBOo0LsDsB4+6oACuPvF2wkX6sbItu0Wriw94SsXswAvzYPRH7uXpRgP08YKa9LMDKc20f2vmxsvW9A3d+7B0RexdKfqy8Rt2DGj/2XsCKMSrA1kXsffWqABti9ccFWNnuN8Cu+LHyOtWE9n5siDWdFWBle3oIHowfK69WD6HWj30oYh+pNwXYRyL2C7DufuwXASvs2wKsWWPP0wrSJ38khhlbRq2dzUosjYBaW+CfZGtLQr5xB+olTD/D9AlzLCJ2M8RuIGI/Q+wHyzXJ7OiE/F2ZSyNDNAIRnWxtwtJUbN/L2mMpCUDUMkRtCVHmkeK7Nn2Zk3dhaiTkNFu5sBTSpzSz31iK9Xgot7wGcS+H4LH9nEb+OkVLGEGhpsqoPc/WeEZGdF+GeE3Rm+ml4SHjpplVcFFvRFTHg+qIqBMP6kREzTyomYiae1BzEWVnvotrBYwAq398Fwu64xHAPnLxFYFXcB1WnVswRyMYP3XwAh9QzT3426DYW7rKJMNoHtdJzHI8yVniMZQWag3qbVRYo/g6oRkWg2Tc8p6O8fEOcxsLPefYCp9mK3mUZUzC6QxInn5GB73FiOZTNTq3qeaUvDsuVcPfyua9KVXD75DGT8mL51I1/FRLPz2D7E2NbZ4B24DZNNLat+WqNDj/wjRM+Tytumhx8a0e6zGD9N5UpL+n38zeGd7LNpVYP7ZcjcbE6d8k178qNKyeJ46eq1FB74m9XlOKKvdkqONeW64qQ0qr6FDLYe+qvhls09NvxpSr0aiDx7VNMffCKVcdvaOsN7Zcjcah4rznKXnyplyNRp/uWR+2XI0GZlvaOs635aqWHTXAsbMtV7XqQ8oCYw6IxzzXWK9oTH7STFMbkH9Qnq1xff7VdQxzNk+zGKGckvVti+l0srWsXCLjL8Rg1aYV5UD/Yub4YHkaC7UlxlcswzS3vq/SsWs8an4ftBjB7Oc9AClnnoCEJieB1jsBipti1JXvmcFtiTgcJUdLqJaunYreouXLWaN83TOqleIy21urxxbZ6wmNvRH5hPukWUkP+4VvuIiipKH9nIZkelV091bP17z2N0TcaAkxykZal3aEeCetPE71ab3h6PiS3uWZwsV7Pnb8Yrb5SFsbjHlSskUoSxlPt53JI7l1uK6uK5vj5mcRvVG0V3OyGgPakZqIUajJFrM3vqB7S/uA9uSQB9PownuMNJWR4l0zzKJjPj0ii+raW4k36stk6Lg8Iatr7HE5uu+g+x509RhnG1aMu1BqQsxwAHfNgCjnfKarlDQ+Vh9nu6MpvcHyiD7JWUhDg+1NnLOQZVH28xyV14DG0cBRejiNZToG31qhJEf9Pnls7Jq3/Jdo59bsb7dpjBeP5uJMTI+4bhHXiGYN7+ry3TIHlmDhfbJF/mt5L5FfFY5oQyWuTx3OrJch7fjHFMGOyDNOaLZJsyPf2s1PLT8xnOrK7J3jbnZKFjIi+xfB+pTSmIzoxz07YHbQ2SIkZCND7M4g8258vs5AHGPWjxsoPtVgx1tMtmxG/A1dd3ZNaCxyxMDrwOnS2DY62SdfMCauY23d7dwuX30Qac9JuKOEKdqxcpn4X6Hf5seMk7WVEYEaxjcw0bbO9z5SillQR21a5cttkGnrSnkxk+Gpltquf1amiznJahRxoTy4WveAc5fumReOkjHJPVlpw+toWTYXKY+W9Ii9PaIonu1+X6/AKPc6rZJrNOdaNEr6MAqmWRRh2kpZ5GW+5bzy1MNoT/4v1K2u81pDipGyGVzWkJTfjylac6VMYFTz+H1Js8mv9fFSq3I+QxqLx85c/gpqP4TfRm5zH0ank7MKN2gMMAV7ZzXCNdFKizBeN3K8zMg0tOy95WfHpGnl1pwlvmbrZmPseWUqdRo1b3TWwpTPQuOFQ+NFoA6btNdotWjqjSV6JsYWTb1bGcqvCrdmBcozkbLskRnUIEBKN5YKo9oTqcoxvkG9FWltiLTaMFvd3QB3zocg/XN9eXZ/la3ukbpJvk2XPDCOX3o0Swfkc5na8kiNKSDna9q+urO/RTXIvUMWFCnzOU6cMbzr1KXrNJP0p3plS8nOW4tgzi291m2MjW1R+ZcryGOaExOalwZxjVrEWn5XjmjJIl11fI6IMv9t8qnY7yiPmd3W9p1EOX/Cxps8qywvjhSGpH8p87a3Er3uOfFrRDHhTHvXHaBV/Q0jBcaYTILfs5zQG8JVjncS2KPtkP1ctVO8izd0JLpKUi/UpwE2hqNeO9bdsWV6bPr2M2iJWrdv3ddC5pcEc5T4nWVHr02r2rH2URdL92ej1darXP6+TA+zJb5WHzNq40YWNsrLY1rqk2AuLFE1LowJ4VKtF1XkryZ5FZl5dyqUsmltKOczDWxjnlO8JJ0DRYTPu7vs9eauCP3orNDrENalxjUSJczGpTo/4FpazEpFSxGSWy+tSYmzHhWtF5aHu2pYO86WMiYrmCgpd8Ot3T60ctGKnI1hCl3FJ3uL4kSX5idw4e9I+aJEwzEkh9gAP/e62lY77+BUxCtd5sxmRDVoE3pLMXhb9zPfolxHrxzqLv0QDuE8BqBrSfoBrahVZWfKsuQu9XD6r8kajFUsSm9bVu+Dy0XuySqnKv0ZkIWTezNQ5pucqn0xHEJ6kucSzof3N6ReHCnzbVO1Phjqcg/yHKrwMOcZwt65bV2dl8upXF+rXEJ58Dpgdl4MDncAi2MW2y7EQo2dN/LuOaB1OCqhblaL/7Ufho/lVJ1XKLcJfXP2IuCtc7tYZ2bRL64+Zyy3kNFczDGcZ5r1znpNfn7s/0WV3lTq9Obd00e/1I4Bw2uhOB8qS8d4dxRZeUOp4P6AT4ZU/Uf9/Zz8VcKrjEaRHFUomf2KYmqmhUzNfHnp6515FiKTpVMkU56ajScadDJ2W+2pm/CznXmAVU+J8jeV/Bex/u9oe1B7RNbDZNM5g9CiupiyIHY3rUf39hxtkcR4ppfP+DahBvfE96kWz/vepfZ45reZ61vxlyQ81++oVPVykcnyLp+dVx3oQX4HjnNB5nvfiM7UczaLT6AdB+wx8jkqjpTM188LQvQoLlyWdEEIM1rKKHe8lDt0JikuoN3J9a1LI3ykd/px3wHP57ez7FKkfkF1bb064EotSVX3SPWYMgMd0v8GRGi/Uuvwd12X/ZLWVySd0DvIS/TGeVZ+EuzUOy7s14yXKA9mMnVz3S6lqN7uHpZnYmuFXPjEezm+X4LvO1I26G29pLh7rMpzh7MSmjMtk7ufO1Qm78l6wGi2nY2P8vh5XsJrHtD/24Xo246kuyBLh7LtEe3njYleonWzQ9LzucryvO2tEmnNV5tM056stOPAnJEs3xNI9Lgrnv18DlLK1cQFdNy5zicypdMiAy8leX6OAk5DtAN6K/c1pKcSlZkoySzgS+R5gCzzADpHgjRHIoW+KIm2D88urG0u/18fq4XDraubv7567f7W2mc39P8D8r76sfqJugxr32/UZzD+6+oAOP1e/VH9Rf219rvaH2p/qv2Zm753TmN+pHL/an/7L1SKSio=</latexit> `(y, y0) = log(1 + e yy0 )
  9. Overview <latexit 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1 layer <latexit sha1_base64="uoH27gkJuuVhju8UQj1RJfFdpdM=">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</latexit> 2 layers <latexit

    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Convolutional <latexit 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ResNet <latexit 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Transformers Networks Architectures Optimization <latexit 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non-linear <latexit 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structured <latexit 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very deep <latexit 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very non-linear Machine Learning <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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x <latexit 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y yi = +1 yi = 1
  10. 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)
  11. 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
  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?
  13. 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.
  14. The Complexity of Gradient Computation Hypothesis: elementary operations (a ⇥

    b, log(a), p a . . . ) and their derivatives cost O(1). Seppo Linnainmaa This algorithm is reverse mode automatic di↵erentiation [Seppo Linnainmaa, 1970] Theorem: there is an algorithm to compute rE in O(K) operations. <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.
  15. Backward Automatic Differentiation return ✓R ✓r = gr(✓Parents(r) ) for

    r = M + 1, . . . , R function `(✓1, . . . , ✓M ) forward for r = R 1, . . . , 1 rR` = 1 rr` = X s2Child(r) @rgs(✓) rs` backward return (r1`, . . . , rM `) function r`(✓1, . . . , ✓M ) computing ` computing r` `(✓1, ✓2) def. = ✓2e✓1 p ✓1 + ✓2e✓1 ✓1 ✓2 input ✓3 def. = e✓1 ✓4 def. = ✓2✓3 ✓5 def. = ✓1 + ✓4 ✓6 def. = p ✓5 output ✓7 def. = ✓4✓6 g3 g4 g5 g7 g6 `
  16. Overview <latexit 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1 layer <latexit sha1_base64="uoH27gkJuuVhju8UQj1RJfFdpdM=">AABE6HictVzbchu5EYU3t41z8yaPeZmN1ilvyuvIinOp2krV2qIsa821ZZOSvWvZriE5ommPODSHpC9c/UMqL6lU8pQfyXfkA1KVPOUX0hdggCEx0xjF8ZQkDAanu9EDNLobGPcm6SifbW7+49x73/jmt779nfe/e/573//BD3904YMfH+bZfNpPDvpZmk0f9uI8SUfj5GA2mqXJw8k0iU96afKg92Ibnz9YJNN8lI27szeT5PFJPByPjkf9eAZVD7eiNH4Dj59e2Ni8skn/ovXCVV3YUPrffvbBh/9UR2qgMtVXc3WiEjVWMyinKlY5XI/UVbWpJlD3WC2hbgqlET1P1Kk6D9g5tEqgRQy1L+D3EO4e6dox3CPNnNB94JLCzxSQkboImAzaTaGM3CJ6PifKWFtFe0k0UbY38LenaZ1A7Uw9g1oJZ1qG4rAvM3Wsfkd9GEGfJlSDvetrKnPSCkoeOb2aAYUJ1GF5AM+nUO4T0ug5IkxOfUfdxvT8X9QSa/G+r9vO1b9JyotwRaqje58VFGK1IPoRvc05PGN5UuA8BAqJ7iOWXpGuT6j3Y2i/hPo7cJ1SyeikB9eSak9rkdtw+ZDbInIXLh9yV0S24fIh2yJyHy4fcl8jETslnfvxHbh8+I7I+R5cPuQ9EXkfLh/yvog8hMuHPBSRX8HlQ34lIm/C5UPeFJG34fIhb4vILlw+ZFdEHsDlQx6IyB24fMgdjayeqVO4MqIzEmbldSiXeaClSKHmuijfDbKOPuyNgDndr8DKs7oFf/3YVoBOkwrsTsC4O67AyiNvF2ykHyvbolu0mviwt0TsHowAP3ZPxH6unldgPw+YaS8qsPJca0M7P1a2vl/AnR/7hYi9AyU/Vl6j7kKNH3s3YMWYVGD3Rew99bICG2L1pxVY2e53wK74sfI61YX2fmyINZ1XYGV7eggejB8rr1YPoNaPfSBiH6rXFdiHIvZLsO5+7JcBK+zbCqxZY8/TCjIkfySBGVtHLS5mJZYmQC0W+KfF2pKSb9yDegkzLDBDwpyIiN0CsRuIaBeIdrBceWFHc/J3ZS6dAtEJRPSKtQlLM7H9oGiPpTQA0SoQrRVEnUeK79r0ZUHehamRkLNi5cJSSJ+ywn5jKdHjod7yGsTdEoLH9jMa+ZcpWsIICjVVR+1ZscYzMqL7OsQrit5MLw0PGTcrrIKLei2ieh5UT0S98aDeiKi5BzUXUQsPaiGi7Mx3cUcBI8DqH9/Fku54BLCPXH1F4BVch1XnFszRCMbPPniB96nmLvztUOwtXXWSYTSP6yRmOR6XLPEUSku1AfU2KmxRfJ3SDEtAMm55V8f4eIe5jaWec2yFT4uVPCoyJuF0RiTPsKCD3mJE86kZndtUc0reHZea4W8V896UmuF3SOOn5MVzqRl+pqWfnUH2rsZ2z4DtwGyaaO3bclManH9hGqZ8nlZdtLj4Vk/0mEF6rxvS39NvZu8M72WbSqwfW25GI3f6l5f614SG1XPu6LkZFfSe2Os1pahxT8Y67rXlpjJktIqOtRz2rumbwTYD/WZMuRmNffC4tinmXjrlpqN3UvTGlpvROFSc9zwlT96Um9EY0j3rw5ab0cBsS6zjfFtuatlRAxw723JTqz6mLDDmgHjMc431iqbkJ801tRH5B/XZGtfnX1/HMGfzpIgR6ilZ37aaTq9Yy+olMv5CAlZt1lAO9C/mjg9WprFUW2J8xTLMSuv7Oh27xqPm26DFCGY/7wFIOfMUJDQ5CbTeKVC8KkZd5Z4Z3JaIw1FyvII60rUz0Vu0fDlrVK57SrVSXGZ7a/V4RPY6p7E3IZ+wTZqV9NCufMNVFCUNtUsakuk10d1bPV/L2t8UcZMVxKQYaX3aEeKdtPo41af1jqPji3qXZwYX7/nY8YvZ5mNtbTDmycgWoSx1PN12Jo/k1uG6elnZHDc/i+iNor1akNUY0Y5ULkahJlvM3viS7i3tA9qTQx5Mow/vMdJUJop3zTCLjvn0iCyqa28l3qgvk6Hjck5W19jjevTQQQ896OYxzjasGHeg1IWY4QDuugFRzvlCVxlpfKo+KXZHM3qD9RF9WrKQhgbbm6RkIeui7GclKq8AjaOBo/RwGqt0DP5ojZIc9fvksbFr2fJfpJ1bs78d0xivHs3VmZgBcd0irhHNGt7V5btVDizB0vtki/zX+l4ivyYc0YZKXJ84nFkvY9rxTyiCnZBnnNJsk2ZHubWbn1p9YjjtK7N3jrvZGVnIiOxfBOtTRmMyoh/37IDZQWeLkJKNDLE7o8K78fk6I3GMWT9upPhUgx1vCdmyOfE3dN3ZldNY5IiB14HTlbFtdNImXzAhrlNt3e3crl99EGnPSbijhCnasXKJ+H9Mv82PGScbayMCNYxvINe2zvc+MopZUEcxrfL1Nsi0daX8qJDhiZbarn9Wpo9KkrUo4kJ5cLUeAOc+3TMvHCVTkjtfa8PraF02FylPVvSIvT2mKJ7t/lCvwCj3ZVolN2jOHdEoGcIomBVRhGkrZZFX+dbzKlMPo53/X6hbXZe1hhQjZTO4rCEpv59QtOZKmcKo5vH7gmaTX+vTlVb1fMY0Fk+cufw11H4Iv43c5j6MTq9kFW7QGGAK9s5qhGuitRZhvG6UeJmRaWjZe8vPjknTyq05S3zN1s3G2IvGVPZp1LzWWQtTPguN5w6N54E67NJeo9WiqTeW6KkYW3T1bmUovybcug0oz0XKskdmUKMAKd1YKozqQKQqx/gG9VaktSnSimG2ursB7pwPQfrn+urs/rpY3SN1k3ybPnlgHL8MaJaOyOcytfWRGlNAzte0fXVn/xHVIPceWVCkzOc4ccbwrlOfrtNC0p/rlS0jO28tgjm39Eq3MTb2iMq/WkOe0JzIaV4axDVqkWj5XTmiFYt0xfE5Isr8x+RTsd9RHzO7re07iUr+hI03eVZZXhwpjEn/UuZtby163XPi14hiwrn2rntAq/kbRgqMMZkEv2eZ0xvCVY53Etij7ZH9XLdTvIs3diS6QlIv1e8DbAxHvXasu2PL9Nj07RfQErVu37qvhcwvDeYo8TvLjl5Mq9qJ9lGXK/dnoxXrVa58X6eH+Qpfq485tXEjCxvllTFH6tNgLixRMy6MCeHSrBdN5G8meROZeXcqlLJpbSiXMw1sY55RvCSdA0WEz7u75PXmPhb60Vuj1yOsS41rJEqYjct0fsC1tJiVilYiJLdeWpNSZz2qWi8sD3fVsHacLWVCVjBVUu6GW7t9OCpFK3I2hin0FZ/srYoTXZqfwoW/I+WLEg3HkBxiB/zc62pb7byDUxEvdZkzmxHVoE0YrMTgse5nuUW9jl461F36IRzCeYxA15L0I1pRm8rOlGXJXerh9F+RNZiqRJTetmzeB5eL3JN1Tk36MyILJ/dmpMw3OU37YjiE9KTMJZwP729IvThW5tumZn0w1OUelDk04WHOM4S9c9u6OS+XU72+1rmE8uB1wOy8GBzuAFbHLLZdiIWaOm/k3XNA63BcQ92sFv9rPwwfy6k5r1BuOX1z9jzgrXO7RGdm0S9uPmcst5DRXM0xnGdW9M56TX5+7P9Fjd5U5vTm3dNHv9SOAcNrqTgfKkvHeHcUWXlDqeD+gE+GTP1H/f2c/FXCy4JGlRxNKJn9impqpoVMzXx56eudeRYik6VTJVOZmo0nOnQydlvtqZvws114gE1PifI3lfwXsf7vaAdQe0zWw2TTOYNwRHUJZUHsbtqA7u052iqJ8Uwvn/HtQg3uibepFs/73qH2eOa3W+pb9ZckPNe/UJkalCKT1V0+O6960IPyDhzngsz3vhGdqedsFp9AOwnYY+RzVBwpma+fl4QYUFy4KumSEGa01FHueSn36ExSUkG7V+pbn0b4RO/0474Dns+Pi+xSpH5JdbFeHXCllqTa90j1iDIDPdL/JkRov1aX4e9lXfZLur8maU7voCzRa+dZ/UmwU++4sF8zXqQ8mMnULXS7jKJ6u3tYn4ltVXLhE+/1+GENfuhI2aG39YLi7qmqzx3Oa2jOtUzufu5Ymbwn6wGj2bgYH/Xx86KG1yKg/7cr0bcdSXdBlh5l2yPaz5sSvVTrZoek53OV9XnbWzXSmq82maY9WWnHgTkjWb8nkOpxVz37+RyklKtJKui4c51PZEqnRUZeSvL8nASchogDeiv3NaSnEpW5KMk84EvkRYAsiwA6x4I0xyKFoSiJtg9PL2xcXf2/PtYLh1tXrv7myrV7Wxuf3dD/D8j76qfqZ+oSrH2/VZ/B+N9XB+Qz/FH9Rf219bz1h9afWn/mpu+d05ifqNK/1t/+C7ZmQos=</latexit> 2 layers <latexit

    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Convolutional <latexit 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ResNet <latexit 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Transformers Networks Architectures Optimization <latexit 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non-linear <latexit 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structured <latexit 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very deep <latexit 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very non-linear Machine Learning <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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x <latexit 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y yi = +1 yi = 1
  17. Linear model (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:
  18. Linear model (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:
  19. Linear model (1 layer) <latexit sha1_base64="fpFKG9ItOKdtii1xLRoQrxil3sU=">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</latexit> f✓(x) = hx, ✓i

    = P k xk✓k <latexit sha1_base64="5ovuZPU8dBWGm61iCoXPGu2QOzU=">AABFEnictVzvchPJER8u/y7kH5d8zJclhhSkCDEO+VO5StWBZYwPHwgkG+4QULvSWl5YaYVWEgad3yKVh0mlKpVKJZ/yAnmAVCWf8grp6Z7ZmZVmt2cdwpbt2dn5dff0zvR098wSTdIkn21u/uPcB1/56te+/o0Pv3n+W9/+zne/d+Gj7x/m2Xzajw/6WZpNn0RhHqfJOD6YJbM0fjKZxuEoSuPH0att+fzxIp7mSTbuzt5O4mejcDhOjpJ+OIOqFxc2e7P4BHDLVhxPgjQOp+NkPAxG8ew4G+S/OaWq4FJvcpxcObl66eKLCxub1zfxX7BeuKEKG0L9a2cfXfyn6ImByERfzMVIxGIsZlBORShyuJ6KG2JTTKDumVhC3RRKCT6Pxak4D9g5tIqhRQi1r+D3EO6eqtox3EuaOaL7wCWFnykgA3EZMBm0m0JZcgvw+Rwpy9oq2kukKWV7C38jRWsEtTNxDLUcTrf0xcm+zMSR+DX2IYE+TbBG9q6vqMxRK1LywOrVDChMoE6WB/B8CuU+IrWeA8Tk2Hep2xCf/wtbylp531dt5+LfKOVluALRUb3PCgqhWCD9AN/mHJ6RPClwHgKFWPVRlt6grkfY+zG0X0L9fbhOsaR1EsG1xNrTWuQ2XC7kNovchcuF3GWR+3C5kPsssg2XC9lWSImdos7d+A5cLnyH5fwQLhfyIYt8BJcL+YhFHsLlQh6yyC/gciG/YJF34HIh77DIe3C5kPdYZBcuF7LLIg/gciEPWOQOXC7kjkJWz9QpXBnSSZhZeQvKZR7SUqRQc4uV7zZaRxf2tsec7ldg+Vndgr9ubMtDp3EFdsdj3B1VYPmRtws20o3lbdFdXE1c2Lssdg9GgBu7x2I/FS8rsJ96zLRXFVh+ru1DOzeWt76fwZ0b+xmLvQ8lN5Zfox5AjRv7wGPFmFRg2yz2oXhdgfWx+tMKLG/3O2BX3Fh+nepCezfWx5rOK7C8PT0ED8aN5Verx1Drxj5msU/ESQX2CYv9HKy7G/u5xwr7rgKr19jzuIIM0R+JYcbWUQuLWSlLE6AWMvzTYm1J0TeOoJ7DDAvMEDEjFrFbIHY9EfsFYt9brrywozn6uzyXToHoeCKiYm2SpRnbflC0l6XUA9EqEK0VRJ1HKt+17ssCvQtdwyFnxcolSz59ygr7LUuxGg/1llcjHpQQNLaPceRfw2hJRlBSU3XUjos1npAB3tch3mD0pnupefC4WWEVbNQJi4ocqIhFvXWg3rKouQM1Z1ELB2rBoszMt3E9jxFg9C/fxRLvaASQj1x9BeAV3IJV5y7M0QDGTxu8wEdY8wD+djD25q46yWQ0L9dJmeV4VrLEUygtxQbUm6iwhfF1ijMsBsmo5QMV48s7mdtYqjlHVvi0WMmDImPiTydBeYYFHektBjifmtG5hzWn6N1RqRn+bjHvdakZfgc1fopePJWa4WdK+tkZZO8qbPcM2A7MponSvik3pUH5F6Khy+dx1ZUWV77VkRozkt5JQ/p76s3sneG9bGOJ9GPKzWjkVv/yUv+a0DB6zi09N6MivSfyenUpaNyTsYp7TbmpDBmuomMlh7lr+mZkm4F6M7rcjEYbPK5tjLmXVrnp6J0UvTHlZjQOBeU9T9GT1+VmNIZ4T/ow5WY0ZLYlVHG+KTe17FIDFDubclOrPsYssMwB0ZinGuMVTdFPmitqCfoH9dka2+dfX8dkzuZ5ESPUUzK+bTWdqFjL6iXS/kIMVm3WUA7pX8wtH6xMYym22PiKZJiV1vd1OmaNl5rfBy0GMPtpD4DLmacgoc5JSOudAsUbbNRV7pnGbbE4OUqOVlA9VTtjvUXDl7JG5boXWMvFZaa3Ro89tNc5jr0J+oT7qFlOD/uVb7iKIqeh/ZKGeHpNdPdOzdey9jdZ3GQFMSlGWh93hGgnrT5OdWm9Y+n4strlmcFFez5m/Mps85GyNjLmydAWSVnqeNrtdB7JrpPr6jVhctz0LMA3Ku3VAq1GgjtSORuF6mwxeeNLvDe0D3BPTvIgGn14j4GiMhG0ayaz6DKfHqBFte0tx1vqS2foqJyj1dX2uB49tNBDB7p5jLMNK8Z9KHUhZjiAu65HlHO+0FWGGp+Knxa7oxm+wfqIPi1ZSE2D7E1cspB1UfZxicobQMvRQFG6P41VOhrfW6PER/0ueUzsWrb8l3HnVu9vhzjGq0dzdSZmgFy3kGuAs4Z2delulQNJsHQ+2UL/tb6Xkl8TjtKGclyfW5xJL2Pc8Y8xgp2gZ5zibONmR7m1nZ9afaI5tYXeO5e72RlayADtXwDrU4ZjMsAf++yA3kEni5CijfSxO0nh3bh8nYQdY8aPSwSdajDjLUZbNkf+mq49u3IcixQx0DpwujK2tU720ReMketUWXczt+tXH4k05yTsUUIUzVi5gvyv4m/9o8fJxtqIkBqWbyBXts71PjKMWaSOQlzl622QbmtLeamQ4bmS2qx/RqZLJclaGHFJeeRqPQDOfbwnXnKUTFHufK0NraN12VxJebKiR9nbI4ziye4P1Qos5b6Gq+QGzrkejpIhjIJZEUXotlwWeZVvPa8ydT/a+f+FutF1WWuSYiBMBpc0xOX3Y4zWbClTGNU0fl/hbHJrfbrSqp7PGMfiyJrLX0LtRfit5db3fnSiklW4jWOAKJg7oxGqCdZa+PG6XeKlR6amZe4NPzMmdSu75izxNVk3E2MvGlNp46g5UVkLXT4LjZcWjZeeOuziXqPRoq7XlugFG1t01W6lL78m3LoNKM9ZyrxHplGJh5R2LOVHdcBS5WN8jXrH0tpkaYUwW+3dAHvO+yDdc311dn9ZrO6BuIO+TR89MIpfBjhLE/S5dG19pEYUJOebyr7as7+HNZJ7hBZUUqZznHLG0K5TH6/TQtIfq5UtQztvLII+t/RGtdE2tofln68hRzgncpyXGnETW8RKfluOYMUiXbd8jgAz/yH6VOR31MfMdmvzToKSP2HiTZpVhhdFCmPUP5d521uLXves+DXAmHCuvOsIaDV/w5ICYXQmwe1Z5viG5CpHOwnk0UZoP9ftFO3ijS2JrqPUS/FbDxtDUa8Z6/bY0j3WffsJtJRaN2/d1YLnl3pz5PidZUcvxFVtpHzU5cr92WiFapUr39fpYb7C1+hjjm3syMJEeWVMT3zszYUkasaFMD5cmvWiifzNJG8iM+1O+VLWrTXlcqaBbMwxxkvcOVCJcHl3V5ze3FWmH9EavQixNjWq4SjJbFym8gO2pZVZqWAlQrLruTUptdajqvXC8LBXDWPHyVLGaAVTweVuqLXdh14pWuGzMUShL+hkb1WcaNP8GC75OxCuKFFz9MkhdsDPvSW2xc57OBXxWpUpsxlgjbQJg5UYPFT9LLeo19Fri7pN34eDP48EdM1Jn+CK2lR2osxLblP3p/8GrcFUxKz0pmXzPthc+J6sc2rSnwQtHN+bROhvcpr2RXPw6UmZiz8f2t/genEk9LdNzfqgqfM9KHNowkOfZ/B756Z1c142p3p9rXPx5UHrgN550Ti5A1gds5h2PhZqar2R989BWoejGup6tfhf+6H5GE7Neflyy/Gbs5ceb53axSozK/3i5nPGcPMZzdUc/XlmRe+M1+TmR/5f0OhNZVZv3j996ZeaMaB5LQXlQ3npCG+PIiOvLxW5P+CSIRP/EX86x3+V8LqgUSVHE0p6v6Kamm7BU9NfXrp6p5/5yGToVMlUpmbiiQ6ejN0We+IO/GwXHmDTU6L0TSX9lVj3d7QDqD1C66Gz6ZRB6GFdjFkQs5s2wHtzjrZKYnmml874dqFG7onvY60873sf28szv91S36q/JKG5/pnIxKAUmazu8pl5FUEPyjtwlAvS3/sGeKaesll0Am3kscdI56goUtJfPy8RMcC4cFXSJSL0aKmjHDkpR3gmKa6gHZX61scRPlE7/XLfQZ7PD4vsUiB+hnWhWh3kSs1J1XZI9RQzAxHqfxMitF+Ia/D3miq7JW2vSZrjOyhLdGI9qz8JduocF+ZrxsuYB9OZuoVql2FUb3YP6zOxrUoudOK9Hj+swQ8tKTv4tl5h3D0V9bnDeQ3NuZLJ3s8dC533JD3IaDYsxkd9/Lyo4bXw6P+9SvQ9S9JdkCXCbHuA+3lTpJcq3eyg9HSusj5ve7dGWv3VJtE0JyvNONBnJOv3BFI17qpnP52D5HI1cQUde67TiUzutEjipMTPz4nHaYjQo7d8X316ylGZs5LMPb5EXnjIsvCgc8RIc8RSGLKSKPvw4sLGjdX/62O9cLh1/cYvr998uLXxyW31/4B8KH4ofiSuwNr3K/EJjP+2OABOvxd/FH8Vf2v9rvWH1p9bf6GmH5xTmB+I0r/W3/8LieJSvQ==</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:
  20. Overview <latexit 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1 layer <latexit sha1_base64="uoH27gkJuuVhju8UQj1RJfFdpdM=">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</latexit> 2 layers <latexit

    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Convolutional <latexit 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ResNet <latexit 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Transformers Networks Architectures Optimization <latexit 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non-linear <latexit 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structured <latexit 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very deep <latexit 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very non-linear Machine Learning <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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x <latexit 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y yi = +1 yi = 1
  21. Multi-layer Perceptron <latexit 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x <latexit 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