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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=">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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
  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=">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</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=">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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
  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=">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.
  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=">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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.
  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=">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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
  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=">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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:
  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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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
  22. 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
  23. 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 sha1_base64="5l068i531rvtROhjU8UjQi74MDA=">AABFMHictVzdc9u4EUeuX9f043LtY194ddLJdXKu46Yf05vOXGI5ji+O40Syk7soyZASrTChREWUFDs6/1OdvvffaPvS6bSdafvSf6G7C4AAJZALumk4tkEQv93FEljsLsBE4zTJpxsbf7rw3te+/o1vfuv9b1/8zne/9/0PLn34g6M8m0168WEvS7PJ4yjM4zQZxYfTZJrGj8eTOBxGafwoerWFzx/N40meZKPO9HQcPx2Gg1FynPTCKVQ9v7TfncYngFvsZ6NPkEg4SaanvzkLLnfz5HIwnOXTIIqDETwdZ+npKBsmYRpMsyAZ9YBPHgfxCTDM82QOuPXnl9Y21jfoX7BauK4Ka0L9O8g+/Ogfoiv6IhM9MRNDEYuRmEI5FaHI4XoirosNMYa6p2IBdRMoJfQ8FmfiImBn0CqGFiHUvoLfA7h7ompHcI80c0L3gEsKPxNABuIKYDJoN4Eycgvo+YwoY20V7QXRRNlO4W+kaA2hdipeQC2H0y19cdiXqTgWv6Y+JNCnMdVg73qKyoy0gpIHVq+mQGEMdVjuw/MJlHuE1HoOCJNT31G3IT3/J7XEWrzvqbYz8S+S8gpcgWir3mcFhVDMiX5Ab3MGz6Q8KXAeAIVY9RFLb0jXQ+r9CNovoH4frjMqaZ1EcC2o9qwWuQWXC7nFInfgciF3WOQeXC7kHos8gMuFPFBIxE5I5258Gy4Xvs1yfgCXC/mART6Ey4V8yCKP4HIhj1jkl3C5kF+yyNtwuZC3WeRduFzIuyyyA5cL2WGRh3C5kIcschsuF3JbIatn6gSujOgkzKy8CeUyD7QUKdTcZOW7RdbRhb3lMad7FVh+Vrfgrxvb8tBpXIHd9hh3xxVYfuTtgI10Y3lbdIdWExf2DovdhRHgxu6y2M/Fywrs5x4z7VUFlp9re9DOjeWt7z24c2Pvsdh9KLmx/Bp1H2rc2PseK8a4AnvAYh+I1xVYH6s/qcDydr8NdsWN5depDrR3Y32s6awCy9vTI/Bg3Fh+tXoEtW7sIxb7WJxUYB+z2C/AuruxX3issG8rsHqNvUgryID8kRhmbB21sJiVWBoDtZDhnxZrS0q+cQT1HGZQYAaEGbKInQKx44nYKxB73nLlhR3Nyd/lubQLRNsTERVrE5ambPt+0R5LqQeiVSBaS4g6jxTfte7LnLwLXcMhp8XKhSWfPmWF/cZSrMZDveXViPslhBzbL2jkX6NoCSMo1FQdtRfFGi+RAd3XId5Q9KZ7qXnwuGlhFWzUCYuKHKiIRZ06UKcsauZAzVjU3IGasygz821c12MEGP3ju1jQnRwB0keuvgLwCm7CqnMH5mgA4+cAvMCHVHMf/rYp9uauOskwmsd1ErMcT0uWeAKlhViDehMVtii+TmmGxSCZbHlfxfh4h7mNhZpz0gqfFSt5UGRM/OkkJM+goIPeYkDzqRmdu1RzRt6dLDXD3ynmvS41w2+Txs/Ii5elZvipkn56Dtk7Cts5B7YNs2mstG/KTWnI/IukocsXadVFi4tvdajGDNI7aUh/V72Z3XO8ly0qSf2YcjMaudW/vNS/JjSMnnNLz82ooPckvV5dChr3ZKTiXlNuKkNGq+hIyWHumr4ZbNNXb0aXm9E4AI9ri2LuhVVuOnrHRW9MuRmNIyHznmfkyetyMxoDupf6MOVmNDDbEqo435SbWnbUgIydTbmpVR9RFhhzQHLMyxrjFU3IT5opagn5B/XZGtvnX13HMGfzrIgR6ikZ37aaTlSsZfUSaX8hBqs2bSgH+hczywcr01iITTa+kjJMS+v7Kh2zxqPm90CLAcx+uQfA5cxTkFDnJNB6p0DxOht1lXumcZssDkfJ8RKqq2qnrLdo+MqsUbnuOdVycZnprdFjl+x1TmNvTD7hHmmW08Ne5RuuoshpaK+kIZ5eE929VfO1rP0NFjdeQoyLkdajHSG5k1Yfp7q03rZ0fEXt8kzhkns+ZvxitvlYWRuMeTKyRShLHU+7nc4j2XW4rl4TJsctnwX0RtFezclqJLQjlbNRqM4WS298QfeG9iHtySEPSaMH7zFQVMZC7pphFh3z6QFZVNvecrxRXzpDJ8s5WV1tj+vRAws9cKCbxzhbsGLsQ6kDMcMh3HU8opyLha4y0vhEfFLsjmb0Busj+rRkITUNaW/ikoWsi7JflKi8ATSOBhml+9NYpqPx3RVKfNTvksfErmXLf4V2bvX+dkhjvHo0V2di+sR1k7gGNGvkrq68W+YgJVg4n2yS/1rfS+TXhCPaUI7rM4uz1MuIdvxjimDH5BmnNNu42VFubeenlp9oTgdC753jbnZGFjIg+xfA+pTRmAzoxz47oHfQpUVIyUb62J2k8G5cvk7CjjHjxyVCnmow4y0mWzYj/pquPbtyGosyYpDrwNnS2NY62SNfMCauE2XdzdyuX30Qac5J2KNEUjRj5Srx/5h+6x89TtZWRgRqGN9Armyd631kFLOgjkJa5ettkG5rS3m5kOGZktqsf0amyyXJWhRxoTy4WveBc4/uJS8cJROSO19pI9fRumwuUh4v6RF7e0xRvLT7A7UCo9zXaJVcoznXpVEygFEwLaII3ZbLIi/zredVpu5HO/+/UDe6LmsNKQbCZHClhrj8fkzRmi1lCqNajt9XNJvcWp8starnM6KxOLTm8ldQ+xH81nLrez86Uckq3KIxICmYO6MRWROstPDjdavES49MTcvcG35mTOpWds154mtp3UyMPW9M5YBGzYnKWujyeWi8tGi89NRhh/YajRZ1vbZEz9nYoqN2K335NeHWaUB5xlLmPTKNSjyktGMpP6p9liof42vUW5bWBksrhNlq7wbYc94H6Z7ry7P7q2J1D8Rt8m165IHJ+KVPszQhn0vX1kdqkgJyvqHsqz37u1SD3COyoEhZnuPEGSN3nXp0nRWS/kStbBnZeWMR9LmlN6qNtrFdKv98BTmkOZHTvNSIG9QiVvLbcgRLFmnd8jkCyvyH5FNJv6M+ZrZbm3cSlPwJE2/KWWV4yUhhRPrnMm+7K9HrrhW/BhQTzpR3HQGt5m8YKUiMziS4Pcuc3hCucnInQXq0EdnPVTsld/FGlkTrJPVC/NbDxsio14x1e2zpHuu+/RRaotbNW3e14Pml3hw5fufZ0QtpVRsqH3WxdH8+WqFa5cr3dXqYLfE1+phRGzuyMFFeGdMVn3pzkRI14yIxPlya9aKJ/M0kbyKz3J3ypaxba8rlTIO0MS8oXuLOgSLC5d1ddXpzHzP9iFboRYS1qckajhJm4zKVH7AtLWalgqUIya7n1qTUWo+q1gvDw141jB2XljImK5gKLncjW9t96JaiFT4bIyn0hDzZWxUn2jQ/hQt/B8IVJWqOPjnENvi5N8WW2H4HpyJeq7LMbAZUgzahvxSDh6qf5Rb1OnptUbfp+3Dw55GArjnpE1pRm8ouKfOS29T96b8hazARMSu9adm8DzYXviernJr0JyELx/cmEfqbnKZ90Rx8elLm4s9H7m9wvTgW+tumZn3Q1PkelDk04aHPM/i9c9O6OS+bU72+Vrn48pDrgN550TjcAayOWUw7Hws1sd7Iu+eA1uG4hrpeLf7Xfmg+hlNzXr7ccvrm7KXHW5ftYpWZRb+4+Zwx3HxGczVHf55Z0TvjNbn5Sf8vaPSmMqs3754++qVmDGheCyHzobx0Em+PIiOvLxXcH3DJkIn/iD9c4L9KeF3QqJKjCSW9X1FNTbfgqekvL1290898ZDJ0qmQqUzPxRJtOxm6JXXEbfrYKD7DpKVH5TaX8i1j3d7R9qD0m66Gz6TKD0KW6mLIgZjetT/fmHG2VxHimV57x7UAN7onvUS2e992n9njmt1PqW/WXJHKu3xOZ6Jcik+VdPjOvIuhBeQdO5oL0974BnamX2Sx5Am3osccoz1HJSEl//bwgRJ/iwmVJF4TQo6WOcuSkHNGZpLiCdlTqW49G+Fjt9OO+A57PD4vsUiB+RnWhWh1wpeakOnBI9YQyAxHpfwMitF+Ia/D3miq7JT1YkTSnd1CW6MR6Vn8S7Mw5LszXjFcoD6YzdXPVLqOo3uwe1mdiW5Vc5In3evygBj+wpGzT23pFcfdE1OcOZzU0Z0omez93JHTeU+oBo9mwGB/18fO8htfco/93K9F3LUl3QJaIsu0B7edNiF6qdLNN0stzlfV52zs10uqvNiVNc7LSjAN9RrJ+TyBV46569stzkFyuJq6gY891eSKTOy2SOCnx83PscRoi9Ogt31efnnJUZqwkM48vkecessw96Bwz0hyzFAasJMo+PL+0dn35//pYLRxtrl//5fqNB5trn91S/w/I++JH4sfiKqx9vxKfwfg/EIfA6ffiz+Jv4u+t37X+2PpL66+y6XsXFOaHovSv9e//AucCYDU=</latexit> Non-linearity: must be non-polynomial to increase expressivity. Frank Rosenblatt
  24. Two Layers Perceptron <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
  25. Two Layers Perceptron <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
  26. Universality of Perceptrons George Cybenko <latexit sha1_base64="1cPL0hVAAPSTiGgmemX25s2L0wU=">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</latexit> Theorem: <latexit sha1_base64="MP3/RiWYOhtA+n0bvU1LGtOpdp8=">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</latexit>

    If f is continuous on a compact ⌦, for all " > 0 <latexit 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! non quantitative . . . no free lunch. <latexit sha1_base64="Jz0OwNMv5k//RDHXco0G6196vK0=">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</latexit> for p large enough, there exists ✓ such that <latexit sha1_base64="xfYOcK/7SLaT3sOdjnYPuyrhJIM=">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</latexit> 8 x 2 ⌦, |f✓(x) f(x)| 6 "
  27. Universality of Perceptrons George Cybenko Andrew Barron <latexit sha1_base64="1cPL0hVAAPSTiGgmemX25s2L0wU=">AABE83ictVxbcxPJFW42tw25QfKYl9l4SbEpQgwhl8pWqhYsY7wYMEg27CKgdBnLgpFGaCRh0PqXpPKSSiVP+Q35HfkBqUqe8hdyLt3TPVLPnB6HMGW7p6e/c06f6T59zukeupNkmM02N/9x7oOvff0b3/zWh98+/53vfu/7P7hw8YeHWTqf9uKDXpqk0yfdThYnw3F8MBvOkvjJZBp3Rt0kftx9tYXPHy/iaTZMx63Z20n8bNQZjIdHw15nBlUvLlxsz+ITwC1bx3E6jUe/O31xYWPz6ib9i9YL13RhQ+l/++nFj/6p2qqvUtVTczVSsRqrGZQT1VEZXE/VNbWpJlD3TC2hbgqlIT2P1ak6D9g5tIqhRQdqX8HvAdw91bVjuEeaGaF7wCWBnykgI3UJMCm0m0IZuUX0fE6UsbaM9pJoomxv4W9X0xpB7UwdQ62EMy1DcdiXmTpSv6U+DKFPE6rB3vU0lTlpBSWPnF7NgMIE6rDch+dTKPcIafQcESajvqNuO/T8X9QSa/G+p9vO1b9JyktwRaqpe5/mFDpqQfQjeptzeMbyJMB5ABRi3UcsvSFdj6j3Y2i/hPr7cJ1SyeikC9eSak8rkVtw+ZBbInIHLh9yR0TuweVD7onIfbh8yH2NROyUdO7HN+Hy4Zsi54dw+ZAPReQjuHzIRyLyEC4f8lBEfgmXD/mliLwNlw95W0TehcuHvCsiW3D5kC0ReQCXD3kgIrfh8iG3NbJ8pk7hSonOUJiVN6Fc5IGWIoGam6J8t8g6+rC3AuZ0rwQrz+oG/PVjGwE6jUuw2wHj7qgEK4+8HbCRfqxsi+7QauLD3hGxuzAC/NhdEfu5elmC/Txgpr0qwcpzbQ/a+bGy9b0Hd37sPRF7H0p+rLxGPYAaP/ZBwIoxKcHui9iH6nUJNsTqT0uwst1vgl3xY+V1qgXt/dgQazovwcr29BA8GD9WXq0eQ60f+1jEPlEnJdgnIvYLsO5+7BcBK+y7EqxZY8/TCjIgfySGGVtFrZPPSixNgFpH4J/ka0tCvnEX6iXMIMcMCDMSETs5YicQsZcj9oLlynI7mpG/K3Np5ohmIKKbr01Ymont+3l7LCUBiEaOaKwgqjxSfNemLwvyLkyNhJzlKxeWQvqU5vYbS7EeD9WW1yAeFBA8to9p5F+haAkjKNRUFbXjfI1nZET3VYg3FL2ZXhoeMm6WWwUXdSKiuh5UV0S99aDeiqi5BzUXUQsPaiGi7Mx3ce2AEWD1j+9iSXc8AthHLr8i8ApuwqpzB+ZoBONnH7zAR1TzAP42KfaWrirJMJrHdRKzHM8KlngKpaXagHobFTYovk5ohsUgGbd8oGN8vMPcxlLPObbCp/lKHuUZk3A6Q5JnkNNBbzGi+VSPzl2qOSXvjkv18HfyeW9K9fDbpPFT8uK5VA8/09LPziB7S2NbZ8A2YTZNtPZtuS4Nzr8wDVM+T6suWlx8qyM9ZpDeSU36u/rN7J7hvWxRifVjy/VoZE7/skL/6tCwes4cPdejgt4Te72mFNXuyVjHvbZcV4aUVtGxlsPe1X0z2Kav34wp16OxDx7XFsXcS6dcd/RO8t7Ycj0ah4rznqfkyZtyPRoDumd92HI9Gpht6eg435brWnbUAMfOtlzXqo8pC4w5IB7zXGO9oin5SXNNbUj+QXW2xvX519cxzNk8z2OEakrWty2n083XsmqJjL8Qg1Wb1ZQD/Yu544MVaSzVdTG+YhlmhfV9nY5d41Hze6DFCGY/7wFIOfMEJDQ5CbTeCVC8JkZdxZ4Z3HURh6PkaAXV1rUz0Vu0fDlrVKx7QbVSXGZ7a/XYJnud0dibkE+4R5qV9LBX+obLKEoa2itoSKZXR3fv9Hwtan9TxE1WEJN8pPVoR4h30qrjVJ/Wm46OL+ldnhlcvOdjxy9mm4+0tcGYJyVbhLJU8XTbmTySW4fr6hVlc9z8LKI3ivZqQVZjSDtSmRiFmmwxe+NLure0D2hPDnkwjR68x0hTmSjeNcMsOubTI7Korr2VeKO+TIaOyxlZXWOPq9EDBz3woOvHOFuwYtyHUgtihgO4awVEOedzXaWk8an6eb47mtIbrI7ok4KFNDTY3sQFC1kVZR8XqLwBNI4GjtLDaazSMfj2GiU56vfJY2PXouW/RDu3Zn+7Q2O8fDSXZ2L6xPU6cY1o1vCuLt+tcmAJlt4n18l/re4l8qvDEW2oxPW5w5n1MqYd/5gi2Al5xgnNNml2FFu7+anVJ4bTvjJ757ibnZKFjMj+RbA+pTQmI/pxzw6YHXS2CAnZyBC7M8y9G5+vMxTHmPXjhopPNdjxFpMtmxN/Q9edXRmNRY4YeB04XRnbRid75AvGxHWqrbud29WrDyLtOQl3lDBFO1YuE/9P6Lf5MeNkY21EoIbxDWTa1vneR0oxC+qoQ6t8tQ0ybV0pP85leK6ltuuflenjgmQNirhQHlyt+8C5R/fMC0fJlOTO1trwOlqVzUXKkxU9Ym+PKIpnuz/QKzDKfYVWyQ2ac20aJQMYBbM8ijBtpSzyKt9qXkXqYbSz/wt1q+ui1pBipGwGlzUk5fdjitZcKRMY1Tx+X9Fs8mt9utKqms+YxuLImctfQe1H8NvIbe7D6HQLVuEWjQGmYO+sRrgmWmsRxutWgZcZmYaWvbf87Jg0rdyas8TXbN1sjL2oTWWfRs2JzlqY8llovHRovAzUYYv2Gq0WTb2xRC/E2KKldytD+dXh1qpBeS5Slj0ygxoGSOnGUmFU+yJVOcY3qHcirU2RVgdmq7sb4M75EKR/rq/O7q/y1T1St8m36ZEHxvFLn2bpkHwuU1sdqTEF5HxD21d39repBrl3yYIiZT7HiTOGd516dJ3mkv5Ur2wp2XlrEcy5pTe6jbGxbSr/cg05ojmR0bw0iBvUItbyu3JEKxbpquNzRJT575BPxX5HdczstrbvJCr4Ezbe5FlleXGkMCb9S5m33bXoddeJXyOKCefau+4CrfpvGCkwxmQS/J5lRm8IVzneSWCPtkv2c91O8S7e2JHoKkm9VL8PsDEc9dqx7o4t02PTt59BS9S6feu+FjK/JJijxO8sO3odWtVG2kddrtyfjVZHr3LF+yo9zFf4Wn3MqY0bWdgor4hpq0+DubBE9bgwJoRLvV7Ukb+e5HVk5t2pUMqmtaFczDSwjTmmeEk6B4oIn3d32evNfSL0o7tGr0tYlxrXSJQwG5fq/IBraTErFa1ESG69tCYlznpUtl5YHu6qYe04W8qYrGCipNwNt3b70C5EK3I2hin0FJ/sLYsTXZqfwoW/I+WLEg3HkBxiE/zcm2pLbb+HUxGvdZkzmxHVoE3or8TgHd3PYotqHb12qLv0QziE8xiCriXph7Si1pWdKcuSu9TD6b8hazBVsSi9bVm/Dy4XuSfrnOr0Z0gWTu7NUJlvcur2xXAI6UmRSzgf3t+QenGkzLdN9fpgqMs9KHKow8OcZwh757Z1fV4up2p9rXMJ5cHrgNl5MTjcASyPWWy7EAs1dd7I++eA1uGogrpZLf7Xfhg+llN9XqHcMvrm7GXAW+d2sc7Mol9cf85YbiGjuZxjOM807531mvz82P+Lar2p1OnN+6ePfqkdA4bXUnE+VJaO8e4osvKGUsH9AZ8MqfqP+vs5+auE1zmNMjnqUDL7FeXUTAuZmvny0tc78yxEJkunTKYiNRtPNOlk7JbaVbfhZyv3AOueEuVvKvkvYv3f0fah9oish8mmcwahTXUxZUHsblqf7u052jKJ8Uwvn/FtQQ3uie9RLZ73vU/t8cxvq9C38i9JeK7fU6nqFyKT1V0+O6+60IPiDhzngsz3vhGdqedsFp9AGwXsMfI5Ko6UzNfPS0L0KS5clXRJCDNaqih3vZS7dCYpLqHdLfStRyN8onf6cd8Bz+d38uxSpH5BdR29OuBKLUm175HqKWUGuqT/TYjQfqWuwN8ruuyXdH9N0ozeQVGiE+dZ9UmwU++4sF8zXqI8mMnULXS7lKJ6u3tYnYltlHLhE+/V+EEFfuBI2aS39Yri7qmqzh3OK2jOtUzufu5Ymbwn6wGj2U4+Pqrj50UFr0VA/++Wou86ku6ALF3Ktke0nzcleonWzTZJz+cqq/O2dyqkNV9tMk17stKOA3NGsnpPINHjrnz28zlIKVcTl9Bx5zqfyJROiwy9lOT5OQk4DdEJ6K3c15CeSlTmoiTzgC+RFwGyLALoHAnSHIkUBqIk2j68uLBxbfX/+lgvHF6/eu3XV288vLHx2S39/4B8qH6sfqIuw9r3G/UZjP99dQCc3qg/qr+ovzbmjT80/tT4Mzf94JzG/EgV/jX+9l9BPEc6</latexit> Theorem:

    <latexit sha1_base64="MP3/RiWYOhtA+n0bvU1LGtOpdp8=">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</latexit> If f is continuous on a compact ⌦, for all " > 0 <latexit 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! non quantitative . . . no free lunch. <latexit sha1_base64="Jz0OwNMv5k//RDHXco0G6196vK0=">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</latexit> for p large enough, there exists ✓ such that <latexit sha1_base64="xfYOcK/7SLaT3sOdjnYPuyrhJIM=">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</latexit> 8 x 2 ⌦, |f✓(x) f(x)| 6 " Barron’s functions: <latexit 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||f||B , R Rd ||!||| ˆ f(!)|d! < +1 <latexit 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||f f✓ ||L2(⌦) 6 2diam(⌦)||f||B p p <latexit sha1_base64="hO/zv54wxyh8DCE0R1zGL/84DG0=">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</latexit> Theorem: for p large, there exists ✓ such that
  28. Universality of Perceptrons George Cybenko Andrew Barron <latexit sha1_base64="1cPL0hVAAPSTiGgmemX25s2L0wU=">AABE83ictVxbcxPJFW42tw25QfKYl9l4SbEpQgwhl8pWqhYsY7wYMEg27CKgdBnLgpFGaCRh0PqXpPKSSiVP+Q35HfkBqUqe8hdyLt3TPVLPnB6HMGW7p6e/c06f6T59zukeupNkmM02N/9x7oOvff0b3/zWh98+/53vfu/7P7hw8YeHWTqf9uKDXpqk0yfdThYnw3F8MBvOkvjJZBp3Rt0kftx9tYXPHy/iaTZMx63Z20n8bNQZjIdHw15nBlUvLlxsz+ITwC1bx3E6jUe/O31xYWPz6ib9i9YL13RhQ+l/++nFj/6p2qqvUtVTczVSsRqrGZQT1VEZXE/VNbWpJlD3TC2hbgqlIT2P1ak6D9g5tIqhRQdqX8HvAdw91bVjuEeaGaF7wCWBnykgI3UJMCm0m0IZuUX0fE6UsbaM9pJoomxv4W9X0xpB7UwdQ62EMy1DcdiXmTpSv6U+DKFPE6rB3vU0lTlpBSWPnF7NgMIE6rDch+dTKPcIafQcESajvqNuO/T8X9QSa/G+p9vO1b9JyktwRaqpe5/mFDpqQfQjeptzeMbyJMB5ABRi3UcsvSFdj6j3Y2i/hPr7cJ1SyeikC9eSak8rkVtw+ZBbInIHLh9yR0TuweVD7onIfbh8yH2NROyUdO7HN+Hy4Zsi54dw+ZAPReQjuHzIRyLyEC4f8lBEfgmXD/mliLwNlw95W0TehcuHvCsiW3D5kC0ReQCXD3kgIrfh8iG3NbJ8pk7hSonOUJiVN6Fc5IGWIoGam6J8t8g6+rC3AuZ0rwQrz+oG/PVjGwE6jUuw2wHj7qgEK4+8HbCRfqxsi+7QauLD3hGxuzAC/NhdEfu5elmC/Txgpr0qwcpzbQ/a+bGy9b0Hd37sPRF7H0p+rLxGPYAaP/ZBwIoxKcHui9iH6nUJNsTqT0uwst1vgl3xY+V1qgXt/dgQazovwcr29BA8GD9WXq0eQ60f+1jEPlEnJdgnIvYLsO5+7BcBK+y7EqxZY8/TCjIgfySGGVtFrZPPSixNgFpH4J/ka0tCvnEX6iXMIMcMCDMSETs5YicQsZcj9oLlynI7mpG/K3Np5ohmIKKbr01Ymont+3l7LCUBiEaOaKwgqjxSfNemLwvyLkyNhJzlKxeWQvqU5vYbS7EeD9WW1yAeFBA8to9p5F+haAkjKNRUFbXjfI1nZET3VYg3FL2ZXhoeMm6WWwUXdSKiuh5UV0S99aDeiqi5BzUXUQsPaiGi7Mx3ce2AEWD1j+9iSXc8AthHLr8i8ApuwqpzB+ZoBONnH7zAR1TzAP42KfaWrirJMJrHdRKzHM8KlngKpaXagHobFTYovk5ohsUgGbd8oGN8vMPcxlLPObbCp/lKHuUZk3A6Q5JnkNNBbzGi+VSPzl2qOSXvjkv18HfyeW9K9fDbpPFT8uK5VA8/09LPziB7S2NbZ8A2YTZNtPZtuS4Nzr8wDVM+T6suWlx8qyM9ZpDeSU36u/rN7J7hvWxRifVjy/VoZE7/skL/6tCwes4cPdejgt4Te72mFNXuyVjHvbZcV4aUVtGxlsPe1X0z2Kav34wp16OxDx7XFsXcS6dcd/RO8t7Ycj0ah4rznqfkyZtyPRoDumd92HI9Gpht6eg435brWnbUAMfOtlzXqo8pC4w5IB7zXGO9oin5SXNNbUj+QXW2xvX519cxzNk8z2OEakrWty2n083XsmqJjL8Qg1Wb1ZQD/Yu544MVaSzVdTG+YhlmhfV9nY5d41Hze6DFCGY/7wFIOfMEJDQ5CbTeCVC8JkZdxZ4Z3HURh6PkaAXV1rUz0Vu0fDlrVKx7QbVSXGZ7a/XYJnud0dibkE+4R5qV9LBX+obLKEoa2itoSKZXR3fv9Hwtan9TxE1WEJN8pPVoR4h30qrjVJ/Wm46OL+ldnhlcvOdjxy9mm4+0tcGYJyVbhLJU8XTbmTySW4fr6hVlc9z8LKI3ivZqQVZjSDtSmRiFmmwxe+NLure0D2hPDnkwjR68x0hTmSjeNcMsOubTI7Korr2VeKO+TIaOyxlZXWOPq9EDBz3woOvHOFuwYtyHUgtihgO4awVEOedzXaWk8an6eb47mtIbrI7ok4KFNDTY3sQFC1kVZR8XqLwBNI4GjtLDaazSMfj2GiU56vfJY2PXouW/RDu3Zn+7Q2O8fDSXZ2L6xPU6cY1o1vCuLt+tcmAJlt4n18l/re4l8qvDEW2oxPW5w5n1MqYd/5gi2Al5xgnNNml2FFu7+anVJ4bTvjJ757ibnZKFjMj+RbA+pTQmI/pxzw6YHXS2CAnZyBC7M8y9G5+vMxTHmPXjhopPNdjxFpMtmxN/Q9edXRmNRY4YeB04XRnbRid75AvGxHWqrbud29WrDyLtOQl3lDBFO1YuE/9P6Lf5MeNkY21EoIbxDWTa1vneR0oxC+qoQ6t8tQ0ybV0pP85leK6ltuuflenjgmQNirhQHlyt+8C5R/fMC0fJlOTO1trwOlqVzUXKkxU9Ym+PKIpnuz/QKzDKfYVWyQ2ac20aJQMYBbM8ijBtpSzyKt9qXkXqYbSz/wt1q+ui1pBipGwGlzUk5fdjitZcKRMY1Tx+X9Fs8mt9utKqms+YxuLImctfQe1H8NvIbe7D6HQLVuEWjQGmYO+sRrgmWmsRxutWgZcZmYaWvbf87Jg0rdyas8TXbN1sjL2oTWWfRs2JzlqY8llovHRovAzUYYv2Gq0WTb2xRC/E2KKldytD+dXh1qpBeS5Slj0ygxoGSOnGUmFU+yJVOcY3qHcirU2RVgdmq7sb4M75EKR/rq/O7q/y1T1St8m36ZEHxvFLn2bpkHwuU1sdqTEF5HxD21d39repBrl3yYIiZT7HiTOGd516dJ3mkv5Ur2wp2XlrEcy5pTe6jbGxbSr/cg05ojmR0bw0iBvUItbyu3JEKxbpquNzRJT575BPxX5HdczstrbvJCr4Ezbe5FlleXGkMCb9S5m33bXoddeJXyOKCefau+4CrfpvGCkwxmQS/J5lRm8IVzneSWCPtkv2c91O8S7e2JHoKkm9VL8PsDEc9dqx7o4t02PTt59BS9S6feu+FjK/JJijxO8sO3odWtVG2kddrtyfjVZHr3LF+yo9zFf4Wn3MqY0bWdgor4hpq0+DubBE9bgwJoRLvV7Ukb+e5HVk5t2pUMqmtaFczDSwjTmmeEk6B4oIn3d32evNfSL0o7tGr0tYlxrXSJQwG5fq/IBraTErFa1ESG69tCYlznpUtl5YHu6qYe04W8qYrGCipNwNt3b70C5EK3I2hin0FJ/sLYsTXZqfwoW/I+WLEg3HkBxiE/zcm2pLbb+HUxGvdZkzmxHVoE3or8TgHd3PYotqHb12qLv0QziE8xiCriXph7Si1pWdKcuSu9TD6b8hazBVsSi9bVm/Dy4XuSfrnOr0Z0gWTu7NUJlvcur2xXAI6UmRSzgf3t+QenGkzLdN9fpgqMs9KHKow8OcZwh757Z1fV4up2p9rXMJ5cHrgNl5MTjcASyPWWy7EAs1dd7I++eA1uGogrpZLf7Xfhg+llN9XqHcMvrm7GXAW+d2sc7Mol9cf85YbiGjuZxjOM807531mvz82P+Lar2p1OnN+6ePfqkdA4bXUnE+VJaO8e4osvKGUsH9AZ8MqfqP+vs5+auE1zmNMjnqUDL7FeXUTAuZmvny0tc78yxEJkunTKYiNRtPNOlk7JbaVbfhZyv3AOueEuVvKvkvYv3f0fah9oish8mmcwahTXUxZUHsblqf7u052jKJ8Uwvn/FtQQ3uie9RLZ73vU/t8cxvq9C38i9JeK7fU6nqFyKT1V0+O6+60IPiDhzngsz3vhGdqedsFp9AGwXsMfI5Ko6UzNfPS0L0KS5clXRJCDNaqih3vZS7dCYpLqHdLfStRyN8onf6cd8Bz+d38uxSpH5BdR29OuBKLUm175HqKWUGuqT/TYjQfqWuwN8ruuyXdH9N0ozeQVGiE+dZ9UmwU++4sF8zXqI8mMnULXS7lKJ6u3tYnYltlHLhE+/V+EEFfuBI2aS39Yri7qmqzh3OK2jOtUzufu5Ymbwn6wGj2U4+Pqrj50UFr0VA/++Wou86ku6ALF3Ktke0nzcleonWzTZJz+cqq/O2dyqkNV9tMk17stKOA3NGsnpPINHjrnz28zlIKVcTl9Bx5zqfyJROiwy9lOT5OQk4DdEJ6K3c15CeSlTmoiTzgC+RFwGyLALoHAnSHIkUBqIk2j68uLBxbfX/+lgvHF6/eu3XV288vLHx2S39/4B8qH6sfqIuw9r3G/UZjP99dQCc3qg/qr+ovzbmjT80/tT4Mzf94JzG/EgV/jX+9l9BPEc6</latexit> Theorem:

    <latexit sha1_base64="MP3/RiWYOhtA+n0bvU1LGtOpdp8=">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</latexit> If f is continuous on a compact ⌦, for all " > 0 <latexit 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! non quantitative . . . no free lunch. <latexit 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! non-constructive. <latexit 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||f f✓ || <latexit sha1_base64="Jz0OwNMv5k//RDHXco0G6196vK0=">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</latexit> for p large enough, there exists ✓ such that <latexit sha1_base64="xfYOcK/7SLaT3sOdjnYPuyrhJIM=">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</latexit> 8 x 2 ⌦, |f✓(x) f(x)| 6 " Barron’s functions: <latexit 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||f||B , R Rd ||!||| ˆ f(!)|d! < +1 <latexit 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||f f✓ ||L2(⌦) 6 2diam(⌦)||f||B p p <latexit sha1_base64="hO/zv54wxyh8DCE0R1zGL/84DG0=">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</latexit> Theorem: for p large, there exists ✓ such that <latexit sha1_base64="h3UjZpHYqKKTnXvhALF0qypbZ8I=">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</latexit> ! for p “large enough” <latexit 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gradient descent works <latexit 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[Chizat-Bach 2018] <latexit 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f✓ <latexit sha1_base64="fF2PPWSzbfCdHr4VM9aT3EanWT0=">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</latexit> f
  29. 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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y yi = +1 yi = 1
  30. Convolutional CNN <latexit 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x <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=">AABFLnictVxtc9u4EUaub9f05XLtx37h1Ukn18m5jpu+TG86c4nlOL4oiRPJTu6iJENKlMyYEhW+yEl0/k+d/oH+i077odNpZzpz/dS/0MUCIEAJ5IJuGo5tEMSzu1gCi90FmGAeR1m+tfXXC+9945vf+vZ33v/uxe99/wc//ODShz86ypIiHYaHwyRO0ieBn4VxNAsP8yiPwyfzNPSnQRw+Dk52+PPHizDNomTWz9/Mw2dTfzKLxtHQz6HqxaXu5UEaTY5zP02T08terwg+yfwpcJ5NfucFQOkk86LZwk8jfzYMvaDI4XYI9VnopeEwnOfRIvTGURiPss0Xlza2Nrfwn7deuC4LG0z+O0g+/OhfbMBGLGFDVrApC9mM5VCOmc8yuJ6y62yLzaHuGVtCXQqlCJ+H7IxdBGwBrUJo4UPtCfyewN1TWTuDe04zQ/QQuMTwkwLSY1cAk0C7FMqcm4fPC6TMa+toL5Eml+0N/A0krSnU5uwYaimcaumK433J2Zj9FvsQQZ/mWMN7N5RUCtQKl9wzepUDhTnU8fIInqdQHiJS6dlDTIZ957r18fnX2JLX8vuhbFuwf6OUV+DyWE/2Pikp+GyB9D18mwU8E/LEwHkCFELZR146RV1PsfczaL+E+vtwnWFJ6SSAa4m1Z43IHbhsyB0SuQeXDblHIrtw2ZBdEnkAlw15IJEcm6LO7fgeXDZ8j+T8EC4b8iGJfASXDfmIRB7BZUMekcgv4bIhvySRt+GyIW+TyLtw2ZB3SWQfLhuyTyIP4bIhD0nkLlw25K5E1s/UFK4E6UTErLwJ5SoPbiliqLlJyncLraMNe8thTg9rsPSs7sBfO7bjoNOwBrvrMO7GNVh65O2BjbRjaVt0B1cTG/YOid2HEWDH7pPYz9nLGuznDjPtpAZLz7UutLNjaet7D+7s2Hsk9j6U7Fh6jXoANXbsA4cVY16DPSCxD9mrGqyL1U9rsLTd74FdsWPpdaoP7e1YF2ta1GBpe3oEHowdS69Wj6HWjn1MYp+w1zXYJyT2C7DuduwXDivs2xqsWmMv4goyQX8khBnbRM0vZyUvzYGaT/CPy7UlRt84gHoKMykxE8RMScReidhzRHRLRNdZrqy0oxn6uzSXXonoOSKCcm3ipZxsPyrb81LsgOiUiM4Koskj5e9a9WWB3oWqoZB5uXLxkkufktJ+81Iox0Oz5VWIBxWEGNvHOPKvYbTEIyiuqSZqx+UaL5Ae3jchTjF6U71UPGhcXloFE/WaRAUWVECi3lhQb0hUYUEVJGphQS1IlJ75Jm7gMAK0/vm7WOKdGAHCR66/PPAKbsKqcwfmqAfj5wC8wEdY8wD+9jD2pq4myXg0z9dJnuV4VrHEKZSWbAPqdVTYwfg6xhkWgmSi5QMZ4/M7nttYyjknrPBZuZJ7ZcbEnU6E8kxKOtxb9HA+taNzF2vO0LsTpXb4O+W8V6V2+F3U+Bl68aLUDp9L6fNzyN6X2P45sD2YTXOpfV1uS0PkXwQNVb6Iqy63uPytTuWY4fRet6S/L9/M/jneyw6WhH50uR2NzOhfVulfGxpaz5mh53ZUuPckvF5V8lr3ZCbjXl1uK0OCq+hMyqHv2r4Z3mYk34wqt6NxAB7XDsbcS6PcdvTOy97ocjsaR0zkPc/Qk1fldjQmeC/0ocvtaPBsiy/jfF1ua9m5BkTsrMttrfoMs8A8ByTGvKjRXlGKflIhqUXoHzRna0yff30d4zmb52WM0ExJ+7b1dIJyLWuWSPkLIVi1vKUc3L8oDB+sSmPJtsn4SsiQV9b3dTp6jeea74IWPZj9Yg+AypnHIKHKSXDrHQPF62TUVe2Zwm2TOD5KxiuogazNSW9R8xVZo2rdC6yl4jLdW63HAdrrDMfeHH3CLmqW0kO39g3XUaQ01K1oiKbXRndv5Xytan+LxM1XEPNypA1xR0jspDXHqTat9wwdX5G7PDlcYs9Hj1+ebR5La8NjngRtEZeliafZTuWRzDq+rl5jOsctnnn4Rrm9WqDViHBHKiOjUJUtFt74Eu817UPck+M8BI0hvEdPUpkzsWvGs+g8n+6hRTXtLcWb60tl6EQ5Q6ur7HEzemKgJxZ0+xhnB1aM+1DqQ8xwCHd9hyjnYqmrBDWesk/K3dEE32BzRB9XLKSiIexNWLGQTVH2cYXKKaD5aBBRujuNVToKP1ijREf9Nnl07Fq1/Fdw51btb/s4xutHc30mZoRct5Grh7NG7OqKu1UOQoKl9ck2+q/NveT82nDkNpTi+tzgLPQywx3/ECPYOXrGMc42anZUW5v5qdUnitMBU3vnfDc7QQvpof3zYH1KcEx6+GOeHVA76MIixGgjXexOVHo3Nl8nIseY9uMiJk416PEWoi0rkL+ia86uDMeiiBjEOnC2MraVTrroC4bINZXWXc/t5tWHI/U5CXOUCIp6rFxF/h/jb/WjxsnG2ojgGuZvIJO2zvY+EoxZuI58XOWbbZBqa0p5uZThuZRar39apssVyToYcXF5+Go9As5DvBe8+ChJUe5srY1YR5uyuZzyfEWPvLdjjOKF3Z/IFZjLfQ1XyQ2ccwMcJRMYBXkZRai2VBZ5lW8zryp1N9rZ/4W61nVVa5yix3QGV2iIyu+HGK2ZUsYwqsX4PcHZZNd6utKqmc8Mx+LUmMtfQe1H8FvJre7d6AQVq3ALx4CgoO+0RkSNt9bCjdetCi81MhUtfa/56TGpWpk154mvhXXTMfaiNZUDHDWvZdZClc9D46VB46WjDvu416i1qOqVJXpBxhZ9uVvpyq8Nt34LygVJmfbIFCpykNKMpdyojkiqdIyvUG9JWlskLR9mq7kbYM55F6R9rq/O7q/K1d1jt9G3GaIHJuKXEc7SCH0uVdscqQkKnPMNaV/N2T/AGs49QAvKKYtznHzGiF2nIV5npaQ/kytbgnZeWwR1bulUtlE2doDlX64hpzgnMpyXCnEDW4RSflMOb8UibRo+h4eZfx99KuF3NMfMZmv9TryKP6HjTTGrNC8RKcxQ/1TmbX8tet034lcPY8JCetcB0Gr/hjkFgVGZBLtnmeEb4quc2EkQHm2A9nPdToldvJkh0SZKvWS/d7AxIurVY90cW6rHqm8/h5Zc6/qt21rQ/GJnjhS/8+zo+biqTaWPuly5Px8tX65y1fsmPRQrfLU+CmxjRhY6yqtiBuxTZy5ConZcBMaFS7tetJG/neRtZBa7U66UVWtFuZppEDbmGOMl6hwoR9i8u6tWb+5joh/BGr0AsSY1UUNR4tm4ROYHTEvLs1LeSoRk1lNrUmysR3XrheZhrhrajgtLGaIVjBmVuxGtzT4MKtEKnY0RFIZMnOytixNNmp/CxX97zBYlKo4uOcQe+Lk32Q7bfQenIl7JsshseljDbcJoJQb3ZT+rLZp19MqgbtJ34eDOIwJdU9JHuKK2lV1QpiU3qbvTP0VrkLKQlF63bN8Hkwvdk3VObfoToYWjexMx9U1O274oDi49qXJx5yP2N6hejJn6tqldHxR1ugdVDm14qPMMbu9ct27Py+TUrK91Lq48xDqgdl4Uju8A1scsup2LhUqNN/LuOXDrMG6grlaL/7Ufio/m1J6XK7cMvzl76fDWRbtQZma5X9x+zmhuLqO5nqM7z6Tsnfaa7PyE/+e1elOJ0Zt3T5/7pXoMKF5LJvKhtHQCb44iLa8rFb4/YJMhYf9hf7pAf5XwqqRRJ0cbSmq/op6aakFTU19e2nqnnrnIpOnUyVSlpuOJHp6M3WH77Db87JQeYNtTouKbSvGXY+3f0Y6gdozWQ2XTRQZhgHUhZkH0btoI7/U52jqJ+Zlecca3DzV8T7yLtfy8731sz8/89it9q/+SRMz1eyxho0pksrrLp+dVAD2o7sCJXJD63tfDM/UimyVOoE0d9hjFOSoRKamvn5eIGGFcuCrpEhFqtDRRDqyUAzyTFNbQDip9G+IIn8udfr7vwM/n+2V2yWO/wDpfrg58paakOrBI9RQzAwHqfwsitF+xa/D3mizbJT1YkzTDd1CV6LXxrPkk2Jl1XOivGa9gHkxl6hayXYJRvd49bM7Edmq5iBPvzfhJA35iSNnDt3WCcXfKmnOHRQPNQspk7ufOmMp7Cj3waNYvx0dz/Lxo4LVw6P/dWvRdQ9I9kCXAbLuH+3kp0oulbnZRenGusjlve6dBWvXVpqCpT1bqcaDOSDbvCcRy3NXPfnEOksrVhDV0zLkuTmRSp0UiKyV6fs4dTkP4Dr2l++rSU4pKQUpSOHyJvHCQZeFAZ0xIMyYpTEhJpH14cWnj+ur/9bFeONrevP7rzRsPtzc+uyX/H5D32U/YT9lVWPt+wz6D8X/ADoHTH9lf2D/YPzt/6Py587fO30XT9y5IzI9Z5V/n6/8CqYle1A==</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
  31. Convolutional CNN <latexit 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x <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
  32. 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
  33. ResNet-type Architectures [He et al’ 16] ResNet-34 <latexit sha1_base64="hGNaHRogJoszxpvRv/VDNWjykms=">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</latexit> image

  34. ResNet-type Architectures [He et al’ 16] ResNet-34 <latexit sha1_base64="hGNaHRogJoszxpvRv/VDNWjykms=">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</latexit> image

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

    <latexit sha1_base64="T6b9HaNHHjCNOHzV6onUoURA+iA=">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</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 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xt = xt 1 + v✓t (xt 1)
  36. 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
  37. 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)
  38. 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.
  39. 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
  40. Transformers … <latexit sha1_base64="l9TysiXzoaJNeK5mklRn5kPJwdQ=">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</latexit> Token extraction <latexit sha1_base64="BdA8v340HbcvEXK7ggTo43G6W3c=">AAA9i3ictVv9chu3EYfTr1htGqf9szOdaxV3koyrkRRP22mqmVgflhQrlmxSspPQ1vDjRJ195NE8UrLN6En6b/scfY6+QftXX6G7C+CAI3G3gOoKIwkH4re7WAD7ARw7ozTJJ6ur/7zx3g9++KMf/+T9m0s//dkHP//w1ke/OMmz6bgbH3ezNBs/7bTzOE2G8fEkmaTx09E4bg86afyk83ILP39yEY/zJBs2J29G8bNBuz9MzpJuewJNp7c+PIm7k2wcxcNu1kuG/dNby6srq/QTLVbWVGVZqJ+j7KNfr4uW6IlMdMVUDEQshmIC9VS0RQ7lO7EmVsUI2p6JGbSNoZbQ57G4EkuAnUKvGHq0ofUl/O3D03eqdQjPSDMndBe4pPA7BmQkbgMmg35jqCO3iD6fEmVsraI9I5oo2xv431G0BtA6EefQyuF0T18cjmUizsSfaAwJjGlELTi6rqIyJa2g5JE1qglQGEEb1nvw+RjqXUJqPUeEyWnsqNs2ff4v6omt+NxVfafi3yTlbSiRaKjRZwWFtrgg+hHN5hQ+k/KkwLkPFGI1Rqxdkq4HNPoh9J9B+0MoV1TTOulAmVHrVS1yC4oLucUid6G4kLss8gCKC3nAIo+guJBHConYMencjW9AceEbLOdHUFzIRyzyMRQX8jGLPIHiQp6wyG+huJDfssj7UFzI+yzyARQX8gGLbEJxIZss8hiKC3nMIneguJA7Clm9U8dQMqKTMLvyHtTLPNBSpNByj5Vvk6yjC7vpsae7FVh+V2/Dfzd220OncQV2x2PdnVVg+ZW3CzbSjeVt0R55Exd2j8XuwwpwY/dZ7FfiRQX2K4+d9rICy++1A+jnxvLW92t4cmO/ZrEPoebG8j7qEFrc2EMPjzGqwB6x2EfiVQXWx+qPK7C83W+AXXFjeT/VhP5urI81nVZgeXt6AhGMG8t7qyfQ6sY+YbFPxesK7FMW+w1Ydzf2Gw8P+7YCq33sEnmQPsUjMezYOmrtYldibQTU2gz/tPAtKcXGHWjnMP0C0yfMgEXsFohdT8RBgTjwlisv7GhO8S7PpVEgGp6ITuGbsDZh+/eK/lhLPRDbBWJ7DlEXkeJc67FcUHShWzjkpPBcWPMZU1bYb6zFaj3UW16NOCwh5No+p5V/h7IlzKBQU3XUzgsfL5ERPdchLil706PUPHjcpLAKNuo1i+o4UB0W9caBesOipg7UlEVdOFAXLMrsfBvX8lgBRv84FzN6kitAxsjVJYKo4B54nT3YoxGsnyOIAh9TyyH8b1DuzZU6yTCbRz+JpxzPSpZ4DLWZWIZ2kxVuU36d0g6LQTLZ81Dl+PiEZxszteekFb4qPHlUnJj400lInn5BB6PFiPZTGJ0H1HJF0Z2sheH3in2va2H4HdL4FUXxshaGnyjpJ9eQvamwzWtgG7CbRkr7ph5KQ56/SBq6vkReFy0uzupArRmk9zqQ/r6amf1rzMsW1aR+TD2MRm6NLy+NL4SG0XNu6TmMCkZPMurVtSh4JEOV95p6qAwZedGhksM8hc4M9umpmdH1MBpHEHFtUc49s+qhq3dUjMbUw2icCHnueUWRvK6H0ejTs9SHqYfRwNOWtsrzTT3UsqMGZO5s6qFWfUinwHgGJNe8bDFR0ZjipKmillB8UH9aY8f8i34Mz2yeFzlCPSUT21bT6RS+rF4iHS/EYNUmgXJgfDG1YrAyjZlYZ/MrKcOk5N8X6Rgfj5o/AC1GsPvlHQB3Zp6ChPpMAq13ChTX2KyrPDKNW2dxuErO5lAt1Tpho0XDV54aldtOqZXLy8xojR5bZK9zWnsjigkPSLOcHg4qZ7iKIqehg5KGeHohunur9mtZ+6ssbjSHGBUrrUs3QvImrT5PdWm9Yen4trrlmUCRdz5m/eJp85myNpjzZGSLUJY6nnY/fY5kt6FfvSPMGbf8LKIZRXt1QVYjoRupnM1C9WmxjMZn9GxoH9OdHPKQNLowj5GiMhLy1gxP0fE8PSKLattbjjfqS5/QyXpOVlfb43p030L3HejwHGcLPMZDqDUhZziGp6ZHlrNU6CojjY/F74vb0YxmsD6jT0sWUtOQ9iYuWci6LPu8ROUS0LgaZJbuT2Oejsa3FijxWb9LHpO7li3/bbq51ffbbVrj1au5+iSmR1zXiWtEu0be6sqneQ5Sgpnzk3WKX+tHifxCOKIN5bg+tzhLvQzpxj+mDHZEkXFKu43bHeXe9vnU/Cea05HQd+d4m52RhYzI/kXgnzJakxH92u8O6Bt0aRFSspE+dicpohtXrJOwa8zEcYmQbzWY9RaTLZsSf03X3l05rUWZMUg/cDW3trVODigWjInrWFl3s7frvQ8izXsS9iqRFM1a+YT4f0p/9a9eJ8sLKwI1jDOQK1vnmo+MchbUUZu8fL0N0n1tKT8uZHiupDb+z8j0cUmybcq4UB701j3g3KVnyQtXyZjkzhf6SD9ad5qLlEdzesTRnlEWL+1+X3lglPsOecll2nMtWiV9WAWTIovQfblT5Hm+9bzK1P1o5/8X6kbXZa0hxUiYE1ypIe58P6ZszZYyhVUt1+9L2k1urY/netXzGdJaHFh7+Xto/Q381XLrZz86nZJV2KQ1ICmYJ6MR2RIt9PDjtVnipVempmWeDT+zJnUvu+U6+bW0bibHvgimckSr5rU6tdD169B4YdF44anDJt01Gi3qdm2JTtncoqluK335hXBrBlCespT5iEyjEg8p7VzKj2qPpcrn+Br1lqW1ytJqw261bwPsPe+DdO/1+d39feHdI3GfYpsuRWAyf+nRLk0o5tKt9ZmapICc7yr7au/+FrUg9w5ZUKQs3+PEHSNvnbpUrgpJf6c8W0Z23lgE/d7SpeqjbWyL6p8vIAe0J3Lalxpxl3rESn5bjmjOIq1YMUdEJ/9tiqlk3FGfM9u9zZxEpXjC5JtyVxleMlMYkv65k7f9hex138pfI8oJpyq67gCt8BlGChKjTxLckWVOM4ReTt4kyIi2Q/Zz0U7JW7yhJdEKST0TGx42Rma9Zq3ba0uPWI/tM+iJWjez7urB80u9OXL8rnOj1yavNlAx6mzu+Xq02srLlZ/r9DCd42v0MaU+dmZhsrwypiW+8OYiJQrjIjE+XMJGESJ/mOQhMsvbKV/KuremXD5pkDbmnPIl7j1QRLiiu0+c0dynzDg6C/Q6hLWpyRaOEp7GZep8wLa0eCp1c8EPydabtd4otTxRlafQ1G1vYey3tJAxWb9UcGc2srcte6uUpfCnMJJCV8g3eqvyQ5vmF1DwbyRc2aHm6HN22ID49p7YEjvv4G2IV6ouTzQjakFb0JvLvdtqnOUe9Tp6ZVG36ftw8OeRgK456RPypKGyS8q85DZ1f/qXZAXGImalNz3Dx2Bz4UeyyClkPAlZNn40idDfxQkdi+bgM5IyF38+8l6DG8WZ0N9pChuDps6PoMwhhId+j8Fvzk3vcF42p3p9LXLx5SG9gL5x0Ti8+avOVUw/Hws1tmbk3XNA63BWQ117i/91HJqP4RTOy5dbTt81e+Ex67JfrE5kMR4O3zOGm89qrubozzMrRmeiJTc/GfdFQTOVWaN59/QxHjVrQPOaCXkOyksn8fYqMvL6UsF7AZcMmfiP+McN/tsIrwoaVXKEUNL3FNXUdA+emv7GpWt0+jMfmQydKpnK1Ewe0aA3YrfEvrgPv1tFBBj6dqj8LqX8j1j392d70HpG1kOfosuTgxa1xXT6YW7RevSszhhPby2vzX8LebFysr6y9oeVu4/Wl7/cVN9Qfl/8SvwW8pI18UfxpdiD8R4L/KbyX8XfxN83Ptj4fOPPG3+RXd+7oTC/FKWfjZ3/AuWq2i4=</latexit> Vector encoding

    <latexit 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Positional encoding + <latexit 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Points cloud <latexit 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x1 <latexit 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x2 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 sha1_base64="Fgw+vWgPriclgLxpVoHDEicBDLw=">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</latexit> {xi }i
  41. Transformers … <latexit 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Token extraction <latexit 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Vector encoding

    <latexit 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Positional encoding + <latexit 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Points cloud <latexit 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x1 <latexit 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x2 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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{xi }i <latexit sha1_base64="TxVb5GdPgjAMGN+p3Ec+OBLQ8cg=">AAA+EXictVvbchy3EQWdm8Xc5OQxVSk4tFK+KCqSUSWpcrHKEqkLLdqitEtKtldi7WW4Gmr2op1d6rLer0jlY/KUVF7zBfmD5CkfkJd0N4ABZhczDTAKp0hiMDinGz1AoxvY7YyzNJ9ubv5j7Z3vfPd73//Bu5fWf/ijH//kp5ff+9lxPppNuslRd5SNJo877TzJ0mFyNE2nWfJ4PEnag06WPOo838Xnj86TSZ6Ohs3p63HyZNDuD9PTtNueQtXJ5T/NW0Qy72SzZNGaplkvka9O0oXckevm2STpLahu/ZP11umk3Z2vS9nKZ4OTM5k8nd+Tyw2vygdAciblQh5jYX0RjlhfSHlyeWPz2ib9yNXCli5sCP1zOHrvl9uiJXpiJLpiJgYiEUMxhXIm2iKH6xuxJTbFGOqeiDnUTaCU0vNELMQ6YGfQKoEWbah9Dn/7cPeNrh3CPXLmhO6ClAx+J4CU4gpgRtBuAmWUJun5jJixtop7Tpyo22v439FcA6idimdQy+FMy1Ac9mUqTsUfqA8p9GlMNdi7rmaZkVVQc+n0agoMY6jDcg+eT6DcJaSxsyRMTn1H27bp+T+pJdbifVe3nYl/kZZX4JKioXs/Khja4pz4Jb3NGTxT+mQguQ8Mie4jll6SrQfU+yG0n0P9l3AtqGRs0oFrTrWLWuQuXD7kLou8A5cPeYdFHsDlQx6wyEO4fMhDjUTshGzuxzfg8uEbrOQHcPmQD1jkQ7h8yIcs8hguH/KYRX4Nlw/5NYu8DZcPeZtF3oPLh7zHIptw+ZBNFnkElw95xCJvweVD3tLI6pk6gWtEPCkzK29AuSwDPUUGNTdY/W6Sd/RhbwbM6W4Flp/Ve/Dfj90LsGlSgb0VMO5OK7D8yLsDPtKP5X3RXVpNfNi7LHYfRoAfu89iPxdnFdjPA2ba8wosP9cOoJ0fy3vfL+DOj/2CxX4JJT+WX6PuQ40fez9gxRhXYA9Z7APxogIb4vUnFVje7zfAr/ix/DrVhPZ+bIg3nVVgeX96DBGMH8uvVo+g1o99xGIfi1cV2Mcs9ivw7n7sVwEr7JsKrFlj12kF6VM8ksCMrWNrF7MSS2NgazPys2JtySg27kA9h+kXmD5hBiziToG4E4g4KBAHwXrlhR/NKd7lpTQKRCMQ0SnWJixN2fa9oj2WsgDEXoHYW0LURaT4rk1fzim6MDUcclqsXFgK6dOo8N9YSvR4qPe8BnG/hFBj+xmN/KuULWEGhZaqY3tWrPEKKem+DvGSsjfTSyODx00Lr+CiXrGojgfVYVGvPajXLGrmQc1Y1LkHdc6i7Mx3ca2AEWDtj+9iTndqBKgYufqSEBXcgFXnLsxRCePnEKLAh1RzH/43KPfmrjrNMJvHdRJ3OZ6UPPEESnOxAfU2K9yj/DqjGZaAZqrlfZ3j4x3ubcz1nFNeeFGs5LLYMQnnSUmffsGD0aKk+RTHc49qFhTdqVIc/m4x700pDn+LLL6gKF6V4vBTrf30Aro3NbZ5AWwDZtNYW9+WYznU/oviMOV1WnXR4+JbHegxg3yvIvn39ZvZv8B72aWSso8tx3HkTv/yUv9iOKydc8fOcSwYPamo15RkdE+GOu+15VgdRrSKDrUe9i72zWCbnn4zphzHcQgR1y7l3HOnHDt6x0VvbDmO41iofc8FRfKmHMfRp3tlD1uO48DdlrbO82051rOjBVTubMuxXn1Iu8C4B6TGvKqxUdGE4qSZZkspPqjfrXFj/tV1DPdsnhY5Qj2TjW2reTrFWlavkYkXEvBq00g9ML6YOTFYmWMuttn8SukwLa3vqzx2jUfLH4AVJcx+dQbA7ZlnoKHZk0DvnQHjFpt1lXtmcNssDkfJ6RKqpWunbLRo5apdo3LdCdVyeZntrbVji/x1TmNvTDHhAVmWs8NB5RuuYuQsdFCyEM8XY7s3er6Wrb/J4sZLiHEx0rp0IqRO0urzVJ/VG46Nr+hTnilc6szHjl/cbT7V3gZznhH5ItSlTqbbzuwjuXW4rl4Vdo9bPZP0RtFfnZPXSOlEKmezULNbrKLxOd1b7iM6k0MZiqML71FqlrFQp2a4i4776ZI8qutvOdloL7NDp8o5eV3jj+vRfQfd96Djc5xdWDG+hFITcoYjuGsGZDnrha1GZPGJ+E1xOjqiN1if0WclD2k4lL9JSh6yLst+VmJ5CWgcDSpLD+dY5jH41goTn/X79LG5a9nzX6GTW3O+3aYxXj2aq3dieiR1m6RKmjXqVFfdLUtQGsy9T7Ypfq3vJcqLkYg+lJP61JGs7DKkE/+EMtgxRcYZzTZudpRbu/tTy0+MpENhzs7xNHtEHlKS/5OwPo1oTEr6dT87YE7QlUfIyEeG+J20iG58sU7KjjEbx6VCfarBjreEfNmM5Bted3blNBZVxqDWgcXS2DY2OaBYMCGpE+3d7dyuX30QaT8n4Y4SxWjHyock/yP6a37NONlYGRFoYXwDufZ1vvcxopwFbdSmVb7eB5m2rpYfFDo81Vrb9c/q9EFJsz3KuFAfXK17ILlL90oWjpIJ6Z2vtFHraN1uLjKPl+yIvT2lLF75/b5egVHvq7RKbtCca9Eo6cMomBZZhGnL7SIvy62XVWYP487/L+zW1mWrIaMUdgdXWYjb308oW3O1zGBUq/H7nGaT3+qTpVb1coY0FgfOXP4Wat+Hv0Zvcx/G0yl5hZs0BhSDvbMWUTVypUWYrJslWWZkGi57b+XZMWlauTUXya+Vd7M59nk0yyGNmld618KUL8Jx5nCcBdqwSWeN1oqm3niiEza3aOrTylB5MdKaEcwzlpmPyAwqDdDSzaXCWHssK5/jG9QblmuT5WrDbHVPA9w5H4L0z/Xl2f1tsbpLcZtimy5FYCp/6dEsTSnmMrX1mZpiQMnXtX91Z3+LalB6hzwoMqvPceKMUadOXboWhaa/1ivbiPy89Qjmc0svdRvjY1tU/u0KckBzIqd5aRDXqUWi9Xf1kEse6ZoTc0ja+W9TTKXijvqc2W1t34ksxRM231SzyspSmcKQ7M/tvO2vZK/7Tv4qKSec6ei6A1zxbxgZFMbsJPgjy5zeEK5y6iRBRbQd8p+rfkqd4g0dja6R1nOxE+BjVNZrx7o7tkyPTd8+hpZodfvWfS14eVmwRE7eRU702rSqDXSMOl+6vxhXW69y5fs6O8yW5Fp7zKiNm1nYLK+MaYlPg6UojeKkKEyIlLhexOgfp3mMzup0KpTZtDbM5Z0G5WOeUb7EfQ4UEb7o7kNvNPcR04/OCl+HsC6bquGYcDdupPcHXE+Lu1KXVtYhVXupdjXKnJWoaqUw7O5qYf238pAJeb9McHs2qrWre6uUpfC7MIqhK9QneqvyQ5fzU7jwrxS+7NBIDNk7bEB8e0Psiltv4dMQL3RZ7WhKqkFf0FvKvdu6n+UW9TZ64bC7/CESwmWkYGtO+5RW0ljdFTOvucsezv+SvMBEJKz2tmV8H1wpfE9WJcX0JyXPxvcmFea7OLF9MRJCelKWEi5HnWtwvTgV5jtNcX0w7HwPyhJiZJjPMYS9c9s6XpYrqd5eq1JCZahVwJy4GBye/FXnKrZdiIeaOG/k7UtA73Baw25Wi/+1H0aOlRQvK1RaTt81Owt466pdondkMR6OnzNWWshorpYYLnNU9M5GS355Ku6TUW9q5PTm7fNjPGrHgJE1F2oflNdO4d1RZPUNZcFzAZ8OI/Fv8Zc1/tsILwqOKj1imMw5RTWbacGzmW9c+npnnoXoZHmqdCqz2TyiQZ+I3RX74jb87hYRYOynQ9V3KdV/xPq/P9uD2lPyHmYXXe0ctKguod0Pe4rWo3u9x3hyeWNr+VvIq4Xj7Wtbv7t2/cH1jc9u6m8ovyt+IX4FecmW+L34TNyF/h6BTv9Ze3/t47VPdv648+edv+78TTV9Z01jfi5KPzt//y8+lQrA</latexit> ˜ xi = xi + P j eKxi,Qxj V xj P j eKxi,Qxj <latexit 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xi <latexit 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Residual <latexit sha1_base64="kiilw7sVPY4P3AU1/BWI0LZRksc=">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</latexit> Replace convolution by attention:
  42. Conclusion <latexit sha1_base64="jbLn2QJrU9w/PEhZ22KB/uNQWBc=">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</latexit> Strong connexion with mathematical concepts: <latexit sha1_base64="u87ECRVWF3t2o9JvpD2AguJghmk=">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</latexit>

    – Going wider ⇠ function approximation. <latexit sha1_base64="1Ff8hMTJC+VC68D/wkUoX8Fs95g=">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</latexit> – Going deeper ⇠ di↵erential equations. <latexit sha1_base64="3/fFVeWYr5E/FNRPF0lCMHCIEpM=">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</latexit> – Attention ⇠ interacting particles. <latexit 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xi <latexit sha1_base64="Z7Rj29tlYwYzn6VgK9Cg5HgF0Ow=">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</latexit> xj
  43. Conclusion <latexit sha1_base64="jbLn2QJrU9w/PEhZ22KB/uNQWBc=">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</latexit> Strong connexion with mathematical concepts: <latexit sha1_base64="u87ECRVWF3t2o9JvpD2AguJghmk=">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</latexit>

    – Going wider ⇠ function approximation. <latexit sha1_base64="1Ff8hMTJC+VC68D/wkUoX8Fs95g=">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</latexit> – Going deeper ⇠ di↵erential equations. <latexit sha1_base64="3/fFVeWYr5E/FNRPF0lCMHCIEpM=">AAA9pnictVvpchy3EYacy1IuOfmZqtQklFKyS2KRjCpJlYtVlkgdtGiJ0i4p2V5J2WO4Gml2ZzUzSx1rvkOeJn+d18gbJL/yCuluAAPMLmYaYBSiSGKw+LobDaAPYHYwS5Oi3Nj457mPfvDDH/34Jx+fv/DTn/38F7+8+Mmvjopsng/jw2GWZvmTQb+I02QaH5ZJmcZPZnncnwzS+PHg1Q5+/vgkzoskm3bLd7P46aQ/nibHybBfQtPzi59duxbdKMt4io/RpV6RTC5FybSM8/6wTKbjaNbPy2SYxsX684trG+sb9BOtVjZVZU2on4Psk99uiZ4YiUwMxVxMRCymooR6KvqigPKt2BQbYgZtT8UC2nKoJfR5LE7FBcDOoVcMPfrQ+gr+juHpW9U6hWekWRB6CFxS+M0BGYnLgMmgXw515BbR53OijK1NtBdEE2V7B/8HitYEWkvxAlo5nO7pi8OxlOJY/IXGkMCYZtSCoxsqKnPSCkoeWaMqgcIM2rA+gs9zqA8JqfUcEaagsaNu+/T5v6gntuLzUPWdi3+TlJehRKKjRp9VFPrihOhHNJtz+EzKkwLnMVCI1Rix9oZ0PaHRT6H/AtrvQzmlmtbJAMqCWk9bkTtQXMgdFnkHigt5h0XuQ3Eh91nkARQX8kAhEZuTzt34DhQXvsNyfgjFhXzIIh9BcSEfscgjKC7kEYv8BooL+Q2LvA3FhbzNIu9BcSHvscguFBeyyyIPobiQhyzyFhQX8pZCNu/UHEpGdBJmV96Aep0HWooUWm6w8t0k6+jC3vTY08MGLL+rd+G/G7vrodO4AXvLY90dN2D5lXcHbKQby9uiu+RNXNi7LHYPVoAbu8divxQvG7Bfeuy0Vw1Yfq/tQz83lre+X8GTG/sVi70PNTeW91EPoMWNfeDhMWYN2AMW+1C8bsD6WP28Acvb/Q7YFTeW91Nd6O/G+ljTeQOWt6dHEMG4sby3egytbuxjFvtEvG3APmGxX4N1d2O/9vCw7xuw2sdeIA8ypngkhh3bRq1f7UqszYBan+GfVr4lpdh4AO0cZlxhxoSZsIg7FeKOJ2K/Qux7y1VUdrSgeJfn0qkQHU/EoPJNWCvZ/qOqP9ZSD8RuhdhdQrRFpDjXeiwnFF3oFg5ZVp4Laz5jyir7jbVYrYd2y6sRD2oIubZf0Mq/StkSZlCoqTZqLyofL5ERPbch3lD2pkepefC4srIKNuotixo4UAMW9c6Besei5g7UnEWdOFAnLMrsfBvX81gBRv84Fwt6kitAxsjNJYKo4AZ4nbuwRyNYPwcQBT6ilgfwv0O5N1faJMNsHv0knnI8rVniHGoLsQbtJivcpfw6pR0Wg2Sy5wOV4+MTnm0s1J6TVvi08uRRdWLiTychecYVHYwWI9pPYXTuUcspRXeyFoa/W+17XQvD3yKNn1IUL2th+FJJX55B9q7Cds+A7cBumintm3ooDXn+Imno+gXyumhxcVYnas0gvbeB9PfUzOydYV52qCb1Y+phNAprfEVtfCE0jJ4LS89hVDB6klGvrkXBI5mqvNfUQ2XIyItOlRzmKXRmsM9IzYyuh9E4gIhrh3LuhVUPXb2zajSmHkbjSMhzz1OK5HU9jMaYnqU+TD2MBp629FWeb+qhlh01IHNnUw+16lM6BcYzILnmZYuJinKKk+aKWkLxQftpjR3zr/oxPLN5VuUI7ZRMbNtMZ1D5snaJdLwQg1UrA+XA+GJuxWB1GguxxeZXUoay5t9X6Rgfj5rfBy1GsPvlHQB3Zp6ChPpMAq13ChQ32ayrPjKN22JxuEqOl1A91Vqy0aLhK0+N6m3PqZXLy8xojR57ZK8LWnszign3SbOcHvYbZ7iJIqeh/ZqGeHohunuv9mtd+xssbraEmFUrbUg3QvImrT1PdWm9Y+n4srrlKaHIOx+zfvG0+VhZG8x5MrJFKEsbT7ufPkey29CvXhXmjFt+FtGMor06IauR0I1UwWah+rRYRuMLeja0D+lODnlIGkOYx0hRmQl5a4an6HieHpFFte0txxv1pU/oZL0gq6vtcTt6bKHHDnR4jrMDHuM+1LqQMxzCU9cjy7lQ6SojjefiWnU7mtEMtmf0ac1CahrS3sQ1C9mWZb+oUXkDaFwNMkv3p7FMR+N7K5T4rN8lj8ld65b/Mt3c6vvtPq3x5tXcfBIzIq5bxDWiXSNvdeXTMgcpwcL5yRbFr+2jRH4hHNGGclyfWZylXqZ04x9TBjujyDil3cbtjnpv+3xq+RPN6UDou3O8zc7IQkZk/yLwTxmtyYh+7XcH9A26tAgp2Ugfu5NU0Y0r1knYNWbiuETItxrMeovJls2Jv6Zr766C1qLMGKQfOF1a21on+xQLxsQ1V9bd7O1274NI856EvUokRbNWrhD/T+mv/tXrZG1lRaCGcQYKZetc85FRzoI66pOXb7dBuq8t5aVKhmdKauP/jEyXapLtUsaF8qC3HgHnIT1LXrhKcpK7WOkj/WjbaS5Sni3pEUd7TFm8tPtj5YFR7qvkJddoz/VolYxhFZRVFqH7cqfIy3zbedWp+9Eu/i/Uja7rWkOKkTAnuFJD3Pl+TNmaLWUKq1qu31e0m9xaz5d6tfOZ0lqcWHv5O2j9HfzVcutnPzqDmlW4SWtAUjBPRiOyJVrp4cfrZo2XXpmalnk2/Mya1L3slrPk19K6mRz7JJjKAa2at+rUQtfPQuOlReOlpw67dNdotKjbtSV6zuYWXXVb6csvhFs3gPKcpcxHZBqVeEhp51J+VEcsVT7H16j3LK0NllYfdqt9G2DveR+ke68v7+7vKu8eidsU2wwpApP5y4h2aUIxl25tz9QkBeR8XdlXe/f3qAW5D8iCImX5HifuGHnrNKRyWkn6B+XZMrLzxiLo95beqD7axvao/scV5IT2REH7UiOuU49YyW/LES1ZpHUr5ojo5L9PMZWMO9pzZru3mZOoFk+YfFPuKsNLZgpT0j938ra3kr3uWflrRDnhXEXXA6AVPsNIQWL0SYI7sixohtDLyZsEGdEOyH6u2il5ize1JFonqRdi28PGyKzXrHV7bekR67F9Bj1R62bWXT14fqk3R47fWW70+uTVJipGXSw9n41WX3m5+nObHuZLfI0+5tTHzixMllfH9MTn3lykRGFcJMaHS9goQuQPkzxEZnk75UtZ99aU6ycN0sa8oHyJew8UEa7o7oozmvuUGcdghd6AsDY12cJRwtO4TJ0P2JYWT6XOr/gh2Xq+1Rullidq8hSauu0tjP2WFjIm65cK7sxG9rZl79WyFP4URlIYCvlGb1N+aNP8HAr+jYQrO9Qcfc4OOxDf3hA74tYHeBvitarLE82IWtAWjJZy774aZ71Hu45eW9Rt+j4c/HkkoGtO+oQ8aajskjIvuU3dn/4bsgK5iFnpTc/wMdhc+JGscgoZT0KWjR9NIvR3cULHojn4jKTOxZ+PvNfgRnEs9HeawsagqfMjqHMI4aHfY/Cbc9M7nJfNqV1fq1x8eUgvoG9cNA5v/ppzFdPPx0Ll1ox8eA5oHY5bqGtv8b+OQ/MxnMJ5+XIr6LtmLz1mXfaL1YksxsPhe8Zw81nNzRz9eWbV6Ey05OYn474oaKYyazQfnj7Go2YNaF4LIc9Beekk3l5FRl5fKngv4JIhE/8R35/jv43wuqLRJEcIJX1P0UxN9+Cp6W9cukanP/ORydBpkqlOzeQRHXojdkfsidvwu1NFgKFvh8rvUsr/iHV/f3YErcdkPfQpujw56FFbTKcf5hZtRM/qjPH5xbXN5W8hr1aOttY3/7R+/eHW2hc31TeUPxa/Eb+HvGRT/Fl8Ie7CeA9Bpr+Jv4vvxT+2r2zf3z7cfiy7fnROYX4taj/bf/0vfazkbA==</latexit> – Attention ⇠ interacting particles. <latexit 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Very limited theoretical understanding: <latexit 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– Why gradient descent works? <latexit 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– Implicit bias of architectures. <latexit sha1_base64="KBGoT+7TvUf0ErVyzTfXTLNE3i4=">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</latexit> – Implicit bias of optimizers. <latexit 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xi <latexit sha1_base64="Z7Rj29tlYwYzn6VgK9Cg5HgF0Ow=">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</latexit> xj
  44. Conclusion <latexit sha1_base64="jbLn2QJrU9w/PEhZ22KB/uNQWBc=">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</latexit> Strong connexion with mathematical concepts: <latexit sha1_base64="u87ECRVWF3t2o9JvpD2AguJghmk=">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</latexit>

    – Going wider ⇠ function approximation. <latexit sha1_base64="1Ff8hMTJC+VC68D/wkUoX8Fs95g=">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</latexit> – Going deeper ⇠ di↵erential equations. <latexit sha1_base64="3/fFVeWYr5E/FNRPF0lCMHCIEpM=">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</latexit> – Attention ⇠ interacting particles. <latexit 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Very limited theoretical understanding: <latexit 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– Why gradient descent works? <latexit 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– Implicit bias of architectures. <latexit sha1_base64="KBGoT+7TvUf0ErVyzTfXTLNE3i4=">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</latexit> – Implicit bias of optimizers. <latexit 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xi <latexit 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xj <latexit 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– Mean field analysis of transformers (optimal transport?) <latexit 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– Why some optimizers (e.g. Adam) works for transformers? <latexit 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Examples of open problems: