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@@ -112,7 +112,7 @@ $$
 \mathcal { K } _ { z } = \mathbb { E } _ { \theta } \left[ z _ { i } z _ { i } ^ { T } \right] , \quad \Theta _ { z } = \mathbb { E } _ { \theta } \left[ \frac { \partial z _ { i } } { \partial \theta } \left( \frac { \partial z _ { i } } { \partial \theta } \right) ^ { T } \right]
 $$
 
-where $z _ { i } ~ \in ~ \mathbb { R } ^ { d }$ is the $i ^ { \mathrm { { t h } } }$ neuron and $\theta$ are the parameters in the network up until $z$ . Here $d$ is the cardinality of the network inputs $\mathcal { X }$ and $n$ is the number of neurons in the $z$ node. We nonlinear interactions between spatially-distant pixels at shallow depths (left). Values are rep• Pooling enforces translation-invariant Figure 4: An example of the translation of a convolutional neural network into a sequence of on a 2K/4K train/validation subsets of CIFAR10. See §A.7.3 for experimental details.• For other architectures we use a Monte Carlo approach. Sampling finite random networks of a given architecture and predictions in both CNNs and CNN-kernel operations. We demonstrate how the compositional nature of a typical NN computation on its empirically computing the output covariance allows to performance.inputs induces a corresponding compositional computation on the NNGP and NT kernels. Presented 1. Global aver(biased) estimate converges to the true CNN-GP covariance in • Shallow CNN models can perform is a 2-hidden-layer 1D CNN with nonlinearity $\phi$ pooling: take h = 1d1d. Then, performing regression on the 10-dimensional outputs $z ^ { 2 }$ nstantiated networks) and #channels, both in terms of variance Frobenius distance and the GP accuracy.rse than fully-connectedernatives due to failing to capture for each of the 4 (1, 2, 3, 4) inputs $x$ from the dataset $\mathcal { X }$ 2  (attaching a fully-connected layer . To declutter notation, unit weight K 1 ⌘ d2Figure 2: Validation accuracy (left) of an MC-Figure 2: Validation accuracy (left) of annon-linear interactions between and zero bias variances are assumed in all layers. Top: recursive output $( z ^ { 2 } )$ ↵,↵0 +  b . P increases with n M (i.e. channNN-GP increases with n M (i.e. c only to the center pixel of the output) computation in the CNN times number of sa(right) to the exactimes numbe(right) to the(top) induces a respective recursive NNGP kernel $( \bar { \tilde { \mathcal { K } } } ^ { 2 } \otimes I _ { 1 0 } )$ roaches that of the exact CNN-GP (not shown), while the des. The dark band in the left plot corresponds to ill-condd approaches that of the exact CNN-GP (not shown), whilecreases. The dark band in the left plot corresponds to ill- computation (NTK computation being tional layer.8 This approach takes all pixel-pixel covariances into consideration and m               of KL+1 when the number of outer products contributing to KL+1 approximately equals i               of KL+1 when the number of outer products contributing to KL+1 approximately eqFigure 1: Different dimensionality collapsing strategies described in §3. Validation accuraUnder review as a conference paper at ICLR 2019Under review as a conference paper at ICLR 2019similar, not shown). Bottom: explicit listing of tensor and corresponding kernel ops in each layer. the kernel translation invariant. However, it requires O  |X |2d2 memory to computValues reported are for a 3-layer model applied to a 2K/4K train/validation subset of CIValues reported are for a 3-layer model applied to a 2K/4K train/validation subset    §         invariance of the kernel. CNN-GP with zero padding ( 3.1) outperforms an analogous C4. Computing the NN-GP covarianceSee Table 1 for operation definitions. Illustration and description adapted from Figure 3 in Novak et al. (2019).
+where $z _ { i } ~ \in ~ \mathbb { R } ^ { d }$ is the $i ^ { \mathrm { { t h } } }$ neuron and $\theta$ are the parameters in the network up until $z$ . Here $d$ is the cardinality of the network inputs $\mathcal { X }$ and $n$ is the number of neurons in the $z$ node. We nonlinear interactions between spatially-distant pixels at shallow depths (left). Values are rep• Pooling enforces translation-invariant Figure 4: An example of the translation of a convolutional neural network into a sequence of on a 2K/4K train/validation subsets of CIFAR10. See §A.7.3 for experimental details.• For other architectures we use a Monte Carlo approach. Sampling finite random networks of a given architecture and predictions in both CNNs and CNN-kernel operations. We demonstrate how the compositional nature of a typical NN computation on its empirically computing the output covariance allows to performance.inputs induces a corresponding compositional computation on the NNGP and NT kernels. Presented 1. Global aver(biased) estimate converges to the true CNN-GP covariance in • Shallow CNN models can perform is a 2-hidden-layer 1D CNN with nonlinearity $\phi$ pooling: take h = 1d1d. Then, performing regression on the 10-dimensional outputs $z ^ { 2 }$ nstantiated networks) and #channels, both in terms of variance Frobenius distance and the GP accuracy.rse than fully-connectedernatives due to failing to capture for each of the 4 (1, 2, 3, 4) inputs $x$ from the dataset $\mathcal { X }$ 2  (attaching a fully-connected layer . To declutter notation, unit weight K 1 ⌘ d2Figure 2: Validation accuracy (left) of an MC-Figure 2: Validation accuracy (left) of annon-linear interactions between and zero bias variances are assumed in all layers. Top: recursive output $( z ^ { 2 } )$ ↵,↵0 +  b . P increases with n M (i.e. channNN-GP increases with n M (i.e. c only to the center pixel of the output) computation in the CNN times number of sa(right) to the exactimes numbe(right) to the(top) induces a respective recursive NNGP kernel $( \bar { \tilde { \mathcal { K } } } ^ { 2 } \otimes I _ { 1 0 } )$ roaches that of the exact CNN-GP (not shown), while the des. The dark band in the left plot corresponds to ill-condd approaches that of the exact CNN-GP (not shown), whilecreases. The dark band in the left plot corresponds to ill- computation (NTK computation being tional layer.8 This approach takes all pixel-pixel covariances into consideration and m               of KL+1 when the number of outer products contributing to KL+1 approximately equals i               of KL+1 when the number of outer products contributing to KL+1 approximately eqFigure 1: Different dimensionality collapsing strategies described in §3. Validation accuraUnder review as a conference paper at ICLR 2019Under review as a conference paper at ICLR 2019similar, not shown). Bottom: explicit listing of tensor and corresponding kernel ops in each layer. the kernel translation invariant. However, it requires O  |X |2d2 memory to computValues reported are for a 3-layer model applied to a 2K/4K train/validation subset of CIValues reported are for a 3-layer model applied to a 2K/4K train/validation subset    §         invariance of the kernel. CNN-GP with zero padding ( 3.1) outperforms an analogous C4. Computing the NN-GP covarianceSee Table 1 for operation definitions. Illustration and description adapted from Figure 3 in Novak et al. (2019).
 
 ![](images/dc33197d54e9fa73da24e373d18f1fc7c1cc2e91400e0870ddf2b1d21bfcc05f.jpg)  
 ubsampling, zero or no padding(not always necessary to track the whole 4x4x10x10 covariance.)
@@ -125,15 +125,15 @@ $$
 h = \mathrm { D e n s e } ( \sigma _ { \omega } , \sigma _ { b } ) ( y ) \equiv \left( 1 / \sqrt { n } \right) \sigma _ { \omega } W y + \sigma _ { b } \beta ,
 $$
 
-covariance Froand the variables $W _ { i j }$ distanceand $\beta _ { i }$ analogous to §2 (Lee et a     the GP accuracy.are i.i.d. Gaussian $\mathcal { N } ( 0 , 1 )$ tthews et al., 2018b).n!1 n,M  1   n!1 n,M  1  pool 2! X ⇥ L+1 ⇤ 2(#NN instantiations). We will compute kernel operations - denoted $\phi ^ { * }$ and Dense $( \sigma _ { \omega } , \sigma _ { b } ) ^ { * }$ For finite width networks, the uncertainty i⇥For finite width networks, the uncerta1 d2Figure 2: Validation accuracy (left) of an MFigure 2: Validation accuracy (left) o - induced by the tensor operations $\phi$ Kn,M is Var K⇤ 1y in Kn,M is V⇥ l ⇤ 1↵0 1 ↵,↵0 -CNN-GP incren MC-CNN-GPand Dense $( \sigma _ { \omega } , \sigma _ { b } )$ ✓ Kn (✓) /M .l= Var✓ Kn (✓)lM (i.e. channh n ⇥ M (i.e. c5. Finally, 4 MONTE CARLO EVALUATION OF INTRACT       ✓ n  / finite n, Kl is also a biased estimate of Kl , wher       ✓ n finite n, Kl is also a biased estimate of Kl ,This approach corresponds to applying global av(right) to the exact kernel decreases. The dark band i(right) to the exact kernel decreases. The dark bwe will compute the kernel operation associated with the composition $( \mathrm { D e n s e } ( \sigma _ { \omega } , \dot { \sigma _ { b } } ) \circ \phi ) ^ { * } \stackrel { \cdot } { = }$ Dense $( \sigma _ { \omega } , \sigma _ { b } ) ^ { * } \circ \phi ^ { * }$ .
+covariance Froand the variables $W _ { i j }$ distanceand $\beta _ { i }$ analogous to §2 (Lee et a     the GP accuracy.are i.i.d. Gaussian $\mathcal { N } ( 0 , 1 )$ tthews et al., 2018b).n!1 n,M  1   n!1 n,M  1  pool 2! X ⇥ L+1 ⇤ 2(#NN instantiations). We will compute kernel operations - denoted $\phi ^ { * }$ and Dense $( \sigma _ { \omega } , \sigma _ { b } ) ^ { * }$ For finite width networks, the uncertainty i⇥For finite width networks, the uncerta1 d2Figure 2: Validation accuracy (left) of an MFigure 2: Validation accuracy (left) o - induced by the tensor operations $\phi$ Kn,M is Var K⇤ 1y in Kn,M is V⇥ l ⇤ 1↵0 1 ↵,↵0 -CNN-GP incren MC-CNN-GPand Dense $( \sigma _ { \omega } , \sigma _ { b } )$ ✓ Kn (✓) /M .l= Var✓ Kn (✓)lM (i.e. channh n ⇥ M (i.e. c5. Finally, 4 MONTE CARLO EVALUATION OF INTRACT       ✓ n  / finite n, Kl is also a biased estimate of Kl , wher       ✓ n finite n, Kl is also a biased estimate of Kl ,This approach corresponds to applying global av(right) to the exact kernel decreases. The dark band i(right) to the exact kernel decreases. The dark bwe will compute the kernel operation associated with the composition $( \mathrm { D e n s e } ( \sigma _ { \omega } , \dot { \sigma _ { b } } ) \circ \phi ) ^ { * } \stackrel { \cdot } { = }$ Dense $( \sigma _ { \omega } , \sigma _ { b } ) ^ { * } \circ \phi ^ { * }$ .
 
-to traditional randomis nearly constais nearly csample-samdownsampled to downsampFirst we compute the NNGP and NT kernels for $y$ ature methods (Rafor constant Mn. stant for constant e covariance of the⇥ 8. See Figure  d to 8 ⇥ 8. See Fi. To compute $\mathcal { K } _ { y }$ & Recht, 2007), the core idea is to instann,M us treat M n as the effective sample size fWe thus treat M n as the effective sampleor O  d2 per covariance entry in an itera similar results with other architectures a7 for similar results with other architectunote that from its definition,
+to traditional randomis nearly constais nearly csample-samdownsampled to downsampFirst we compute the NNGP and NT kernels for $y$ ature methods (Rafor constant Mn. stant for constant e covariance of the⇥ 8. See Figure  d to 8 ⇥ 8. See Fi. To compute $\mathcal { K } _ { y }$ & Recht, 2007), the core idea is to instann,M us treat M n as the effective sample size fWe thus treat M n as the effective sampleor O  d2 per covariance entry in an itera similar results with other architectures a7 for similar results with other architectunote that from its definition,
 
 $$
 \begin{array} { r } { \mathcal { K } _ { y } = \mathcal { K } _ { \phi ( z ) } = \mathbb { E } _ { \theta } \left[ \phi ( z ) _ { i } \phi ( z ) _ { i } ^ { T } \right] = \mathbb { E } _ { \theta } [ \phi ( z _ { i } ) \phi ( z _ { i } ) ^ { T } ] = \mathcal { T } ( \mathcal { K } _ { z } ) . } \end{array}
 $$
 
-Since $\phi$ O ⇣ |X |2 d2 O ⇣ |Xthe Monte Cthe Mdoes not introduce any new variables $\Theta _ { y }$ ⇣ |X |2 + n2 + nd ⌘ , making th⌘ to O ⇣ |X |2 + n2 + nd ⌘, makKe↵1 ⌘ 2! ⇥GP (MC-GP) kernel,Carlo-GP (MC-GP) kernel,can be computed as,
+Since $\phi$ O ⇣ |X |2 d2 O ⇣ |Xthe Monte Cthe Mdoes not introduce any new variables $\Theta _ { y }$ ⇣ |X |2 + n2 + nd ⌘ , making th⌘ to O ⇣ |X |2 + n2 + nd ⌘, makKe↵1 ⌘ 2! ⇥GP (MC-GP) kernel,Carlo-GP (MC-GP) kernel,can be computed as,
 
 $$
 \begin{array} { r } { \Theta _ { y } = \mathbb { E } _ { \theta } \left[ \frac { \partial \phi ( z _ { i } ) } { \partial \theta } \left( \frac { \partial \phi ( z _ { i } ) } { \partial \theta } \right) ^ { T } \right] = \mathbb { E } _ { \theta } \left[ \mathrm { d i a g } ( \dot { \phi } ( z _ { i } ) ) \frac { \partial z _ { i } } { \partial \theta } \left( \frac { \partial z _ { i } } { \partial \theta } \right) ^ { T } \mathrm { d i a g } ( \dot { \phi } ( z _ { i } ) ) \right] = \dot { \mathcal { T } } ( K _ { z } ) \odot \Theta _ { z } . } \end{array}