Title: BAYESIAN DEEP CONVOLUTIONAL NETWORKS WITH MANY CHANNELS ARE GAUSSIAN PROCESSES

Abstract: There is a previously identified equivalence between wide fully connected neural networks (FCNs) and Gaussian processes (GPs). This equivalence enables, for instance, test set predictions that would have resulted from a fully Bayesian, infinitely wide trained FCN to be computed without ever instantiating the FCN, but by instead evaluating the corresponding GP. In this work, we derive an analogous equivalence for multi-layer convolutional neural networks (CNNs) both with and without pooling layers, and achieve state of the art results on CIFAR10 for GPs without trainable kernels. We also introduce a Monte Carlo method to estimate the GP corresponding to a given neural network architecture, even in cases where the analytic form has too many terms to be computationally feasible. Surprisingly, in the absence of pooling layers, the GPs corresponding to CNNs with and without weight sharing are identical. As a consequence, translation equivariance, beneficial in finite channel CNNs trained with stochastic gradient descent (SGD), is guaranteed to play no role in the Bayesian treatment of the infinite channel limit -a qualitative difference between the two regimes that is not present in the FCN case. We confirm experimentally, that while in some scenarios the performance of SGD-trained finite CNNs approaches that of the corresponding GPs as the channel count increases, with careful tuning SGD-trained CNNs can significantly outperform their corresponding GPs, suggesting advantages from SGD training compared to fully Bayesian parameter estimation.

1. INTRODUCTION:

Neural networks (NNs) demonstrate remarkable performance (He et al., 2016; Oord et al., 2016; Silver et al., 2017; Vaswani et al., 2017) , but are still only poorly understood from a theoretical perspective (Goodfellow et al., 2015; Choromanska et al., 2015; Pascanu et al., 2014; Zhang et al., 2017) . NN performance is often motivated in terms of model architectures, initializations, and training procedures together specifying biases, constraints, or implicit priors over the class of functions learned by a network. This induced structure in learned functions is believed to be well matched to structure inherent in many practical machine learning tasks, and in many real-world datasets. For instance, properties of NNs which are believed to make them well suited to modeling the world include: hierarchy and compositionality (Lin et al., 2017; Poggio et al., 2017) , Markovian dynamics (Tiňo et al., 2004; 2007) , and equivariances in time and space for RNNs (Werbos, 1988) and CNNs (Fukushima & Miyake, 1982; Rumelhart et al., 1985) respectively.

The recent discovery of an equivalence between deep neural networks and GPs (Lee et al., 2018; Matthews et al., 2018b) allow us to express an analytic form for the prior over functions encoded by deep NN architectures and initializations. This transforms an implicit prior over functions into an explicit prior, which can be analytically interrogated and reasoned about. 1 Previous work studying these Neural Network-equivalent Gaussian Processes (NN-GPs) has established the correspondence only for fully connected networks (FCNs). Additionally, previous work has not used analysis of NN-GPs to gain specific insights into the equivalent NNs. SGD-trained finite model, allowing for a significant improvement. (d) Finally, invariance enabled by weight sharing and pooling allows for the best performance. Due to computational limitations, the performance of CNN-GPs with pooling -which also possess the property of invariance -remains an open question. Values are reported for 8-layer ReLU models. See §G.6 for experimental details, Figure 7 , and Table 1 for more model comparisons.

In the present work, we extend the equivalence between NNs and NN-GPs to deep Convolutional Neural Networks (CNNs), both with and without pooling. CNNs are a particularly interesting architecture for study, since they are frequently held forth as a success of motivating NN design based on invariances and equivariances of the physical world (Cohen & Welling, 2016 ) -specifically, designing a NN to respect translation equivariance (Fukushima & Miyake, 1982; Rumelhart et al., 1985) . As we will see in this work, absent pooling, this quality of equivariance has no impact in the Bayesian treatment of the infinite channel (number of convolutional filters) limit (Figure 1 ).

The specific novel contributions of the present work are:

1. We show analytically that CNNs with many channels, trained in a fully Bayesian fashion, correspond to an NN-GP ( §2, §3). We show this for CNNs both with and without pooling, with arbitrary convolutional striding, and with both same and valid padding. We prove convergence as the number of channels in hidden layers approach infinity simultaneously (i.e. min n 1 , . . . , n L → ∞, see §E.4 for details), extending the result of Matthews et al. (2018a) under mild conditions on the nonlinearity derivative. Our results also provide a rigorous proof of the assumption made in Xiao et al. (2018) that pre-activations in hidden layers are i.i.d. Gaussian.

2. We show that in the absence of pooling, the NN-GP for a CNN and a Locally Connected Network (LCN) are identical (Figure 1 , §5.1). An LCN has the same local connectivity pattern as a CNN, but without weight sharing or translation equivariance.

3. We experimentally compare trained CNNs and LCNs and find that under certain conditions both perform similarly to the respective NN-GP (Figure 6 , b, c). Moreover, both architectures tend to perform better with increased channel count, suggesting that similarly to FCNs (Neyshabur et al., 2015; Novak et al., 2018) CNNs benefit from overparameterization (Figure 6 , a, b), corroborating a similar trend observed in Canziani et al. (2016, Figure 2) . However, we also show that careful tuning of hyperparameters allows finite CNNs trained with SGD to outperform their corresponding NN-GPs by a significant margin. We ex-perimentally disentangle and quantify the contributions stemming from local connectivity, equivariance, and invariance in a convolutional model in one such setting (Figure 1 ).

4. We introduce a Monte Carlo method to compute NN-GP kernels for situations (such as CNNs with pooling) where evaluating the NN-GP is otherwise computationally infeasible ( §4).

We stress that we do not evaluate finite width Bayesian networks nor do we make any claims about their performance relative to the infinite width GP studied here or finite width SGD-trained networks. While this is an interesting subject to pursue (see Matthews et al. (2018a) ; Neal (1994)), it is outside of the scope of this paper.

1.1. RELATED WORK:

In early work on neural network priors, Neal (1994) demonstrated that, in a fully connected network with a single hidden layer, certain natural priors over network parameters give rise to a Gaussian process prior over functions when the number of hidden units is taken to be infinite. Follow-up work extended the conditions under which this correspondence applied (Williams, 1997; Le Roux & Bengio, 2007; Hazan & Jaakkola, 2015) . An exactly analogous correspondence for infinite width, finite depth deep fully connected networks was developed recently in Lee et al. (2018) ; Matthews et al. (2018b) , with Matthews et al. (2018a) extending the convergence guarantees from ReLU to any linearly bounded nonlinearities and monotonic width growth rates. In this work we further relax the conditions to absolutely continuous nonlinearities with exponentially bounded derivative and any width growth rates.

The line of work examining signal propagation in random deep networks (Poole et al., 2016; Schoenholz et al., 2017; Yang & Schoenholz, 2017; 2018; Hanin & Rolnick, 2018; Chen et al., 2018; Yang et al., 2018) is related to the construction of the GPs we consider. They apply a mean field approximation in which the pre-activation signal is replaced with a Gaussian, and the derivation of the covariance function with depth is the same as for the kernel function of a corresponding GP. Recently, Xiao et al. (2017b; 2018) extended this to convolutional architectures without pooling. Xiao et al. (2018) also analyzed properties of the convolutional kernel at large depths to construct a phase diagram which will be relevant to NN-GP performance, as discussed in §B.

Compositional kernels coming from wide convolutional and fully connected layers also appeared outside of the GP context. Cho & Saul (2009) derived closed-form compositional kernels for rectified polynomial activations (including ReLU). Daniely et al. (2016) proved approximation guarantees between a network and its corresponding kernel, and show that empirical kernels will converge as the number of channels increases.

There is a line of work considering stacking of GPs, such as deep GPs (Lawrence & Moore, 2007; Damianou & Lawrence, 2013) . These no longer correspond to GPs, though they can describe a rich class of probabilistic models beyond GPs. Alternatively, deep kernel learning (Wilson et al., 2016b; a; Bradshaw et al., 2017) utilizes GPs with base kernels which take in features produced by a deep neural network (often a CNN), and train the resulting model end-to-end. Finally, van der Wilk et al. (2017) incorporates convolutional structure into GP kernels, with follow-up work stacking multiple such GPs (Kumar et al., 2018; Blomqvist et al., 2018) to produce a deep convolutional GP (which is no longer a GP). Our work differs from all of these in that our GP corresponds exactly to a fully Bayesian CNN in the infinite channel limit, when all layers are taken to be of infinite size. We remark that while alternative models, such as deep GPs, do include infinite-sized layers in their construction, they do not treat all layers in this way -for instance, through insertion of bottleneck layers which are kept finite. While it remains to be seen exactly which limit is applicable for understanding realistic CNN architectures in practice, the limit we consider is natural for a large class of CNNs, namely those for which all layers sizes are large and rather comparable in size.

Deep GPs, on the other hand, correspond to a potentially richer class of models, but are difficult to analytically characterize and suffer from higher inference cost. Borovykh (2018) analyzes the convergence of CNN outputs at different spatial locations (or different timepoints for a temporal CNN) to a GP for a single input example. Thus, while they also consider a GP limit (and perform an experiment comparing posterior GP predictions to an SGD-trained CNN), they do not address the dependence of network outputs on multiple input examples, and thus their model is unable to generate predictions on a test set consisting of new input examples.

In concurrent work, Garriga-Alonso et al. ( 2018) derive an NN-GP kernel equivalent to one of the kernels considered in our work. In addition to explicitly specifying kernels corresponding to pooling and vectorizing, we also compare the NN-GP performance to finite width SGD-trained CNNs and analyze the differences between the two models.

2 MANY-CHANNEL BAYESIAN CNNS ARE GAUSSIAN PROCESSES 2.1 PRELIMINARIES General setup. For simplicity of presentation we consider 1D convolutional networks with circularly-padded activations (identically to Xiao et al. (2018) ). Unless specified otherwise, no pooling anywhere in the network is used. If a model (NN or GP) is mentioned explicitly as "with pooling", it always corresponds to a single global average pooling layer at the top. Our analysis is straightforward to extend to higher dimensions, using zero (same) or no (valid) padding, strided convolutions, and pooling in intermediary layers ( §C). We consider a series of L + 1 convolutional layers, l = 0, . . . , L.

Random weights and biases. The parameters of the network are the convolutional filters and biases, ω l ij,β and b l i , respectively, with outgoing (incoming) channel index i (j) and filter relative spatial location β ∈ [±k] ≡ {-k, . . . , 0, . . . , k}. 2 Assume a Gaussian prior on both the filter weights and biases,

EQUATION

The weight and bias variances are σ 2 ω , σ 2 b , respectively. n l is the number of channels (filters) in layer l, 2k + 1 is the filter size, and v β is the fraction of the receptive field variance at location β (with β v β = 1). In experiments we utilize uniform v β = 1/(2k + 1), but nonuniform v β = 1/(2k + 1) should enable kernel properties that are better suited for ultra-deep networks, as in Xiao et al. (2018) .

Inputs, pre-activations, and activations. Let X denote a set of input images (training set, validation set, or both). The network has activations y l (x) and pre-activations z l (x) for each input image

x ∈ X ⊂ R n 0 d , with input channel count n 0 ∈ N, number of pixels d ∈ N, where

y l i,α (x) ≡ x i,α l = 0 φ z l-1 i,α (x) l > 0 , z l i,α (x) ≡ n l j=1 k β=-k ω l ij,β y l j,α+β (x) + b l i . (2)

We emphasize the dependence of y l i,α (x) and z l i,α (x) on the input x. φ : R → R is a nonlinearity (with elementwise application to higher-dimensional inputs). Similarly to Xiao et al. (2018) , y l is assumed to be circularly-padded and the spatial size d hence remains constant throughout the network in the main text (a condition relaxed in §C and Remark E.3). See Figures 2 and 3 for a visual depiction of our notation.

Activation covariance. A recurring quantity in this work will be the empirical uncentered covariance matrix K l of the activations y l , defined as

EQUATION

)

K l is a random variable indexed by two inputs x, x and two spatial locations α, α (the dependence on layer widths n 1 , . . . , n l , as well as weights and biases, is implied and by default not stated explicitly). K 0 , the empirical uncentered covariance of inputs, is deterministic.

Shapes and indexing. Whenever an index is omitted, the variable is assumed to contain all possible entries along the respective dimension. For example, y 0 is a vector of size |X | n 0 d, K l α,α is a matrix of shape |X | × |X |, and z l j is a vector of size |X | d.

EQUATION

)

d 1 = (6 ,6 ) d 2 = (4 ,4 ) ϕ Figure 2: A sample 2D CNN classifier annotated according to notation in §2.1, §3. The network transforms n 0 × d 0 = 3 × 8 × 8-dimensional inputs y 0 (x) = x ∈ X into n3 = 10-dimensional logits z2 (x).

Model has two convolutional layers with k = (3, 3)-shaped filters, nonlinearity φ, and a fully connected layer at the top (y 2 (x) → z2 (x), §3.1). Hidden (pre-)activations have n 1 = n 2 = 12 filters. As min n 1 , n 2 → ∞, the prior of this CNN will approach that of a GP indexed by inputs x and target class indices from 1 to n3 = 10. The covariance of such GP can be computed as

σ 2 ω 6×6 d 2 α=(1,1) (C • A) 2 K 0 α,α + σ 2 b ⊗ I n3

, where the sum is over the {1, . . . , 4} 

d 0 = d 1 = d 2 = d is assumed

in the text, yet our formalism easily extends to the model displayed ( §C). Further, while displayed (pre-)activations have 3D shapes, in the text we treat them as 1D vectors ( §2.1).

Our work concerns proving that the top-layer pre-activations z L converge in distribution to an |X | n L+1 d-variate normal random vector with a particular covariance matrix of shape

|X | n L+1 d × |X | n L+1 d as min n 1 , . . . , n L → ∞.

We emphasize that only the channels in hidden layers are taken to infinity, and n L+1 , the number of channels in the top-layer pre-activations z L , remains fixed. For convergence proofs, we always consider z l , y l , as well as any of their indexed subsets like z l j , y l i,α to be 1D vector random variables, while K l , as well as any of its indexed subsets (when applicable, e.g. K l α,α , K l (x, x )) to be 2D matrix random variables.

2.2. CORRESPONDENCE BETWEEN GAUSSIAN PROCESSES AND BAYESIAN DEEP CNNS WITH INFINITELY MANY CHANNELS:

We next consider the prior over outputs z L computed by a CNN in the limit of infinitely many channels in the hidden (excluding input and output) layers, min n 1 , . . . , n L → ∞, and derive its equivalence to a GP with a compositional kernel. This section outlines an argument showing that z L is normally distributed conditioned on previous layer activation covariance K L , which itself becomes deterministic in the infinite limit. This allows to conclude convergence in distribution of the outputs to a Gaussian with the respective deterministic covariance limit. This section omits many technical details elaborated in §E.4.

2.2.1. A SINGLE CONVOLUTIONAL LAYER IS A GP CONDITIONED ON THE UNCENTERED COVARIANCE MATRIX OF THE PREVIOUS LAYER'S ACTIVATIONS:

As can be seen in Equation 2, the pre-activations z l are a linear transformation of the multivariate Gaussian ω l , b l , specified by the previous layer's activations y l . A linear transformation of a multivariate Gaussian is itself a Gaussian with a covariance matrix that can be derived straightforwardly. Specifically,

EQUATION

where I n l+1 is an n l+1 × n l+1 identity matrix, and A K l is the covariance of the pre-activations z l i and is derived in Xiao et al. (2018) . Precisely, A : PSD |X |d → PSD |X |d is an affine transformation (a cross-correlation operator followed by a shifting operator) on the space of positive semi-definite |X | d × |X | d matrices defined as follows:

EQUATION

A preserves positive semi-definiteness due to Equation 4. Notice that the covariance matrix in Equation 4 is block diagonal due to the fact that separate channels z l i n l+1

i=1 are i.i.d. conditioned on y l , due to i.i.d. weights and biases ω l i , b l i n l+1

i=1 . We further remark that per Equation 4 the normal distribution of z l |y l only depends on K l , hence the random variable z l |K l has the same distribution by the law of total expectation:

z l |K l ∼ N 0, A K l ⊗ I n l+1 . (6) 2.2.2 ACTIVATION COVARIANCE MATRIX BECOMES DETERMINISTIC WITH INCREASING CHANNEL COUNT

It follows from Equation 6that the summands in Equation 3 are i.i.d. conditioned on fixed K l-1 . Subject to weak restrictions on the nonlinearity φ, we can apply the weak law of large numbers and conclude that the covariance matrix K l becomes deterministic in the infinite channel limit in layer l (note that pre-activations z l remain stochastic). Precisely,

EQUATION

)

where C is defined for any

EQUATION

The decoupling of the kernel "propagation" into C and A is highly convenient since A is a simple affine transformation of the kernel (see Equation 5), and C is a well-studied map in literature (see §G.4), and for nonlinearities such as ReLU (Nair & Hinton, 2010) and the error function (erf) C can be computed in closed form as derived in Cho & Saul (2009) and Williams (1997) respectively. We refer the reader to Xiao et al. (2018, Lemma A.1) for complete derivation of the limiting value in Equation 7.

A less obvious result is that, under slightly stronger assumptions on φ, the top-layer activation covariance K L becomes unconditionally (dependence on observed deterministic inputs y 0 is implied) deterministic as channels in all hidden layers grow to infinity simultaneously:

EQUATION

i.e. K L ∞ is (C • A) applied L times to K 0 , the deterministic input covariance. We prove this in §E.4 (Theorem E.5). See Figure 3 for a depiction of the correspondence between neural networks and their infinite width limit covariances K l ∞ .

2.2.3. A CONDITIONALLY NORMAL RANDOM VARIABLE BECOMES NORMAL IF ITS:

COVARIANCE BECOMES DETERMINISTIC §2.2.1 established that z L |K L is Gaussian, and §2.2.2 established that its covariance matrix

A K L ⊗ I n L+1 converges in probability to a deterministic A K L ∞ ⊗ I n l+1 in the infinite channel limit (since K L P → K L ∞ , and A (•) ⊗ I n L+1 : R |X |d×|X |d → R n L+1 |X |d×n L+1

|X |d is continuous). As we establish in §E.4 (Theorem E.6), this is sufficient to conclude with the following result.

Result. If φ : R → R is absolutely continuous and has an exponentially bounded derivative, i.e. ∃ a, b ∈ R : |φ (x)| ≤ a exp (bx) a.e. (almost everywhere), then the following convergence in distribution holds:

EQUATION

)

For more intuition behind Equation 10and an informal proof please consult §E.3.

Under review as a conference paper at ICLR 2019 Figure 1 : Different dimensionality collapsing strategies described in §3. Validation accuracy of an MC-CNN-GP with pooling ( §3.2.1) is consistently better than other models due to translation invariance of the kernel. CNN-GP with zero padding ( §3.1) outperforms an analogous CNN-GP without padding as depth increases. At depth 15 the spatial dimension of the output without padding is reduced to 1 ⇥ 1, making the CNN-GP without padding equivalent to the center pixel selection strategy ( §3.2.2) -which also performs worse than the CNN-GP (we conjecture, due to overfitting to centrally-located features) but approaches the latter (right) in the limit of large depth, as information becomes more uniformly spatially distributed (Xiao et al., 2018) . CNN-GPs generally outperform FCN-GP, presumably due to the local connectivity prior, but can fail to capture nonlinear interactions between spatially-distant pixels at shallow depths (left). Values are reported on a 2K/4K train/validation subsets of CIFAR10. See §A.7.3 for experimental details.

1.. Global average pooling: take:

EQUATION

This approach corresponds to applying global average pooling right after the last convolutional layer. 8 This approach takes all pixel-pixel covariances into consideration and makes the kernel translation invariant. However, it requires O |X | 2 d 2 memory to compute the sample-sample covariance of the GP (or O d 2 per covariance entry in an iterative or distributed setting). It is impractical to use this method to analytically evaluate the GP, and we propose to use a Monte Carlo approach (see §4).

2. Subsampling one particular pixel: take h = e ↵ ,

EQUATION

)

This approach makes use of only one pixel-pixel covariance, and requires the same amount • CNN-GP exhibits similar behavior. In from the phase boundary, the covari all different inputs becomes asympto constant and makes learning infeasib

3.. Different NN-GP architectures:

• Many different NN architectures converge to a GP.

• Some admit a tractable analytic form. E.g. the covariance of a CNN-GP with no pooling can be computed using O(#samples^2 x #pixels) resources.

• For other architectures we use a Monte Carlo approach.

Sampling finite random networks of a given architecture and empirically computing the output covariance allows to approximate the CNN-GP covariance. In our experiments this (biased) estimate converges to the true CNN-GP covariance in #(instantiated networks) and #channels, both in terms of covariance Frobenius distance and the GP accuracy.

(a) SGD-trained finite (b) Performance of SG CNN-GP with increasi (c) However, the best significant margin.

• Various CNN architectural decisions like pooling / vectorizing / subsampling, zero or no padding have a corresponding NN-GP.

• Pooling enforces translation-invariant predictions in both CNNs and CNN-GPs and allows for the best performance.

• Shallow CNN models can perform worse than fully-connected alternatives due to failing to capture non-linear interactions between distant pixels. NN 

7.. Disentangl:

n 1 → ∞ n 1 → ∞ n 2 → ∞ n 2 → ∞ 4x4 covariance per class y 0 z 0 z 1 y 2 = ϕ ( z 1 ) z2 4 3 2 1 10 classes per input 4 inputs, 3 features 4 3 2 1 4 3 2 1 ⊗ I10 K 0 K 0 K 1 K 2 K 1 K 2 σ 2 ω K 0 + σ 2 b σ 2 ω K 1 + σ 2 b ( K 1 ) ( K 0 )

K 0 K 1 K 2 ( K 1 ) ( K 0 )

affine transform affine transform affine transform convolution convolution matrix-multiply nonlinearity nonlinearity 

y 2 = ϕ ( z 1 ) y 1 = ϕ ( z 0 ) n 1 → ∞ n 1 → ∞ n 2 → ∞ n 2 → ∞ C (K) = E x⇠N (0,K) h (x) (x) T i < l a t e x i t s h a 1 _ b a s e 6 4 = " L K S S 7 i N m 7 W J o + g 4 I + B j N B l P b a w o = " > A A A C e H i c b V F d S 8 M w F E 3 r 9 / y a H 2 + + B K e o I N L u R V 8 E U Q R B E A W n w l p H k q V b M G l L c i u O M v + n z / o r f D J b q + j m h c D h n H O 5 N + f S V A o D n v f m u B O T U 9 M z s 3 O V + Y X F p e X q y u q d S T L N e I M l M t E P l B g u R c w b I E D y h 1 R z o q j k 9 / T p b K D f P 3 N t R B L f Q i / l o S K d W E S C E b B U q / q 6 F S g C X U Z k f t Y P J I 9 g 9 z L Q o t O F P X y M h x q l + X m / l b 8 E R q g f 8 1 V p 9 v a / 7 Q X R D N K u K K S X U h h n H m 8 L E G 6 1 q j X v w B s W H g d + C W q o r O t W 9 T 1 o J y x T P A Y m i T F N 3 0 s h z I k G w S T v V 4 L M 8 J S w J 9 L h T Q t j o r g J 8 2 F O f b x t m T a O E m 1 f D H j I / u 7 I i T K m p 6 h 1 D v 5 p R r U B + a 9 G q R o Z D d F R m I s 4 z Y D H r J g c Z R J D g g d X w G 2 h O Q P Z s 4 A w L e z y m H W J J g z s r S o 2 F X 8 0 g 3 F w V z / w L b 6 p 1 0 5 O y 3 x m 0 Q b a R L v I R 4 f o B F 2 g a 9 R A D H 0 4 F W f N W X c + X e z u u H u F 1 X X K n j X 0 p 9 z 6 F 6 y i w S Y = < / l a t e x i t > < l a t e x i t s h a 1 _ b a s e 6 4 = " L K S S 7 i N m 7 W J o + g 4 I + B j N B l P b a w o = " > A A A C e H i c b V F d S 8 M w F E 3 r 9 / y a H 2 + + B K e o I N L u R V 8 E U Q R B E A W n w l p H k q V b M G l L c i u O M v + n z / o r f D J b q + j m h c D h n H O 5 N + f S V A o D n v f m u B O T U 9 M z s 3 O V + Y X F p e X q y u q d S T L N e I M l M t E P l B g u R c w b I E D y h 1 R z o q j k 9 / T p b K D f P 3 N t R B L f Q i / l o S K d W E S C E b B U q / q 6 F S g C X U Z k f t Y P J I 9 g 9 z L Q o t O F P X y M h x q l + X m / l b 8 E R q g f 8 1 V p 9 v a / 7 Q X R D N K u K K S X U h h n H m 8 L E G 6 1 q j X v w B s W H g d + C W q o r O t W 9 T 1 o J y x T P A Y m i T F N 3 0 s h z I k G w S T v V 4 L M 8 J S w J 9 L h T Q t j o r g J 8 2 F O f b x t m T a O E m 1 f D H j I / u 7 I i T K m p 6 h 1 D v 5 p R r U B + a 9 G q R o Z D d F R m I s 4 z Y D H r J g c Z R J D g g d X w G 2 h O Q P Z s 4 A w L e z y m H W J J g z s r S o 2 F X 8 0 g 3 F w V z / w L b 6 p 1 0 5 O y 3 x m 0 Q b a R L v I R 4 f o B F 2 g a 9 R A D H 0 4 F W f N W X c + X e z u u H u F 1 X X K n j X 0 p 9 z 6 F 6 y i w S Y = < / l a t e x i t > < l a t e x i t s h a 1 _ b a s e 6 4 = " L K S S 7 i N m 7 W J o + g 4 I + B j N B l P b a w o = " > A A A C e H i c b V F d S 8 M w F E 3 r 9 / y a H 2 + + B K e o I N L u R V 8 E U Q R B E A W n w l p H k q V b M G l L c i u O M v + n z / o r f D J b q + j m h c D h n H O 5 N + f S V A o D n v f m u B O T U 9 M z s 3 O V + Y X F p e X q y u q d S T L N e I M l M t E P l B g u R c w b I E D y h 1 R z o q j k 9 / T p b K D f P 3 N t R B L f Q i / l o S K d W E S C E b B U q / q 6 F S g C X U Z k f t Y P J I 9 g 9 z L Q o t O F P X y M h x q l + X m / l b 8 E R q g f 8 1 V p 9 v a / 7 Q X R D N K u K K S X U h h n H m 8 L E G 6 1 q j X v w B s W H g d + C W q o r O t W 9 T 1 o J y x T P A Y m i T F N 3 0 s h z I k G w S T v V 4 L M 8 J S w J 9 L h T Q t j o r g J 8 2 F O f b x t m T a O E m 1 f D H j I / u 7 I i T K m p 6 h 1 D v 5 p R r U B + a 9 G q R o Z D d F R m I s 4 z Y D H r J g c Z R J D g g d X w G 2 h O Q P Z s 4 A w L e z y m H W J J g z s r S o 2 F X 8 0 g 3 F w V z / w L b 6 p 1 0 5 O y 3 x m 0 Q b a R L v I R 4 f o B F 2 g a 9 R A D H 0 4 F W f N W X c + X e z u u H u F 1 X X K n j X 0 p 9 z 6 F 6 y i w S Y = < / l a t e x i t > < l a t e x i t s h a 1 _ b a s e 6 4 = " L K S S 7 i N m 7 W J o + g 4 I + B j N B l P b a w o = " > A A A C e H i c b V F d S 8 M w F E 3 r 9 / y a H 2 + + B K e o I N L u R V 8 E U Q R B E A W n w l p H k q V b M G l L c i u O M v + n z / o r f D J b q + j m h c D h n H O 5 N + f S V A o D n v f m u B O T U 9 M z s 3 O V + Y X F p e X q y u q d S T L N e I M l M t E P l B g u R c w b I E D y h 1 R z o q j k 9 / T p b K D f P 3 N t R B L f Q i / l o S K d W E S C E b B U q / q 6 F S g C X U Z k f t Y P J I 9 g 9 z L Q o t O F P X y M h x q l + X m / l b 8 E R q g f 8 1 V p 9 v a / 7 Q X R D N K u K K S X U h h n H m 8 L E G 6 1 q j X v w B s W H g d + C W q o r O t W 9 T 1 o J y x T P A Y m i T F N 3 0 s h z I k G w S T v V 4 L M 8 J S w J 9 L h T Q t j o r g J 8 2 F O f b x t m T a O E m 1 f D H j I / u 7 I i T K m p 6 h 1 D v 5 p R r U B + a 9 G q R o Z D d F R m I s 4 z Y D H r J g c Z R J D g g d X w G 2 h O Q P Z s 4 A w L e z y m H W J J g z s r S o 2 F X 8 0 g 3 F w V z / w L b 6 p 1 0 5 O y 3 x m 0 Q b a R L v I R 4 f o B F 2 g a 9 R A D H 0 4 F W f N W X c + X e z u u H u F 1 X X K n j X

EQUATION

Under review as a conference paper at ICLR 2019 MC-CNN-GP Figure 2 : Validation accuracy (left) of an MC-CNN-GP increases with n ⇥ M (i.e. channel count times number of samples) and approaches that of the exact CNN-GP (not shown), while the distance (right) to the exact kernel decreases. The dark band in the left plot corresponds to ill-conditioning of K L+1 n,M when the number of outer products contributing to K L+1 n,M approximately equals its rank. Values reported are for a 3-layer model applied to a 2K/4K train/validation subset of CIFAR10 downsampled to 8 ⇥ 8. See Figure 7 for similar results with other architectures and §A.7.2 for experimental details.

random finite width networks and use the empirical uncentered covariances of activations to estimate the Monte Carlo-GP (MC-GP) kernel,

EQUATION

(#NN instantiations)

• Outputs of a CNN converge in distribution to a Gaussian process with a compositional kernel (under mild conditions on the weights and biases distribution and nonlinearity).

(#channels in hidden layers)

(convergence in distribution)

(an affine transformation) (inputs covariance)

(computes covariance of post-nonlinearity activations given normal pre-activations)

outputs inputs

D ! min{n 1 ,...,n L }!1 N ⇣ 0, A (C A) L K 0 ⌦ I n L+1 ⌘ < l a t e x i t s h a 1 _ b a s e 6 4 = " r S e f Q 0 I B g T G o B S F h X r b E 0 x A 5 J b k = " > A A A C 4 n i c b V J N b 9 Q w E H X C V 1 k + u s C R i 8 U W q Y h V l f Q C x 0 I 5 g K i q I r F t p X V 2 5 X i d X a u O H d k T 6 C q Y H 8 A

N c e W P c e b C z 8 D 5 2 K p s G S n y y 5 s 3 e u M Z p 4 U U F q L o V x B e u 3 7 j 5 q 2 N 2 7 0 7 d

+ / d 3 + w / e H h s d W k Y H z E t t T l N q e V S K D 4 C A Z K f F o b T P J X 8 J D 3 b r / M n n 7 i x Q q u P s C x 4 k t O 5 E p l g F D w 1 7 f / Z I p J n s E 2 A n 4 P J K 1 1 C U Y L F J B X z L 1 i o + s c R I + Y L e E b O m 5 M a o z + P S S 5 U U 0 o q N Y m H Z K b B D t X k o N U S R 0 A T o T J Y J h X J K S w Y l d U b 5 y 7 w o W u N o y G + 4 F 4 5 w o R h X U c r d r 9 j L 6 n a f r x Z o 3 w / i V Y d a h A 5 t / j d 1 D d V H T y P 3 U q 7 N e 0 P o p 2 o C X w V x B 0 Y o C 6 O p v 3 f / k 6 s z L k C J q m 1 4 z g q I K m o A c E k d z 1 S W l 5 Q d k b n f O y h o t 4 4 q Z q N O P z U M z O c a e M / B b h h L 1 d U N L d 2 m a d e W V / L r u d q 8 r + 5 N M 3 X r C F 7 m V T N o r h i r X N W S g w a 1 / v G M 2 E 4 A 7 n 0 g D I j f P O Y L a i h D P y r 6 P m p x O s z u A q O d 3 d i j z / s D v Z e d / P Z Q I / R E 7 S N Y v Q C 7 a G 3 6 A i N E A s O A w h c 8 D W c h d / C 7 + G P V h o G X

c 0 j 9 E + E P / 8 C 5 J r t n g = = < / l a t e x i t > < l a t e x i t s h a 1 _ b a s e 6 4 = " r S e f Q 0

I B g T G o B S F h X r b E 0 x A 5 J b k = " > A A A C 4 n i c b V J N b 9 Q w E H X C V 1 k + u s C R i 8 U W q Y h V l f Q C x 0 I 5 g K i q I r F t p X V 2 5 X i d X a u O H d k T 6 C q Y H 8 A

N c e W P c e b C z 8 D 5 2 K p s G S n y y 5 s 3 e u M Z p 4 U U F q L o V x B e u 3 7 j 5 q 2 N 2 7 0 7 d

+ / d 3 + w / e H h s d W k Y H z E t t T l N q e V S K D 4 C A Z K f F o b T P J X 8 J D 3 b r / M n n 7 i x Q q u P s C x 4 k t O 5 E p l g F D w 1 7 f / Z I p J n s E 2 A n 4 P J K 1 1 C U Y L F J B X z L 1 i o + s c R I + Y L e E b O m 5 M a o z + P S S 5 U U 0 o q N Y m H Z K b B D t X k o N U S R 0 A T o T J Y J h X J K S w Y l d U b 5 y 7 w o W u N o y G + 4 F 4 5 w o R h X U c r d r 9 j L 6 n a f r x Z o 3 w / i V Y d a h A 5 t / j d 1 D d V H T y P 3 U q 7 N e 0 P o p 2 o C X w V x B 0 Y o C 6 O p v 3 f / k 6 s z L k C J q m 1 4 z g q I K m o A c E k d z 1 S W l 5 Q d k b n f O y h o t 4 4 q Z q N O P z U M z O c a e M / B b h h L 1 d U N L d 2 m a d e W V / L r u d q 8 r + 5 N M 3 X r C F 7 m V T N o r h i r X N W S g w a 1 / v G M 2 E 4 A 7 n 0 g D I j f P O Y L a i h D P y r 6 P m p x O s z u A q O d 3 d i j z / s D v Z e d / P Z Q I / R E 7 S N Y v Q C 7 a G 3 6 A i N E A s O A w h c 8 D W c h d / C 7 + G P V h o G X

c 0 j 9 E + E P / 8 C 5 J r t n g = = < / l a t e x i t > < l a t e x i t s h a 1 _ b a s e 6 4 = " r S e f Q 0

I B g T G o B S F h X r b E 0 x A 5 J b k = " > A A A C 4 n i c b V J N b 9 Q w E H X C V 1 k + u s C R i 8 U W q Y h V l f Q C x 0 I 5 g K i q I r F t p X V 2 5 X i d X a u O H d k T 6 C q Y H 8 A

N c e W P c e b C z 8 D 5 2 K p s G S n y y 5 s 3 e u M Z p 4 U U F q L o V x B e u 3 7 j 5 q 2 N 2 7 0 7 d

+ / d 3 + w / e H h s d W k Y H z E t t T l N q e V S K D 4 C A Z K f F o b T P J X 8 J D 3 b r / M n n 7 i x Q q u P s C x 4 k t O 5 E p l g F D w 1 7 f / Z I p J n s E 2 A n 4 P J K 1 1 C U Y L F J B X z L 1 i o + s c R I + Y L e E b O m 5 M a o z + P S S 5 U U 0 o q N Y m H Z K b B D t X k o N U S R 0 A T o T J Y J h X J K S w Y l d U b 5 y 7 w o W u N o y G + 4 F 4 5 w o R h X U c r d r 9 j L 6 n a f r x Z o 3 w / i V Y d a h A 5 t / j d 1 D d V H T y P 3 U q 7 N e 0 P o p 2 o C X w V x B 0 Y o C 6 O p v 3 f / k 6 s z L k C J q m 1 4 z g q I K m o A c E k d z 1 S W l 5 Q d k b n f O y h o t 4 4 q Z q N O P z U M z O c a e M / B b h h L 1 d U N L d 2 m a d e W V / L r u d q 8 r + 5 N M 3 X r C F 7 m V T N o r h i r X N W S g w a 1 / v G M 2 E 4 A 7 n 0 g D I j f P O Y L a i h D P y r 6 P m p x O s z u A q O d 3 d i j z / s D v Z e d / P Z Q I / R E 7 S N Y v Q C 7 a G 3 6 A i N E A s O A w h c 8 D W c h d / C 7 + G P V h o G X

c 0 j 9 E + E P / 8 C 5 J r t n g = = < / l a t e x i t > < l a t e x i t s h a 1 _ b a s e 6 4 = " r S e f Q 0

I B g T G o B S F h X r b E 0 x A 5 J b k = " > A A A C 4 n i c b V J N b 9 Q w E H X C V 1 k + u s C R i 8 U W q Y h V l f Q C x 0 I 5 g K i q I r F t p X V 2 5 X i d X a u O H d k T 6 C q Y H 8 A

N c e W P c e b C z 8 D 5 2 K p s G S n y y 5 s 3 e u M Z p 4 U U F q L o V x B e u 3 7 j 5 q 2 N 2 7 0 7 d

+ / d 3 + w / e H h s d W k Y H z E t t T l N q e V S K D 4 C A Z K f F o b T P J X 8 J D 3 b r / M n n 7 i x Q q u P s C x 4 k t O 5 E p l g F D w 1 7 f / Z I p J n s E 2 A n 4 P J K 1 1 C U Y L F J B X z L 1 i o + s c R I + Y L e E b O m 5 M a o z + P S S 5 U U 0 o q N Y m H Z K b B D t X k o N U S R 0 A T o T J Y J h X J K S w Y l d U b 5 y 7 w o W u N o y G + 4 F 4 5 w o R h X U c r d r 9 j L 6 n a f r x Z o 3 w / i V Y d a h A 5 t / j d 1 D d V H T y P 3 U q 7 N e 0 P o p 2 o C X w V x B 0 Y o C 6 O p v 3 f / k 6 s z L k C J q m 1 4 z g q I K m o A c E k d z 1 S W l 5 Q d k b n f O y h o t 4 4 q Z q N O P z U M z O c a e M / B b h h L 1 d U N L d 2 m a d e W V / L r u d q 8 r + 5 N M 3 X r C F 7 m V T N o r h i r X N W S g w a 1 / v G M 2 E 4 A 7 n 0 g D I j f P O Y L a i h D P y r 6 P m p x O s z u A q O d 3 d i j z / s D v Z e d / P Z Q I / R E 7 S N Y v Q C 7 a G 3 6 A i N E A s O A w h c 8 D W c h d / C 7 + G P V h o G X

c 0 j 9 E + E P / 8 C 5 J r t n g = = < / l a t e x i t > (#channels in the outputs)

(not always necessary to track the whole 4x4x10x10 covariance.)

(attaching a fully-connected layer only to the center pixel of the output)

n 1 → ∞ n 1 → ∞ n 2 → ∞ n 2 → ∞ FCN activations FCN-GP covariance K 0 < l a t e x i t s h a _ b a s e = " d j K o C a j Z g k r w u K p B m p = " > A A A B + i c b V C S g N B F L b X F R S t B o N g F W a E Y M g g E c D k j X M T m a T I T O y y s E J Z g q + Q D q x R v B L H R w k F p p M L h n H s l x M k g m u D a e T W h c W l J r p r x u b W X t n Z q O U V Z l c Y i V o A a C Z x K q G G E a i W J E B o L V g / l K / M K V H N Z Q c J S b o R D z k l x k q / e X S j i E p A z R P v h x T P z Z P T h V t q F r Y n p q l k k a G C a N c G L j C j D q W B D t V q l h D a J W t D Q i k m k / m w R A d W a A w V n Y i g y b q u M S K H M r C b k p i e n v X G r e E M i Z a B O e + h m P k t S w i E T w Q g E N x E a j D F a N G D C w h V H H P K I o g g t i X t u L N d j B P a k c l i E b C x f A F T G E P u E Q P D i B M l x B B a p A o Q u P A T P z t A Z O S / O Q / z c M I f O G / f T W X v g = = < / l a t e x i t > C < l a t e x i t s h a 1 _ b a s e 6 4 = " B p 3 s 0 N X a z I 8 d Q f K 7 i 5 e d A + N 1 h F s = " > A A A C A 3 i c b V C 7 S g N B F L 0 b X 3 F 9 R S 1 t B o N g F X a 1 0 E Y M p r G M Y B 6 Q L G F 2 M k m G z M w u M 7 N C W F L 6 C b Z a W N q J r b W f I N b + i L N J C k 0 8 c O F w z r 3 c w w l j z r T x v C 8 n t 7 S 8 s r q W X 3 c 3 N r e 2 d w q 7 e 3 U d J Y r Q G o l 4 p J o h 1 p Q z S W u G G U 6 b s a J Y h J w 2 w m E l 8 x t 3 V G k W y V s z i m k g c F + y H i P Y W K n V F t g M C O Z p Z d w p F L 2 S N w F a J P 6 M F C 8 / 3 I v 4 + d O t d g r f 7 W 5 E E k G l I R x r 3 f K 9 2 A Q p V o Y R T s d u O 9 E 0 x m S I + 7 R l q c S C 6 i C d R B 6 j I 6 t 0 U S 9 S d q R B E / X 3 R Y q F 1 i M R 2 s 0 s o p 7 3 M v F f L w z F 3 G v T O w 9 S J u P E U E m m n 3 s J R y Z C W S G o y x Q l h o 8 s w U Q x G x 6 R A V a Y G F u b a 1 v x 5 z t Y J P W T k n 9 a 8 m 6 8 Y v k K p s j D A R z C M f h w B m W 4 h i r U g E A E D / A I T 8 6 9 8 + K 8 O m / T 1 Z w z u 9 m H P 3 D e f w A 8 Y 5 u m < / l a t e x i t > K 1 1 < l a t e x i t s h a 1 _ b a s e 6 4 = " X v v O k S j q l J 9 2 P o c D U d m z M o T Y 5 m 8 = " > A A A C B H i c b V C 7 S g N B F L 0 b X z G + o p Y 2 Q 0 R I F X a 1 i G X Q R r C J Y B 6 Q X c P s Z D Y Z M j u 7 z M w K y 5 L W T 7 C x 0 A + w E 1 t 7 P 0 G 0 8 0 e c P A p N P H D h c M 6 9 n M v x Y 8 6 U t u 0 P K 7 e 0 v L K 6 l l 8 v b G x u b e 8 U d / e a K k o k o Q 0 S 8 U i 2 f a w o Z 4 I 2 N N O c t m N J c e h z 2 v K H 5 2 O / d U u l Y p G 4 1 m l M v R D 3 B Q s Y w d p I 7 u W N 0 8 1 c J g K d j r r F Q 7 t i T 4 A W i T M j h 7 V S + e u z + v 5 Q 7 x a / 3 V 5 E k p A K T T h W q u P Y s f Y y L D U j n I 4 K b q J o j M k Q 9 2 n H U I F D q r x s 8 v M I H R m l h 4 J I m h E a T d T f F x k O l U p D 3 2 y G W A / U v D c W / / V 8 P 5 y L 1 s G p l z E R J 5 o K M k 0 O E o 5 0 h M a N o B 6 T l G i e G o K J Z O Z 5 R A Z Y Y q J N b w X T i j P f w S J p H l e c k 4 p 9 Z e o 5 g y n y c A A l K I M D V a j B B d S h A Q R i u I d H e L L u r G f r x X q d r u a s 2 c 0 + / I H 1 9 g O q A 5 y R < / l a t e x i t > K 2 1 < l a t e x i t s h a 1 _ b a s e 6 4 = " O U e U K e + l I o b d s 6 D 6 x E X 6 i k d e j z A = " > A A A C B H i c b V C 7 S g N B F J 2 N r x h f U U u b I U F I F X Z j E c u g j W A T w T w g G 8 P s Z D Y Z M j u 7 z N w V l i W t n 2 B j o R 9 g J 7 b 2 f o J o 5 4 8 4 e R S a e O D C 4 Z x 7 O Z f j R Y J r s O 0 P K 7 O y u r a + k d 3 M b W 3 v 7 O 7 l 9 w + a O o w V Z Q 0 a i l C 1 P a K Z 4 J I 1 g I N g 7 U g x E n i C t b z R + c R v 3 T K l e S i v I Y l Y N y A D y X 1 O C R j J v b y p 9 F K X S x + S c S 9 f t M v 2 F H i Z O H N S r B V K X 5 / V 9 4 d 6 L / / t 9 k M a B 0 w C F U T r j m N H 0 E 2 J A k 4 F G + f c W L O I 0 B E Z s I 6 h k g R M d 9 P p z 2 N 8 b J Q + 9 k N l R g K e q r 8 v U h J o n Q S e 2 Q w I D P W i N x H / 9 T w v W I g G / 7 S b c h n F w C S d J f u x w B D i S S O 4 z x W j I B J D C F X c P I / p k C h C w f S W M 6 0 4 i x 0 s k 2 a l 7 J y U 7 S t T z x m a I Y u O U A G V k I O q q I Y u U B 0 1 E E U R

u k e P 6 M m 6 s 5 6 t F + t 1 t p q x 5 j e H 6 A + s t x + r o Z y S < / l a t e x i t > ) is consistently better than other models due to translation invariance of the kernel. CNN-GP with zero padding ( §3.1) outperforms an analogous CNN-GP without padding as depth increases. At depth 15 the spatial dimension of the output without padding is reduced to 1 ⇥ 1, making the CNN-GP without padding equivalent to the center pixel selection strategy ( §3.2.2) -which also performs worse than the CNN-GP (we conjecture, due to overfitting to centrally-located features) but approaches the latter (right) in the limit of large depth, as information becomes more uniformly spatially distributed (Xiao et al., 2018) . CNN-GPs generally outperform FCN-GP, presumably due to the local connectivity prior, but can fail to capture nonlinear interactions between spatially-distant pixels at shallow depths (left). Values are reported on a 2K/4K train/validation subsets of CIFAR10. See §A.7.3 for experimental details.

K vec 1 < l a t e x i t s h a 1 _ b a s e 6 4 = " 6 K 3 m C N D g h m e g 3 o p A x p s E W l s n F s k = " > A A A C H X i c b V C 7 S g N B F J 2 N r x h f U U u b R R G s w q 4 W s Q z a C D Y R T B T c G G Y n d 3 V w Z n a Z u R s M y 3 6 A 3 2 D h J 9 j q B 1 g I Y i u i n T / i J L H Q x A M D h 3 P O 5 d 4 5 Y S K 4 Q c 9 7 d w o T k 1 P T M 8 X Z 0 t z 8 w u J S e X m l a e J U M 2 i w W M T 6 N K Q G B F f Q Q I 4 C T h M N V I Y C T s K r / b 5 / 0 g V t e K y O s Z d A S 9 I L x S P O K F q p X d 4 I J M V L R k V 2 m L e z g K s I e / l 5 F i B c o 5 Z Z F 1 i e 2 5 R X 8 Q Z w x 4 n / Q z Z q 6 1 u f H 9 X n 2 3 q 7 / B V 0 Y p Z K U M g E N e b M 9 x J s Z V Q j Z w L y U p A a S C i 7 o h d w Z q m i E k w r G 3 w m d z e t 0 n G j W N u n 0 B 2 o v y c y K o 3 p y d A m + 6 e b U a 8 v / u u F o R x Z j d F u K + M q S R E U G 2 6 O U u F i 7 P a r c j t c A 0 P R s 4 Q y z e 3 x L r u k m j K 0 h Z Z s K / 5 o B + O k u V 3 x d y r e k a 1 n j w x R J G t k n W w R n 1 R J j R y Q O m k Q R m 7 I P X k g j 8 6 d 8 + S 8 O K / D a M H 5 m V k l f + C 8 f Q N y + K f T < / l a t e x i t > C < l a t e x i t s h a 1 _ b a s e 6 4 = " B p 3 s 0 N X a z I 8 d Q f K 7 i 5 e d A + N 1 h F s = " > A A A C A 3 i c b V C 7 S g N B F L 0 b X 3 F 9 R S 1 t B o N g F X a 1 0 E Y M p r G M Y B 6 Q L G F 2 M k m G z M w u M 7 N C W F L 6 C b Z a W N q J r b W f I N b + i L N J C k 0 8 c O F w z r 3 c w w l j z r T x v C 8 n t 7 S 8 s r q W X 3 c 3 N r e 2 d w q 7 e 3 U d J Y r Q G o l 4 p J o h 1 p Q z S W u G G U 6 b s a J Y h J w 2 w m E l 8 x t 3 V G k W y V s z i m k g c F + y H i P Y W K n V F t g M C O Z p Z d w p F L 2 S N w F a J P 6 M F C 8 / 3 I v 4 + d O t d g r f 7 W 5 E E k G l I R x r 3 f K 9 2 A Q p V o Y R T s d u O 9 E 0 x m S I + 7 R l q c S C 6 i C d R B 6 j I 6 t 0 U S 9 S d q R B E / X 3 R Y q F 1 i M R 2 s 0 s o p 7 3 M v F f L w z F 3 G v T O w 9 S J u P E U E m m n 3 s J R y Z C W S G o y x Q l h o 8 s w U Q x G x 6 R A V a Y G F u b a 1 v x 5 z t Y J P W T k n 9 a 8 m 6 8 Y v k K p s j D A R z C M f h w B m W 4 h i r U g E A E D / A I T 8 6 9 8 + K 8 O m / T 1 Z w z u 9 m H P 3 D e f w A 8 Y 5 u m < / l a t e x i t > 2 ! K 0 + 2 b < l a t e x i t s h a 1 _ b a s e 6 4 = " G M B / 3 b m l A t k y M c o 7 q w g 6 6 s 6 G n K I = " > A A A C H H i c b V D L S s N A F J 3 U V 6 2 v q E s 3 Q 4 t Q E E p S F 3 V Z d C O 4 q W A f 0 K R l M p 2 m Q 2 e S M D M R S s j e f x D 8 B L f 6 A W 5 E 3 A q i O 3 / E 6 Q P U 1 g M X D u f c y 7 3 3 e B G j U l n W u 5 F Z W l 5 Z X c u u 5 z Y 2 t 7 Z 3 z N 2 9 h g x j g U k d h y w U L Q 9 J w m h A 6 o o q R l q R I I h 7 j D S 9 4 d n Y b 1 4 T I W k Y X K l R R F y O / I D 2 K U Z K S 1 0 z 7 0 j q c 9 Q p d x M n 5 M R H K b z o W E c / q p d 2 z Y J V s i a A i 8 S e k U I 1 X / z 8 q D z f 1 r r m l 9 M L c c x J o D B D U r Z t K 1 J u g o S i m J E 0 5 8 S S R A g P k U / a m g a I E + k m k 1 9 S e K i V H u y H Q l e g 4 E T 9 P Z E g L u W I e 7 q T I z W Q 8 9 5 Y / N f z P D 6 3 W v V P 3 I Q G U a x I g K e b + z G D K o T j p G C P C o I V G 2 m C s K D 6 e I g H S C C s d J 4 5 n Y o 9 n 8 E i a Z R L 9 n H J u t T x n I I p s u A A 5 E E R 2 K A C q u A c 1 E A d Y H A D 7 s E D e D T u j C f j x X i d t m a M 2 c w + + A P j 7 R u A R q V 6 < / l a t e x i t > 2 ! K 1 1 + 2 b < l a t e x i t s h a 1 _ b a s e 6 4 = " F Q 8 u o E k 4 e C S 4 T j i q X D n M F w j N C X g = " > A A A C J X i c b V D L S g M x F M 3 U V 6 2 v U Z d u Q o t Q E M p M X d R l 0 Y 3 g p o J 9 Q F 9 k 0 k w b m m S G J C O U Y T 7 C t R / g J 7 j V p Q t 3 I o i C C 3 / E 9 L G o r Q c u n J x z L z f 3 e C G j S j v O h 5 V a W V 1 b 3 0 h v Z r a 2 d 3 b 3 7 P 2 D m g o i i U k V B y y Q D Q 8 p w q g g V U 0 1 I 4 1 Q E s Q 9 R u r e 8 G L s 1 2 + J V D Q Q N 3 o U k j Z H f U F 9 i p E 2 U t c + a S n a 5 6 h T 7 M a t g J M + S u B V x z U P K n w 9 S u Z s L + n a O a f g T A C X i T s j u X I 2 / / V Z e r m v d O 2 f V i / A E S d C Y 4 a U a r p O q N s x k p p i R p J M K 1 I k R H i I + q R p q E C c q H Y 8 O S q B x 0 b p Q T + Q p o S G E 3 V + I k Z c q R H 3 T C d H e q A W v b H 4 r + d 5 f G G 1 9 s / a M R V h p I n A 0 8 1 + x K A O 4 D g y 2 K O S Y M 1 G h i A s q f k 8 x A M k E d Y m 2 I x J x V 3 M Y J n U i g X 3 t O B c m 3 j O w R R p c A S y I A 9 c U A J l c A k q o A o w u A O P 4 A k 8 W w / W q / V m v U 9 b U 9 Z s 5 h D 8 g f X 9 C 3 W o q b I = < / l a t e x i t > < l a t e x i t s h a 1 _ b a s e 6 4 = " S 9 7 h A c 1 X q 6 o + h n y b y N 3 E F c 8 G O U A = " > A A A B / H i c b V C 7 S g N B F L 3 r M 8 Z X 1 F K R w S B Y h V 0 t t A z a W C Z g H p A s Y X Y y m w y Z m V 1 m Z o W w x M 7 W V j / A T m w l v 2 J t 6 U 8 4 m 6 T Q x A M X D u f c y 7 m c I O Z M G 9 f 9 d J a W V 1 b X 1 n M b + c 2 t 7 Z 3 d w t 5 + X U e J I r R G I h 6 p Z o A 1 5 U z S m m G G 0 2 a s K B Y B p 4 1 g c J P 5 j X u q N I v k n R n G 1 B e 4 J 1 n I C D a Z 1 I 7 7 r F M o u i V 3 A r R I v B k p l o / G 1 e / H 4 3 G l U / h q d y O S C C o N 4 V j r l u f G x k + x M o x w O s q 3 E 0 1 j T A a 4 R 1 u W S i y o 9 t P J r y N 0 a p U u C i N l R x o 0 U X 9 f p F h o P R S B 3 R T Y 9 P W 8 l 4 n / e k E g 5 q J N e O W n T M a J o Z J M k 8 O E I x O h r A n U Z Y o S w 4 e W Y K K Y f R 6 R P l a Y G N t X 3 r b i z X e w S O r n J e + i 5 F Z t P d c w R Q 4 O 4 Q T O w I N L K M M t V K A G B P r w B M / w 4 j w 4 r 8 6 b 8 z 5 d X X J m N w f w B 8 7 H D 0 B 7 m P Y = < / l a t e x i t > < l a t e x i t s h a 1 _ b a s e 6 4 = " S 9 7 h A c 1 X q 6 o + h n y b y N 3 E F c 8 G O U A = " > A A A B / H i c b V C 7 S g N B F L 3 r M 8 Z X 1 F K R w S B Y h V 0 t t A z a W C Z g H p A s Y X Y y m w y Z m V 1 m Z o W w x M 7 W V j / A T m w l v 2 J t 6 U 8 4 m 6 T Q x A M X D u f c y 7 m c I O Z M G 9 f 9 d J a W V 1 b X 1 n M b + c 2 t 7 Z 3 d w t 5 + X U e J I r R G I h 6 p Z o A 1 5 U z S m m G G 0 2 a s K B Y B p 4 1 g c J P 5 j X u q N I v k n R n G 1 B e 4 J 1 n I C D a Z 1 I 7 7 r F M o u i V 3 A r R I v B k p l o / G 1 e / H 4 3 G l U / h q d y O S C C o N 4 V j r l u f G x k + x M o x w O s q 3 E 0 1 j T A a 4 R 1 u W S i y o 9 t P J r y N 0 a p U u C i N l R x o 0 U X 9 f p F h o P R S B 3 R T Y 9 P W 8 l 4 n / e k E g 5 q J N e O W n T M a J o Z J M k 8 O E I x O h r A n U Z Y o S w 4 e W Y K K Y f R 6 R P l a Y G N t X 3 r b i z X e w S O r n J e + i 5 F Z t P d c w R Q 4 O 4 Q T O w I N L K M M t V K A G B P r w B M / w 4 j w 4 r 8 6 b 8 z 5 d X X J m N w f w B 8 7 H D 0 B 7 m P Y = < / l a t e x i t > X = y 0 < l a t e x i t s h a 1 _ b a s e 6 4 = " w 7 f O K h z B 6 3 i 5 9 c d F S 6 X E C o b + p D w = " > A A A C C X i c b V C 7 S g N B F L 3 r M 8 Z H V i 0 V G Q y C V d j V Q h s h a G O Z g H l A s o b Z y W w y Z G Z 3 m Z k V l i W l l Z 9 g q x 9 g F 2 z 9 C m t L f 8 L J o 9 D E A x c O 5 9 z L P R w / 5 k x p x / m 0 l p Z X V t f W c x v 5 z a 3 t n Y K 9 u 1 d X U S I J r Z G I R 7 L p Y 0 U 5 C 2 l N M 8 1 p M 5 Y U C 5 / T h j + 4 G f u N B y o V i 8 I 7 n c b U E 7 g X s o A R r I 3 U s Q t t g X W f Y J 4 1 h 1 f p v d O x i 0 7 J m Q A t E n d G i u X D U f X 7 8 W h U 6 d h f 7 W 5 E E k F D T T h W q u U 6 s f Y y L D U j n A 7 z 7 U T R G J M B 7 t G W o S E W V H n Z J P g Q n R i l i 4 J I m g k 1 m q i / L z I s l E q F b z b H M d W 8 N x b / 9 X x f z L 3 W w a W X s T B O N A 3 J 9 H O Q c K Q j N K 4 F d Z m k R P P U E E w k M + E R 6 W O J i T b l 5 U 0 r 7 n w H i 6 R + V n L P S 0 7 V 1 H M N U + T g A I 7 h F F y 4 g D L c Q g V q Q C C B Z 3 i B V + v J e r N G 1 v t 0 d c m a 3 e z D H 1 g f P x U H n c I = < / l a t e x i t > z 0 < l a t e x i t s h a 1 _ b a s e 6 4 = " U 1 l 7 D t q 3 3 F g r R s 5 4 b T L L s j N b C y Q = " > A A A B + 3 i c b V C 7 S g N B F L 0 b X 3 F 9 R S 1 t B o N g F W a 1 0 E Y M 2 l h G N A 9 I 1 j A 7 m U 2 G z O w u M 7 N C X P I J t v o B 6 c T W b / A T x N o f c Z J Y a O K B C 4 d z 7 u V c T p A I r g 3 G n 0 5 u Y X F p e S W / 6 q 6 t b 2 x u F b Z 3 a j p O F W V V G o t Y N Q K i m e A R q x p u B G s k i h E Z C F Y P + p d j v 3 7 P l O Z x d G s G C f M l 6 U Y 8 5 J Q Y K 9 0 8 3 O F 2 o Y h L e A I 0 T 7 w f U j x / d 8 + S 0 Y d b a R e + W p 2 Y p p J F h g q i d d P D i f E z o g y n g g 3 d V q p Z Q m i f d F n T 0 o h I p v 1 s 8 u o Q H V i l g 8 J Y 2 Y k M m q i / L z I i t R 7 I w G 5 K Y n p 6 1 h u L / 3 p B I G e i T X j q Z z x K U s M i O k 0 O U 4 F M j M Z F o A 5 X j B o x s I R Q x e 3 z i P a I I t T Y u l z b i j f b w T y p H Z W 8 4 x K + x s X y B U y R h z 3 Y h 0 P w 4 A T K c A U V q A K F L j z C E z w 7 Q 2 f k v D i v 0 9 W c 8 3 O z C 3 / g v H 0 D m i 6 X 7 Q = = < / l a t e x i t > z 1 < l a t e x i t s h a 1 _ b a s e 6 4 = " V 7 A S t E 6 F L E t g S P N 8 7 a p V g z m L N D U = " > A A A B + 3 i c b V C 7 S g N B F L 0 b X 3 F 9 R S 1 t B o N g F X a 1 0 E Y M 2 l h G N A 9 I 1 j A 7 m U 2 G z M w u M 7 N C X P I J t v o B 6 c T W b / A T x N o f c Z J Y a O K B C 4 d z 7 u V c T p h w p o 3 n f T q 5 h c W l 5 Z X 8 q r u 2 v r G 5 V d j e q e k 4 V Y R W S c x j 1 Q i x p p x J W j X M c N p I F M U i 5 L Q e 9 i / H f v 2 e K s 1 i e W s G C Q 0 E 7 k o W M Y K N l W 4 e 7 v x 2 o e i V v A n Q P P F / S P H 8 3 T 1 L R h 9 u p V 3 4 a n V i k g o q D e F Y 6 6 b v J S b I s D K M c D p 0 W 6 m m C S Z 9 3 K V N S y U W V A f Z 5 N U h O r B K B 0 W x s i M N m q i / L z I s t B 6 I 0 G 4 K b H p 6 1 h u L / 3 p h K G a i T X Q a Z E w m q a G S T J O j l C M T o 3 E R q M M U J Y Y P L M F E M f s 8 I j 2 s M D G 2 L t e 2 4 s 9 2 M E 9 q R y X / u O R d e 8 X y B U y R h z 3 Y h 0 P w 4 Q T K c A U V q A K B L j z C E z w 7 Q 2 f k v D i v 0 9 W c 8 3 O z C 3 / g v H 0 D m 8 O X 7 g = = < / l a t e x i t > y 1 < l a t e x i t s h a 1 _ b a s e 6 4 = " 1 Y 9 j t I A c X c 6 W D 6 d G 1 4 l q r S Z U M k Y = " > A A A B + 3 i c b V C 7 S g N B F L 0 b X z G + o p a K D A b B K u x q o W X Q x j J B 8 4 B k D b O T S T J k Z n a Z m R W W J a W l r X 6 A n d g K + R V r S 3 / C y a P Q x A M X D u f c y 7 m c I O J M G 9 f 9 d D J L y y u r a 9 n 1 3 M b m 1 v Z O f n e v p s N Y E V o l I Q 9 V I 8 C a c i Z p 1 T D D a S N S F I u A 0 3 o w u B 7 7 9 Q e q N A v l n U k i 6 g v c k 6 z L C D Z W u k 3 u v X a + 4 B b d C d A i 8 W a k U D o c V b 4 f j 0 b l d v 6 r 1 Q l J L K g 0 h G O t m 5 4 b G T / F y j D C 6 T D X i j W N M B n g H m 1 a K r G g 2 k 8 n r w 7 R i V U 6 q B s q O 9 K g i f r 7 I s V C 6 0 Q E d l N g 0 9 f z 3 l j 8 1 w s C M R d t u p d + y m Q U G y r J N L k b c 2 R C N C 4 C d Z i i x P D E E k w U s 8 8 j 0 s c K E 2 P r y t l W v P k O F k n t r O i d F 9 2 K r e c K p s j C A R z D K X h w A S W 4 g T J U g U A P n u A Z X p y h 8 + q 8 O e / T 1 Y w z u 9 m H P 3 A + f g A u r p h X < / l a t e x i t > y 2 < l a t e x i t s h a 1 _ b a s e 6 4 = " M 1 N z n f h X n M F o s L H 4 H X e 5 0 7 v F G X w = " > A A A B + 3 i c b V C 7 S g N B F L 3 r M 8 Z X 1 F K R w S B Y h d 1 Y a B m 0 s U z Q P C B Z w + x k k g y Z m V 1 m Z o V l S W l p q x 9 g J 7 Z C f s X a 0 p 9 w 8 i g 0 8 c C F w z n 3 c i 4 n i D j T x n U / n a X l l d W 1 9 c x G d n N r e 2 c 3 t 7 d f 0 2 G s C K 2 S k I e q E W B N O Z O 0 a p j h t B E p i k X A a T 0 Y X I / 9 + g N V m o X y z i Q R 9 Q X u S d Z l B B s r 3 S b 3 x X Y u 7 x b c C d A i 8 W Y k X z o a V b 4 f j 0 f l d u 6 r 1 Q l J L K g 0 h G O t m 5 4 b G T / F y j D C 6 T D b i j W N M B n g H m 1 a K r G g 2 k 8 n r w 7 R q V U 6 q B s q O 9 K g i f r 7 I s V C 6 0 Q E d l N g 0 9 f z 3 l j 8 1 w s C M R d t u p d + y m Q U G y r J N L k b c 2 R C N C 4 C d Z i i x P D E E k w U s 8 8 j 0 s c K E 2 P r y t p W v P k O F k m t W P D O C 2 7 F 1 n M F U 2 T g E E 7 g D D y 4 g B L c Q B m q Q K A H T / A M L 8 7 Q e X X e n P f p 6 p I z u z m A P 3 A + f g A w Q 5 h Y < / l a t e x i t > z2 < l a t e x i t s h a 1 _ b a s e 6 4 = " 4 w e G o m N j p m J m W Z L 3 O 9 4 Y 1 O s Z s 9 o = " > A A A C A H i c b V C 7 S g N B F L 0 b X 3 F 9 R S 1 t B o N g F X Z j o Y 0 Y t L G M Y B 6 Q r G F m M p s M m Z 1 d Z m a F G N L 4 C b b a 2 d i J r T / g J 4 i 1 P + L k U W j i g Q u H c + 7 l X A 5 J B N f G 8 7 6 c z M L i 0 v J K d t V d W 9 / Y 3 M p t 7 1 R 1 n C r K K j Q W s a o T r J n g k l U M N 4 L V E 8 V w R A S r k d 7 F y K / d M q V 5 L K 9 N P 2 F B h D u S h 5 x i Y 6 V 6 k 2 C F 7 m 6 K r V z e K 3 h j o H n i T 0 n + 7 M M 9 T Z 4 / 3 X I r 9 9 1 s x z S N m D R U Y K 0 b v p e Y Y I C V 4 V S w o d t M N U s w 7 e E O a 1 g q c c R 0 M B j / O 0 Q H V m m j M F Z 2 p E F j 9 f f F A E d a 9 y N i N y N s u n r W G 4 n / e o R E M 9 E m P A k G X C a p Y Z J O k s N U I B O j U R u o z R W j R v Q t w V R x + z y i X a w w N b Y z 1 7 b i z 3 Y w T 6 r F g n 9 U 8 K 6 8 f O k c J s j C H u z D I f h w D C W 4 h D J U g I K A B 3 i E J + f e e X F

EQUATION

)

This approach corresponds to applying global average pooling right after the last convolutional layer. 8 This approach takes all pixel-pixel covariances into consideration and makes the kernel translation invariant. However, it requires O |X | 2 d 2 memory to compute the sample-sample covariance of the GP (or O d 2 per covariance entry in an iterative or distributed setting). It is impractical to use this method to analytically evaluate the GP, and we propose to use a Monte Carlo approach (see §4).

2. Subsampling one particular pixel: take h = e ↵ ,

EQUATION

This approach makes use of only one pixel-pixel covariance, and requires the same amount of memory as vectorization ( §3.1) to compute.

We compare the performance of presented strategies in Figure 1 . Note that all described strategies admit stacking additional FC layers on top while retaining the GP equivalence, using a derivation analogous to §2 (Lee et al., 2018; Matthews et al., 2018b) .

4. MONTE CARLO EVALUATION OF INTRACTABLE GP KERNELS:

We introduce a Monte Carlo estimation method for NN-GP kernels which are computationally impractical to compute analytically, or for which we do not know the analytic form. Similar in spirit to traditional random feature methods (Rahimi & Recht, 2007) , the core idea is to instantiate many 8 Spatially local average pooling in intermediary layers can be constructed in a similar fashion ( §A.3). We • CNN-GP exhibits similar behavio from the phase boundary, the cov all different inputs becomes asym constant and makes learning infe

• Many different NN architectures converge to a GP.

• Some admit a tractable analytic form. E.g. the covariance of a CNN-GP with no pooling can be computed using O(#samples^2 x #pixels) resources.

• For other architectures we use a Monte Carlo approach.

Sampling finite random networks of a given architecture and empirically computing the output covariance allows to approximate the CNN-GP covariance. In our experiments this (biased) estimate converges to the true CNN-GP covariance in #(instantiated networks) and #channels, both in terms of covariance Frobenius distance and the GP accuracy.

(a) SGD-trained finit (b) Performance of S CNN-GP with increa (c) However, the bes significant margin.

• Various CNN architectural decisions like pooling / vectorizing / subsampling, zero or no padding have a corresponding NN-GP.

• Pooling enforces translation-invariant predictions in both CNNs and CNN-GPs and allows for the best performance.

• Shallow CNN models can perform worse than fully-connected alternatives due to failing to capture non-linear interactions between distant pixels. 

7.. Disentang:

EQUATION

⊗ I10 K 0 K 0 K 1 K 2 K 1 K 2 σ 2 ω K 0 + σ 2 b σ 2 ω K 1 + σ 2 b ( K 1 ) ( K 0 )

K 0 K 1 K 2 ( K 1 ) ( K 0 )

affine transform affine transform affine transform convolution convolution matrix-multiply nonlinearity nonlinearity 

y 2 = ϕ ( z 1 ) y 1 = ϕ ( z 0 ) n 1 → ∞ n 1 → ∞ n 2 → ∞ n 2 → ∞ C (K) = E x⇠N (0,K) h (x) (x) T i < l a t e x i t s h a 1 _ b a s e 6 4 = " L K S S 7 i N m 7 W J o + g 4 I + B j N B l P b a w o = " > A A A C e H i c b V F d S 8 M w F E 3 r 9 / y a H 2 + + B K e o I N L u R V 8 E U Q R B E A W n w l p H k q V b M G l L c i u O M v + n z / o r f D J b q + j m h c D h n H O 5 N + f S V A o D n v f m u B O T U 9 M z s 3 O V + Y X F p e X q y u q d S T L N e I M l M t E P l B g u R c w b I E D y h 1 R z o q j k 9 / T p b K D f P 3 N t R B L f Q i / l o S K d W E S C E b B U q / q 6 F S g C X U Z k f t Y P J I 9 g 9 z L Q o t O F P X y M h x q l + X m / l b 8 E R q g f 8 1 V p 9 v a / 7 Q X R D N K u K K S X U h h n H m 8 L E G 6 1 q j X v w B s W H g d + C W q o r O t W 9 T 1 o J y x T P A Y m i T F N 3 0 s h z I k G w S T v V 4 L M 8 J S w J 9 L h T Q t j o r g J 8 2 F O f b x t m T a O E m 1 f D H j I / u 7 I i T K m p 6 h 1 D v 5 p R r U B + a 9 G q R o Z D d F R m I s 4 z Y D H r J g c Z R J D g g d X w G 2 h O Q P Z s 4 A w L e z y m H W J J g z s r S o 2 F X 8 0 g 3 F w V z / w L b 6 p 1 0 5 O y 3 x m 0 Q b a R L v I R 4 f o B F 2 g a 9 R A D H 0 4 F W f N W X c + X e z u u H u F 1 X X K n j X 0 p 9 z 6 F 6 y i w S Y = < / l a t e x i t > < l a t e x i t s h a 1 _ b a s e 6 4 = " L K S S 7 i N m 7 W J o + g 4 I + B j N B l P b a w o = " > A A A C e H i c b V F d S 8 M w F E 3 r 9 / y a H 2 + + B K e o I N L u R V 8 E U Q R B E A W n w l p H k q V b M G l L c i u O M v + n z / o r f D J b q + j m h c D h n H O 5 N + f S V A o D n v f m u B O T U 9 M z s 3 O V + Y X F p e X q y u q d S T L N e I M l M t E P l B g u R c w b I E D y h 1 R z o q j k 9 / T p b K D f P 3 N t R B L f Q i / l o S K d W E S C E b B U q / q 6 F S g C X U Z k f t Y P J I 9 g 9 z L Q o t O F P X y M h x q l + X m / l b 8 E R q g f 8 1 V p 9 v a / 7 Q X R D N K u K K S X U h h n H m 8 L E G 6 1 q j X v w B s W H g d + C W q o r O t W 9 T 1 o J y x T P A Y m i T F N 3 0 s h z I k G w S T v V 4 L M 8 J S w J 9 L h T Q t j o r g J 8 2 F O f b x t m T a O E m 1 f D H j I / u 7 I i T K m p 6 h 1 D v 5 p R r U B + a 9 G q R o Z D d F R m I s 4 z Y D H r J g c Z R J D g g d X w G 2 h O Q P Z s 4 A w L e z y m H W J J g z s r S o 2 F X 8 0 g 3 F w V z / w L b 6 p 1 0 5 O y 3 x m 0 Q b a R L v I R 4 f o B F 2 g a 9 R A D H 0 4 F W f N W X c + X e z u u H u F 1 X X K n j X 0 p 9 z 6 F 6 y i w S Y = < / l a t e x i t > < l a t e x i t s h a 1 _ b a s e 6 4 = " L K S S 7 i N m 7 W J o + g 4 I + B j N B l P b a w o = " > A A A C e H i c b V F d S 8 M w F E 3 r 9 / y a H 2 + + B K e o I N L u R V 8 E U Q R B E A W n w l p H k q V b M G l L c i u O M v + n z / o r f D J b q + j m h c D h n H O 5 N + f S V A o D n v f m u B O T U 9 M z s 3 O V + Y X F p e X q y u q d S T L N e I M l M t E P l B g u R c w b I E D y h 1 R z o q j k 9 / T p b K D f P 3 N t R B L f Q i / l o S K d W E S C E b B U q / q 6 F S g C X U Z k f t Y P J I 9 g 9 z L Q o t O F P X y M h x q l + X m / l b 8 E R q g f 8 1 V p 9 v a / 7 Q X R D N K u K K S X U h h n H m 8 L E G 6 1 q j X v w B s W H g d + C W q o r O t W 9 T 1 o J y x T P A Y m i T F N 3 0 s h z I k G w S T v V 4 L M 8 J S w J 9 L h T Q t j o r g J 8 2 F O f b x t m T a O E m 1 f D H j I / u 7 I i T K m p 6 h 1 D v 5 p R r U B + a 9 G q R o Z D d F R m I s 4 z Y D H r J g c Z R J D g g d X w G 2 h O Q P Z s 4 A w L e z y m H W J J g z s r S o 2 F X 8 0 g 3 F w V z / w L b 6 p 1 0 5 O y 3 x m 0 Q b a R L v I R 4 f o B F 2 g a 9 R A D H 0 4 F W f N W X c + X e z u u H u F 1 X X K n j X 0 p 9 z 6 F 6 y i w S Y = < / l a t e x i t > < l a t e x i t s h a 1 _ b a s e 6 4 = " L K S S 7 i N m 7 W J o + g 4 I + B j N B l P b a w o = " > A A A C e H i c b V F d S 8 M w F E 3 r 9 / y a H 2 + + B K e o I N L u R V 8 E U Q R B E A W n w l p H k q V b M G l L c i u O M v + n z / o r f D J b q + j m h c D h n H O 5 N + f S V A o D n v f m u B O T U 9 M z s 3 O V + Y X F p e X q y u q d S T L N e I M l M t E P l B g u R c w b I E D y h 1 R z o q j k 9 / T p b K D f P 3 N t R B L f Q i / l o S K d W E S C E b B U q / q 6 F S g C X U Z k f t Y P J I 9 g 9 z L Q o t O F P X y M h x q l + X m / l b 8 E R q g f 8 1 V p 9 v a / 7 Q X R D N K u K K S X U h h n H m 8 L E G 6 1 q j X v w B s W H g d + C W q o r O t W 9 T 1 o J y x T P A Y m i T F N 3 0 s h z I k G w S T v V 4 L M 8 J S w J 9 L h T Q t j o r g J 8 2 F O f b x t m T a O E m 1 f D H j I / u 7 I i T K m p 6 h 1 D v 5 p R r U B + a 9 G q R o Z D d F R m I s 4 z Y D H r J g c Z R J D g g d X w G 2 h O Q P Z s 4 A w L e z y m H W J J g z s r S o 2 F X 8 0 g 3 F w V z / w L b 6 p 1 0 5 O y 3 x m 0 Q b a R L v I R 4 f o B F 2 g a 9 R A D H 0 4 F W f N W X c + X e z u u H u F 1 X X K n j X

EQUATION

where ✓ consists of M draws of the weights and biases from their prior distribution, ✓ m ⇠ p (✓), and n is the width or number of channels in hidden layers. The MC-GP kernel converges to the analytic kernel with increasing width, lim n!1 K l n,M = K l 1 in probability. For finite width networks, the uncertainty in K l n,M is Var

⇥ K l n,M ⇤ = Var ✓ ⇥ K l n (✓) ⇤ /M . From Daniely et al. (2016), we know that Var ✓ ⇥ K l n (✓) ⇤ / 1 n , which leads to Var ✓ [K l n,M ] / 1 Mn . For finite n, K l n,M is also a biased estimate of K l 1 ,

where the bias depends solely on network width. We do not currently have an analytic form for this bias, but we can see in Figures 2 and 7 that for the hyperparameters we probe it is small relative to the variance. In particular,

K l n,M (✓) K L 1 2 F

is nearly constant for constant Mn. We thus treat Mn as the effective sample size for the Monte Carlo kernel estimate. Increasing M and reducing n can reduce memory cost, though potentially at the expense of increased compute time and bias.

Under review as a conference paper at ICLR 2019 MC-CNN-GP Figure 2 : Validation accuracy (left) of an MC-CNN-GP increases with n ⇥ M (i.e. channel count times number of samples) and approaches that of the exact CNN-GP (not shown), while the distance (right) to the exact kernel decreases. The dark band in the left plot corresponds to ill-conditioning of K L+1 n,M when the number of outer products contributing to K L+1 n,M approximately equals its rank. Values reported are for a 3-layer model applied to a 2K/4K train/validation subset of CIFAR10 downsampled to 8 ⇥ 8. See Figure 7 for similar results with other architectures and §A.7.2 for experimental details. random finite width networks and use the empirical uncentered covariances of activations to estimate the Monte Carlo-GP (MC-GP) kernel,

EQUATION

where ✓ consists of M draws of the weights and biases from their prior distribution, ✓ m ⇠ p (✓), and n is the width or number of channels in hidden layers. The MC-GP kernel converges to the analytic kernel with increasing width, lim n!1 K l n,M = K l 1 in probability. For finite width networks, the uncertainty in K l n,M is Var

⇥ K l n,M ⇤ = Var ✓ ⇥ K l n (✓) ⇤ /M . From Daniely et al. (2016), we know that Var ✓ ⇥ K l n (✓) ⇤ / 1 n , which leads to Var ✓ [K l n,M ] / 1 Mn . For finite n, K l n,M is also a biased estimate of K l 1 ,

where the bias depends solely on network width. We do not currently have an analytic form for this bias, but we can see in Figures 2 and 7 that for the hyperparameters we probe it is small relative to the variance. In particular,

K l n,M (✓) K L 1 2 F

is nearly constant for constant Mn. We thus treat Mn as the effective sample size for the Monte Carlo kernel estimate. Increasing M and reducing n can reduce memory cost, though potentially at the expense of increased compute time and bias.

(#NN instantiations)

(not always necessary to track the whole 4x4x10x10 covariance.)

(attaching a fully-connected layer only to the center pixel of the output)

n 1 → ∞ n 1 → ∞ n 2 → ∞ n 2 → ∞

CNN (1D) activations:



CNN-GP covariance:

Under review as a conference paper at ICLR 2019 2) -which also performs worse than the CNN-GP (we conjecture, due to overfitting to centrally-located features) but approaches the latter (right) in the limit of large depth, as information becomes more uniformly spatially distributed (Xiao et al., 2018) . CNN-GPs generally outperform FCN-GP, presumably due to the local connectivity prior, but can fail to capture nonlinear interactions between spatially-distant pixels at shallow depths (left). Values are reported on a 2K/4K train/validation subsets of CIFAR10. See §A.7.3 for experimental details.

EQUATION

This approach corresponds to applying global average pooling right after the last convolutional layer. 8 This approach takes all pixel-pixel covariances into consideration and makes the kernel translation invariant. • CNN-GP exhibits similar behavio from the phase boundary, the cov all different inputs becomes asym constant and makes learning infe

• Many different NN architectures converge to a GP.

• Some admit a tractable analytic form. E.g. the covariance of a CNN-GP with no pooling can be computed using O(#samples^2 x #pixels) resources.

• For other architectures we use a Monte Carlo approach.

Sampling finite random networks of a given architecture and empirically computing the output covariance allows to approximate the CNN-GP covariance. In our experiments this (biased) estimate converges to the true CNN-GP covariance in #(instantiated networks) and #channels, both in terms of covariance Frobenius distance and the GP accuracy.

(a) SGD-trained finit (b) Performance of S CNN-GP with increa (c) However, the bes significant margin.

• Various CNN architectural decisions like pooling / vectorizing / subsampling, zero or no padding have a corresponding NN-GP.

• Pooling enforces translation-invariant predictions in both CNNs and CNN-GPs and allows for the best performance.

• Shallow CNN models can perform worse than fully-connected alternatives due to failing to capture non-linear interactions between distant pixels. N 

n 1 → ∞ n 1 → ∞ n 2 → ∞ n 2 → ∞ 4x4 covariance per class y 0 z 0 z 1 y 2 = ϕ ( z 1 ) z2 4 3 2 1 10 classes per input 4 inputs, 3 features 4 3 2 1 4 3 2 1 ⊗ I10 K 0 K 0 K 1 K 2 K 1 K 2 σ 2 ω K 0 + σ 2 b σ 2 ω K 1 + σ 2 b ( K 1 ) ( K 0

K 0 K 1 K 2 ( K 1 ) ( K 0 )

affine transform affine transform affine transform convolution convolution matrix-multiply nonlinearity nonlinearity

y 2 = ϕ ( z 1 ) y 1 = ϕ ( z 0 ) n 1 → ∞ n 1 → ∞ n 2 → ∞ n 2 → ∞ C (K) = E x⇠N (0,K) h (x) (x) T i < l a t e x i t s h a 1 _ b a s e 6 4 = " L K S S 7 i N m 7 W J o + g 4 I + B j N B l P b a w o = " > A A A C e H i c b V F d S 8 M w F E 3 r 9 / y a H 2 + + B K e o I N L u R V 8 E U Q R B E A W n w l p H k q V b M G l L c i u O M v + n z / o r f D J b q + j m h c D h n H O 5 N + f S V A o D n v f m u B O T U 9 M z s 3 O V + Y X F p e X q y u q d S T L N e I M l M t E P l B g u R c w b I E D y h 1 R z o q j k 9 / T p b K D f P 3 N t R B L f Q i / l o S K d W E S C E b B U q / q 6 F S g C X U Z k f t Y P J I 9 g 9 z L Q o t O F P X y M h x q l + X m / l b 8 E R q g f 8 1 V p 9 v a / 7 Q X R D N K u K K S X U h h n H m 8 L E G 6 1 q j X v w B s W H g d + C W q o r O t W 9 T 1 o J y x T P A Y m i T F N 3 0 s h z I k G w S T v V 4 L M 8 J S w J 9 L h T Q t j o r g J 8 2 F O f b x t m T a O E m 1 f D H j I / u 7 I i T K m p 6 h 1 D v 5 p R r U B + a 9 G q R o Z D d F R m I s 4 z Y D H r J g c Z R J D g g d X w G 2 h O Q P Z s 4 A w L e z y m H W J J g z s r S o 2 F X 8 0 g 3 F w V z / w L b 6 p 1 0 5 O y 3 x m 0 Q b a R L v I R 4 f o B F 2 g a 9 R A D H 0 4 F W f N W X c + X e z u u H u F 1 X X K n j X 0 p 9 z 6 F 6 y i w S Y = < / l a t e x i t > < l a t e x i t s h a 1 _ b a s e 6 4 = " L K S S 7 i N m 7 W J o + g 4 I + B j N B l P b a w o = " > A A A C e H i c b V F d S 8 M w F E 3 r 9 / y a H 2 + + B K e o I N L u R V 8 E U Q R B E A W n w l p H k q V b M G l L c i u O M v + n z / o r f D J b q + j m h c D h n H O 5 N + f S V A o D n v f m u B O T U 9 M z s 3 O V + Y X F p e X q y u q d S T L N e I M l M t E P l B g u R c w b I E D y h 1 R z o q j k 9 / T p b K D f P 3 N t R B L f Q i / l o S K d W E S C E b B U q / q 6 F S g C X U Z k f t Y P J I 9 g 9 z L Q o t O F P X y M h x q l + X m / l b 8 E R q g f 8 1 V p 9 v a / 7 Q X R D N K u K K S X U h h n H m 8 L E G 6 1 q j X v w B s W H g d + C W q o r O t W 9 T 1 o J y x T P A Y m i T F N 3 0 s h z I k G w S T v V 4 L M 8 J S w J 9 L h T Q t j o r g J 8 2 F O f b x t m T a O E m 1 f D H j I / u 7 I i T K m p 6 h 1 D v 5 p R r U B + a 9 G q R o Z D d F R m I s 4 z Y D H r J g c Z R J D g g d X w G 2 h O Q P Z s 4 A w L e z y m H W J J g z s r S o 2 F X 8 0 g 3 F w V z / w L b 6 p 1 0 5 O y 3 x m 0 Q b a R L v I R 4 f o B F 2 g a 9 R A D H 0 4 F W f N W X c + X e z u u H u F 1 X X K n j X 0 p 9 z 6 F 6 y i w S Y = < / l a t e x i t > < l a t e x i t s h a 1 _ b a s e 6 4 = " L K S S 7 i N m 7 W J o + g 4 I + B j N B l P b a w o = " > A A A C e H i c b V F d S 8 M w F E 3 r 9 / y a H 2 + + B K e o I N L u R V 8 E U Q R B E A W n w l p H k q V b M G l L c i u O M v + n z / o r f D J b q + j m h c D h n H O 5 N + f S V A o D n v f m u B O T U 9 M z s 3 O V + Y X F p e X q y u q d S T L N e I M l M t E P l B g u R c w b I E D y h 1 R z o q j k 9 / T p b K D f P 3 N t R B L f Q i / l o S K d W E S C E b B U q / q 6 F S g C X U Z k f t Y P J I 9 g 9 z L Q o t O F P X y M h x q l + X m / l b 8 E R q g f 8 1 V p 9 v a / 7 Q X R D N K u K K S X U h h n H m 8 L E G 6 1 q j X v w B s W H g d + C W q o r O t W 9 T 1 o J y x T P A Y m i T F N 3 0 s h z I k G w S T v V 4 L M 8 J S w J 9 L h T Q t j o r g J 8 2 F O f b x t m T a O E m 1 f D H j I / u 7 I i T K m p 6 h 1 D v 5 p R r U B + a 9 G q R o Z D d F R m I s 4 z Y D H r J g c Z R J D g g d X w G 2 h O Q P Z s 4 A w L e z y m H W J J g z s r S o 2 F X 8 0 g 3 F w V z / w L b 6 p 1 0 5 O y 3 x m 0 Q b a R L v I R 4 f o B F 2 g a 9 R A D H 0 4 F W f N W X c

+ X e z u u H u F 1 X X K n j X 0 p 9 z 6 F 6 y i w S Y = < / l a t e x i t > < l a t e x i t s h a 1 _ b a s e 6 4 = " L K S S 7 i N m 7 W J o + g 4 n,M when the number of outer products contributing to K L+1 n,M approximately equals its rank. Values reported are for a 3-layer model applied to a 2K/4K train/validation subset of CIFAR10 downsampled to 8 ⇥ 8. See Figure 7 for similar results with other architectures and §A.7.2 for experimental details.

I + B j N B l P b a w o = " > A A A C e H i c b V F d S 8 M w F E 3 r 9 / y a H 2 + + B K e o I N L u R V 8 E U Q R B E A W n w l p H k q V b M G l L c i u O M v + n z / o r f D J b q + j m h c D h n H O 5 N + f S V A o D n v f m u B O T U 9 M z s 3 O V + Y X F p e X q y u q d S T L N e I M l M t E P l B g u R c w b I E D y h 1 R z o q j k 9 / T p b K D f P 3 N t R B L f Q i / l o S K d W E S C E b B U q / q 6 F S g C X U Z k f t Y P J I 9 g 9 z L Q o t O F P X y M h x q l + X m / l b 8 E R q g f 8 1 V p 9 v a / 7 Q X R D N K u K K S X U h h n H m 8 L E G 6 1 q j X v w B s W H g d + C W q o r O t W 9 T 1 o J y x T P A Y m i T F N 3 0 s h z I k G w S T v V 4 L M 8 J S w J 9 L h T Q t j o r g J 8 2 F O f b x t m T a O E m 1 f D H j I / u 7 I i T K m p 6 h 1 D v 5 p R r U B + a 9 G q R o Z D d F R m I s 4 z Y D H r J g c Z R J D g g d X w G 2 h O Q P Z s 4 A w L e z y m H W J J g z s r S o 2 F X 8 0 g 3 F w V z / w L b 6 p 1 0 5 O y 3 x m 0 Q b a R L v I R 4 f o B F 2 g a 9 R A D H 0 4 F W f N W X c + X e z u u H u F 1 X X K n j X

(#NN instantiations)

• Outputs of a CNN converge in distribution to a Gaussian process with a compositional kernel (under mild conditions on the weights and biases distribution and nonlinearity). 

outputs inputs D ! min{n 1 ,...,n L }!1 N ⇣ 0, A (C A) L K 0 ⌦ I n L+1 ⌘ < l a t e x i t s h a 1 _ b a s e 6 4 = " r S e f Q 0 I B g T G o B S F h X r b E 0 x A 5 J b k = " > A A A C 4 n i c b V J N b 9 Q w E H X C V 1 k + u s C R i 8 U W q Y h V l f Q C x 0 I 5 g K i q I r F t p X V 2 5 X i d X a u O H d k T 6 C q Y H 8 A N c

W k Y H z E t t T l N q e V S K D 4 C A Z K f F o b T P J X 8 J D 3 b r / M n n 7 i x Q q u P s C x 4 k t O 5 E p l g F D w 1 7 f / Z I p J n s E 2 A n 4 P J K 1 1 C U Y L F J B X z L 1 i o + s c R I + Y L e E b O m 5 M a o z + P S S 5 U U 0 o q N Y m H Z K b B D t X k o N U S R 0 A T o T J Y J h X J K S w Y l d U b 5 y 7 w o W u N o y G + 4 F 4 5 w o R h X U c r d r 9 j L 6 n a f r x Z o 3 w / i V Y d a h A 5 t / j d 1 D d V H T y P 3 U q 7 N e 0 P o p 2 o C X w V x B 0 Y o C 6 O p v 3 f / k 6 s z L k C J q m 1 4 z g q I K m o A c E k d z 1 S W l 5 Q d k b n f O y h o t 4 4 q Z q N O P z U M z O c a e M / B b h h L 1 d U N L d 2 m a d e W V / L r u d q 8 r + 5 N M 3 X r C F 7 m V T N o r h i r X N W S g w a 1 / v G M 2 E 4 A 7 n 0 g D I j f P O Y L a i h D P y r 6 P m p x O s z u A q O d 3 d i j z / s D v Z e d / P Z Q I / R E 7 S N Y v Q C 7 a G 3 6 A i N E A s O A w h c 8 D W c h d / C 7 + G P V h o G X

c 0 j 9 E + E P / 8 C 5 J r t n g = = < / l a t e x i t > < l a t e x i t s h a 1 _ b a s e 6 4 = " r S e f Q 0 

I B g T G o B S F h X r b E 0 x A 5 J b k = " > A A A C 4 n i c b V J N b 9 Q w E H X C V 1 k + u s C R i 8 U W q Y h V l f Q C x 0 I 5 g K i q I r F t p X V 2 5 X i d X a u O H d k T 6 C q Y H 8 A N c

W k Y H z E t t T l N q e V S K D 4 C A Z K f F o b T P J X 8 J D 3 b r / M n n 7 i x Q q u P s C x 4 k t O 5 E p l g F D w 1 7 f / Z I p J n s E 2 A n 4 P J K 1 1 C U Y L F J B X z L 1 i o + s c R I + Y L e E b O m 5 M a o z + P S S 5 U U 0 o q N Y m H Z K b B D t X k o N U S R 0 A T o T J Y J h X J K S w Y l d U b 5 y 7 w o W u N o y G + 4 F 4 5 w o R h X U c r d r 9 j L 6 n a f r x Z o 3 w / i V Y d a h A 5 t / j d 1 D d V H T y P 3 U q 7 N e 0 P o p 2 o C X w V x B 0 Y o C 6 O p v 3 f / k 6 s z L k C J q m 1 4 z g q I K m o A c E k d z 1 S W l 5 Q d k b n f O y h o t 4 4 q Z q N O P z U M z O c a e M / B b h h L 1 d U N L d 2 m a d e W V / L r u d q 8 r + 5 N M 3 X r C F 7 m V T N o r h i r X N W S g w a 1 / v G M 2 E 4 A 7 n 0 g D I j f P O Y L a i h D P y r 6 P m p x O s z u A q O d 3 d i j z / s D v Z e d / P Z Q I / R E 7 S N Y v Q C 7 a G 3 6 A i N E A s O A w h c 8 D W c h d / C 7 + G P V h o G X

c 0 j 9 E + E P / 8 C 5 J r t n g = = < / l a t e x i t > < l a t e x i t s h a 1 _ b a s e 6 4 = " r S e f Q 0 

I B g T G o B S F h X r b E 0 x A 5 J b k = " > A A A C 4 n i c b V J N b 9 Q w E H X C V 1 k + u s C R i 8 U W q Y h V l f Q C x 0 I 5 g K i q I r F t p X V 2 5 X i d X a u O H d k T 6 C q Y H 8 A N c

W k Y H z E t t T l N q e V S K D 4 C A Z K f F o b T P J X 8 J D 3 b r / M n n 7 i x Q q u P s C x 4 k t O 5 E p l g F D w 1 7 f / Z I p J n s E 2 A n 4 P J K 1 1 C U Y L F J B X z L 1 i o + s c R I + Y L e E b O m 5 M a o z + P S S 5 U U 0 o q N Y m H Z K b B D t X k o N U S R 0 A T o T J Y J h X J K S w Y l d U b 5 y 7 w o W u N o y G + 4 F 4 5 w o R h X U c r d r 9 j L 6 n a f r x Z o 3 w / i V Y d a h A 5 t / j d 1 D d V H T y P 3 U q 7 N e 0 P o p 2 o C X w V x B 0 Y o C 6 O p v 3 f / k 6 s z L k C J q m 1 4 z g q I K m o A c E k d z 1 S W l 5 Q d k b n f O y h o t 4 4 q Z q N O P z U M z O c a e M / B b h h L 1 d U N L d 2 m a d e W V / L r u d q 8 r + 5 N M 3 X r C F 7 m V T N o r h i r X N W S g w a 1 / v G M 2 E 4 A 7 n 0 g D I j f P O Y L a i h D P y r 6 P m p x O s z u A q O d 3 d i j z / s D v Z e d / P Z Q I / R E 7 S N Y v Q C 7 a G 3 6 A i N E A s O A w h c 8 D W c h d / C 7 + G P V h o G X

c 0 j 9 E + E P / 8 C 5 J r t n g = = < / l a t e x i t > < l a t e x i t s h a 1 _ b a s e 6 4 = " r S e f Q 0 

I B g T G o B S F h X r b E 0 x A 5 J b k = " > A A A C 4 n i c b V J N b 9 Q w E H X C V 1 k + u s C R i 8 U W q Y h V l f Q C x 0 I 5 g K i q I r F t p X V 2 5 X i d X a u O H d k T 6 C q Y H 8 A N c

W k Y H z E t t T l N q e V S K D 4 C A Z K f F o b T P J X 8 J D 3 b r / M n n 7 i x Q q u P s C x 4 k t O 5 E p l g F D w 1 7 f / Z I p J n s E 2 A n 4 P J K 1 1 C U Y L F J B X z L 1 i o + s c R I + Y L e E b O m 5 M a o z + P S S 5 U U 0 o q N Y m H Z K b B D t X k o N U S R 0 A T o T J Y J h X J K S w Y l d U b 5 y 7 w o W u N o y G + 4 F 4 5 w o R h X U c r d r 9 j L 6 n a f r x Z o 3 w / i V Y d a h A 5 t / j d 1 D d V H T y P 3 U q 7 N e 0 P o p 2 o C X w V x B 0 Y o C 6 O p v 3 f / k 6 s z L k C J q m 1 4 z g q I K m o A c E k d z 1 S W l 5 Q d k b n f O y h o t 4 4 q Z q N O P z U M z O c a e M / B b h h L 1 d U N L d 2 m a d e W V / L r u d q 8 r + 5 N M 3 X r C F 7 m V T N o r h i r X N W S g w a 1 / v G M 2 E 4 A 7 n 0 g D I j f P O Y L a i h D P y r 6 P m p x O s z u A q O d 3 d i j z / s D v Z e d / P Z Q I / R E 7 S N Y v Q C 7 a G 3 6 A i N E A s O A w h c 8 D W c h d / C 7 + G P V h o G X

c 0 j 9 E + E P / 8 C 5 J r t n g = = < / l a t e x i t > (#channels in the outputs)

(not always necessary to track the whole 4x4x10x10 covariance.)

(attaching a fully-connected layer only to the center pixel of the output)

n 1 → ∞ n 1 → ∞ n 2 → ∞ n 2 → ∞ K 0 < l a t e x i t s h a 1 _ b a s e 6 4 = " d j K o C 8 a 3 j 3 Z g k 4 r w u 2 K p B m 0 0 p 3 8 = " > A A A B + 3 i c b V C 7 S g N B F L 0 b X 3 F 9 R S 1 t B o N g F W a 1 0 E Y M 2 g g 2 E c 0 D k j X M T m a T I T O 7 y 8 y s E J Z 8 g q 1 + Q D q x 9 R v 8 B L H 2 R 5 w k F p p 4 4 M L h n H s 5 l x M k g m u D 8 a e T W 1 h c W l 7 J r 7 p r 6 x u b W 4 X t n Z q O U 0 V Z l c Y i V o 2 A a C Z 4 x K q G G 8 E a i W J E B o L V g / 7 l 2 K 8 / M K V 5 H N 2 Z Q c J 8 S b o R D z k l x k q 3 1 / e 4 X S j i E p 4 A z R P v h x T P 3 9 2 z Z P T h V t q F r 1 Y n p q l k k a G C a N 3 0 c G L 8 j C j D q W B D t 5 V q l h D a J 1 3 W t D Q i k m k / m 7 w 6 R A d W 6 a A w V n Y i g y b q 7 4 u M S K 0 H M r C b k p i e n v X G 4 r 9 e E M i Z a B O e + h m P k t S w i E 6 T w 1 Q g E 6 N x E a j D F a N G D C w h V H H 7 P K I 9 o g g 1 t i 7 X t u L N d j B P a k c l 7 7 i E b 3 C x f A F T 5 G E P 9 u E Q P D i B M l x B B a p A o Q u P 8 A T P z t A Z O S / O 6 3 Q 1 5 / z c 7 M I f O G / f T 3 W X v g = = < / l a t e x i t > A < l a t e x i t s h a 1 _ b a s e 6 4 = " O 6 6 A T y 1 a w 1 t Y L Y U p v p J 8 F g N J C H 8 = " > A A A C A 3 i c b V C 7 S g N B F L 0 b X 3 F 9 R S 1 t B o N g F X a 1 0 E a M 2 l h G M A 9 I l j A 7 m S R D Z m e W m V k h L C n 9 B F s t L O 3 E 1 t p P E G t / x N k k h S Y e u H A 4 5 1 7 u 4 Y Q x Z 9 p 4 3 p e T W 1 h c W l 7 J r 7 p r 6 x u b W 4 X t n Z q W i S K 0 S i S X q h F i T T k T t G q Y 4 b Q R K 4 q j k N N 6 O L j K / P o d V Z p J c W u G M Q 0 i 3 B O s y w g 2 V m q 2 I m z 6 B P P 0 Y t Q u F L 2 S N w a a J / 6 U F M 8 / 3 L P 4 + d O t t A v f r Y 4 k S U S F I R x r 3 f S 9 2 A Q p V o Y R T k d u K 9 E 0 x m S A e 7 R p q c A R 1 U E 6 j j x C B 1 b p o K 5 U d o R B Y / X 3 R Y o j r Y d R a D e z i H r W y 8 R / v T C M Z l 6 b 7 m m Q M h E n h g o y + d x N O D I S Z Y W g D l O U G D 6 0 B B P F b H h E + l h h Y m x t r m 3 F n + 1 g n t S O S v 5 x y b v x i u V L m C A P e 7 A P h + D D C Z T h G i p Q B Q I S H u A R n p x 7 5 8 V 5 d d 4 m q z l n e r M L f + C 8 / w A 5 N 5 u k < / l a t e x i t > C < l a t e x i t s h a 1 _ b a s e 6 4 = " B p 3 s 0 N X a z I 8 d Q f K 7 i 5 e d A + N 1 h F s = " > A A A C A 3 i c b V C 7 S g N B F L 0 b X 3 F 9 R S 1 t B o N g F X a 1 0 E Y M p r G M Y B 6 Q L G F 2 M k m G z M w u M 7 N C W F L 6 C b Z a W N q J r b W f I N b + i L N J C k 0 8 c O F w z r 3 c w w l j z r T x v C 8 n t 7 S 8 s r q W X 3 c 3 N r e 2 d w q 7 e 3 U d J Y r Q G o l 4 p J o h 1 p Q z S W u G G U 6 b s a J Y h J w 2 w m E l 8 x t 3 V G k W y V s z i m k g c F + y H i P Y W K n V F t g M C O Z p Z d w p F L 2 S N w F a J P 6 M F C 8 / 3 I v 4 + d O t d g r f 7 W 5 E E k G l I R x r 3 f K 9 2 A Q p V o Y R T s d u O 9 E 0 x m S I + 7 R l q c S C 6 i C d R B 6 j I 6 t 0 U S 9 S d q R B E / X 3 R Y q F 1 i M R 2 s 0 s o p 7 3 M v F f L w z F 3 G v T O w 9 S J u P E U E m m n 3 s J R y Z C W S G o y x Q l h o 8 s w U Q x G x 6 R A V a Y G F u b a 1 v x 5 z t Y J P W T k n 9 a 8 m 6 8 Y v k K p s j D A R z C M f h w B m W 4 h i r U g E A E D / A I T 8 6 9 8 + K 8 O m / T 1 Z w z u 9 m H P 3 D e f w A 8 Y 5 u m < / l a t e x i t > K 1 1 < l a t e x i t s h a 1 _ b a s e 6 4 = " X v v O k S j q l J 9 2 P o c D U d m z M o T Y 5 m 8 = " > A A A C B H i c b V C 7 S g N B F L 0 b X z G + o p Y 2 Q 0 R I F X a 1 i G X Q R r C J Y B 6 Q X c P s Z D Y Z M j u 7 z M w K y 5 L W T 7 C x 0 A + w E 1 t 7 P 0 G 0 8 0 e c P A p N P H D h c M 6 9 n M v x Y 8 6 U t u 0 P K 7 e 0 v L K 6 l l 8 v b G x u b e 8 U d / e a K k o k o Q 0 S 8 U i 2 f a w o Z 4 I 2 N N O c t m N J c e h z 2 v K H 5 2 O / d U u l Y p G 4 1 m l M v R D 3 B Q s Y w d p I 7 u W N 0 8 1 c J g K d j r r F Q 7 t i T 4 A W i T M j h 7 V S + e u z + v 5 Q 7 x a / 3 V 5 E k p A K T T h W q u P Y s f Y y L D U j n I 4 K b q J o j M k Q 9 2 n H U I F D q r x s 8 v M I H R m l h 4 J I m h E a T d T f F x k O l U p D 3 2 y G W A / U v D c W / / V 8 P 5 y L 1 s G p l z E R J 5 o K M k 0 O E o 5 0 h M a N o B 6 T l G i e G o K J Z O Z 5 R A Z Y Y q J N b w X T i j P f w S J p H l e c k 4 p 9 Z e o 5 g y n y c A A l K I M D V a j B B d S h A Q R i u I d H e L L u r G f r x X q d r u a s 2 c 0 + / I H 1 9 g O q A 5 y R < / l a t e x i t > K 2 1 < l a t e x i t s h a 1 _ b a s e 6 4 = " O U e U K e + l I o b d s 6 D 6 x E X 6 i k d e j z A = " > A A A C B H i c b V C 7 S g N B F J 2 N r x h f U U u b I U F I F X Z j E c u g j W A T w T w g G 8 P s Z D Y Z M j u 7 z N w V l i W t n 2 B j o R 9 g J 7 b 2 f o J o 5 4 8 4 e R S a e O D C 4 Z x 7 O Z f j R Y J r s O 0 P K 7 O y u r a + k d 3 M b W 3 v 7 O 7 l 9 w + a O o w V Z Q 0 a i l C 1 P a K Z 4 J I 1 g I N g 7 U g x E n i C t b z R + c R v 3 T K l e S i v I Y l Y N y A D y X 1 O C R j J v b y p 9 F K X S x + S c S 9 f t M v 2 F H i Z O H N S r B V K X 5 / V 9 4 d 6 L / / t 9 k M a B 0 w C F U T r j m N H 0 E 2 J A k 4 F G + f c W L O I 0 B E Z s I 6 h k g R M d 9 P p z 2 N 8 b J Q + 9 k N l R g K e q r 8 v U h J o n Q S e 2 Q w I D P W i N x H / 9 T w v W I g G / 7 S b c h n F w C S d J f u x w B D i S S O 4 z x W j I B J D C F X c P I / p k C h C w f S W M 6 0 4 i x 0 s k 2 a l 7 J y U 7 S t T z x m a I Y u O U A G V k I O q q I Y u U B 0 1 E E U R

u k e P 6 M m 6 s 5 6 t F + t 1 t p q x 5 j e H 6 A + s t x + r o Z y S < / l a t e x i t > 

K vec 1 < l a t e x i t s h a 1 _ b a s e 6 4 = " 6 K 3 m C N D g h m e g 3 o p A x p s E W l s n F s k = " > A A A C H X i c b V C 7 S g N B F J 2 N r x h f U U u b R R G s w q 4 W s Q z a C D Y R T B T c G G Y n d 3 V w Z n a Z u R s M y 3 6 A 3 2 D h J 9 j q B 1 g I Y i u i n T / i J L H Q x A M D h 3 P O 5 d 4 5 Y S K 4 Q c 9 7 d w o T k 1 P T M 8 X Z 0 t z 8 w u J S e X m l a e J U M 2 i w W M T 6 N K Q G B F f Q Q I 4 C T h M N V I Y C T s K r / b 5 / 0 g V t e K y O s Z d A S 9 I L x S P O K F q p X d 4 I J M V L R k V 2 m L e z g K s I e / l 5 F i B c o 5 Z Z F 1 i e 2 5 R X 8 Q Z w x 4 n / Q z Z q 6 1 u f H 9 X n 2 3 q 7 / B V 0 Y p Z K U M g E N e b M 9 x J s Z V Q j Z w L y U p A a S C i 7 o h d w Z q m i E k w r G 3 w m d z e t 0 n G j W N u n 0 B 2 o v y c y K o 3 p y d A m + 6 e b U a 8 v / u u F o R x Z j d F u K + M q S R E U G 2 6 O U u F i 7 P a r c j t c A 0 P R s 4 Q y z e 3 x L r u k m j K 0 h Z Z s K / 5 o B + O k u V 3 x d y r e k a 1 n j w x R J G t k n W w R n 1 R J j R y Q O m k Q R m 7 I P X k g j 8 6 d 8 + S 8 O K / D a M H 5 m V k l f + C 8 f Q N y + K f T < / l a t e x i t > A K 0 < l a t e x i t s h a 1 _ b a s e 6 4 = " h 4 C + k v b R l W T U q o d P H M 9 c 9 E a / p z A = " > A A A C F X i c b V C 7 T g J B F J 3 F F + J r 1 Z J m I j G B h u x q o S V q Y 2 K D i T w S F s n s M A s T Z h + Z u W u C G w q / w F K / Q F v 9 A D t j a 2 3 t j z g L F A q e 5 C Y n 5 9 y b e + 9 x I 8 E V W N a X k V l Y X F p e y a 7 m 1 t Y 3 N r f M 7 Z 2 6 C m N J W Y 2 G I p R N l y g m e M B q w E G w Z i Q Z 8 V 3 B G u 7 g L P U b N 0 w q H g Z X M I x Y 2 y e 9 g H u c E t B S x 8 w 7 P o E + J S I 5 G T m C e V C 8 u L Y c y X t 9 K H X M g l W 2 x s D z x J 6 S Q q V 0 f x t l n x 6 r H f P b 6 Y Y 0 9 l k A V B C l W r Y V Q T s h E j g V b J R z Y s U i Q g e k x 1 q a B s R n q p 2 M n x j h f a 1 0 s R d K X Q H g s f p 7 I i G + U k P f 1 Z 3 p y W r W S 8 V / P d f 1 Z 1 a D d 9 x O e B D F w A I 6 2 e z F A k O I 0 4 h w l 0 t G Q Q w 1 I V R y f T y m f S I J B R 1 k T q d i z 2 Y w T + o H Z f u w b F 3 q e E 7 R B F m U R 3 u o i G x 0 h C r o H F V R D V F 0 h 5 7 R C 3 o 1 H o w 3 4 9 3 4 m L R m j O n M L v o D 4 / M H S 8 C i B w = = < / l a t e x i t > A < l a t e x i t s h a 1 _ b a s e 6 4 = " O 6 6 A T y 1 a w 1 t Y L Y U p v p J 8 F g N J C H 8 = " > A A A C A 3 i c b V C 7 S g N B F L 0 b X 3 F 9 R S 1 t B o N g F X a 1 0 E a M 2 l h G M A 9 I l j A 7 m S R D Z m e W m V k h L C n 9 B F s t L O 3 E 1 t p P E G t / x N k k h S Y e u H A 4 5 1 7 u 4 Y Q x Z 9 p 4 3 p e T W 1 h c W l 7 J r 7 p r 6 x u b W 4 X t n Z q W i S K 0 S i S X q h F i T T k T t G q Y 4 b Q R K 4 q j k N N 6 O L j K / P o d V Z p J c W u G M Q 0 i 3 B O s y w g 2 V m q 2 I m z 6 B P P 0 Y t Q u F L 2 S N w a a J / 6 U F M 8 / 3 L P 4 + d O t t A v f r Y 4 k S U S F I R x r 3 f S 9 2 A Q p V o Y R T k d u K 9 E 0 x m S A e 7 R p q c A R 1 U E 6 j j x C B 1 b p o K 5 U d o R B Y / X 3 R Y o j r Y d R a D e z i H r W y 8 R / v T C M Z l 6 b 7 m m Q M h E n h g o y + d x N O D I S Z Y W g D l O U G D 6 0 B B P F b H h E + l h h Y m x t r m 3 F n + 1 g n t S O S v 5 x y b v x i u V L m C A P e 7 A P h + D D C Z T h G i p Q B Q I S H u A R n p x 7 5 8 V 5 d d 4 m q z l n e r M L f + C 8 / w A 5 N 5 u k < / l a t e x i t > C < l a t e x i t s h a 1 _ b a s e 6 4 = " B p 3 s 0 N X a z I 8 d Q f K 7 i 5 e d A + N 1 h F s = " > A A A C A 3 i c b V C 7 S g N B F L 0 b X 3 F 9 R S 1 t B o N g F X a 1 0 E Y M p r G M Y B 6 Q L G F 2 M k m G z M w u M 7 N C W F L 6 C b Z a W N q J r b W f I N b + i L N J C k 0 8 c O F w z r 3 c w w l j z r T x v C 8 n t 7 S 8 s r q W X 3 c 3 N r e 2 d w q 7 e 3 U d J Y r Q G o l 4 p J o h 1 p Q z S W u G G U 6 b s a J Y h J w 2 w m E l 8 x t 3 V G k W y V s z i m k g c F + y H i P Y W K n V F t g M C O Z p Z d w p F L 2 S N w F a J P 6 M F C 8 / 3 I v 4 + d O t d g r f 7 W 5 E E k G l I R x r 3 f K 9 2 A Q p V o Y R T s d u O 9 E 0 x m S I + 7 R l q c S C 6 i C d R B 6 j I 6 t 0 U S 9 S d q R B E / X 3 R Y q F 1 i M R 2 s 0 s o p 7 3 M v F f L w z F 3 G v T O w 9 S J u P E U E m m n 3 s J R y Z C W S G o y x Q l h o 8 s w U Q x G x 6 R A V a Y G F u b a 1 v x 5 z t Y J P W T k n 9 a 8 m 6 8 Y v k K p s j D A R z C M f h w B m W 4 h i r U g E A E D / A I T 8 6 9 8 + K 8 O m / T 1 Z w z u 9 m H P 3 D e f w A 8 Y 5 u m < / l a t e x i t > < l a t e x i t s h a 1 _ b a s e 6 4 = " S 9 7 h A c 1 X q 6 o + h n y b y N 3 E F c 8 G O U A = " > A A A B / H i c b V C 7 S g N B F L 3 r M 8 Z X 1 F K R w S B Y h V 0 t t A z a W C Z g H p A s Y X Y y m w y Z m V 1 m Z o W w x M 7 W V j / A T m w l v 2 J t 6 U 8 4 m 6 T Q x A M X D u f c y 7 m c I O Z M G 9 f 9 d J a W V 1 b X 1 n M b + c 2 t 7 Z 3 d w t 5 + X U e J I r R G I h 6 p Z o A 1 5 U z S m m G G 0 2 a s K B Y B p 4 1 g c J P 5 j X u q N I v k n R n G 1 B e 4 J 1 n I C D a Z 1 I 7 7 r F M o u i V 3 A r R I v B k p l o / G 1 e / H 4 3 G l U / h q d y O S C C o N 4 V j r l u f G x k + x M o x w O s q 3 E 0 1 j T A a 4 R 1 u W S i y o 9 t P J r y N 0 a p U u C i N l R x o 0 U X 9 f p F h o P R S B 3 R T Y 9 P W 8 l 4 n / e k E g 5 q J N e O W n T M a J o Z J M k 8 O E I x O h r A n U Z Y o S w 4 e W Y K K Y f R 6 R P l a Y G N t X 3 r b i z X e w S O r n J e + i 5 F Z t P d c w R Q 4 O 4 Q T O w I N L K M M t V K A G B P

o + h n y b y N 3 E F c 8 G O U A = " > A A A B / H i c b V C 7 S g N B F L 3 r M 8 Z X 1 F K R w S B Y h V 0 t t A z a W C Z g H p A s Y X Y y m w y Z m V 1 m Z o W w x M 7 W V j / A T m w l v 2 J t 6 U 8 4 m 6 T Q x A M X D u f c y 7 m c I O Z M G 9 f 9 d J a W V 1 b X 1 n M b + c 2 t 7 Z 3 d w t 5 + X U e J I r R G I h 6 p Z o A 1 5 U z S m m G G 0 2 a s K B Y B p 4 1 g c J P 5 j X u q N I v k n R n G 1 B e 4 J 1 n I C D a Z 1 I 7 7 r F M o u i V 3 A r R I v B k p l o / G 1 e / H 4 3 G l U / h q d y O S C C o N 4 V j r l u f G x k + x M o x w O s q 3 E 0 1 j T A a 4 R 1 u W S i y o 9 t P J r y N 0 a p U u C i N l R x o 0 U X 9 f p F h o P R S B 3 R T Y 9 P W 8 l 4 n / e k E g 5 q J N e O W n T M a J o Z J M k 8 O E I x O h r A n U Z Y o S w 4 e W Y K K Y f R 6 R P l a Y G N t X 3 r b i z X e w S O r n J e + i 5 F Z t P d c w R Q 4 O 4 Q T O w I N L K M M t V K A G B P r w B M / w 4 j w 4 r 8 6 b 8 z 5 d X X J m N w f w B 8 7 H D 0 B 7 m P Y = < / l a t e x i t > X = y 0 < l a t e x i t s h a _ b a s e = " w f O K h z B i c d F S X E C o b + p D w = " > A A A C C X i c b V C S g N B F L r M Z H V i V G Q y C V d j V Q h s h a G O Z g H l A s o b Z y W w y Z G Z m Z k V l i W l l Z g q x g F z C m t L f L J o D E A x c O z L P R w / k x p x / m l p Z X V t f W c x v z a t n Y K u d X U S I J r Z G I R L p Y U C l N M p M Y U C / T h j + G f u N B y o V i I n c b U E g X s o A R r I U s Q t t g X W f Y J h f p v d O x i J m Q A t E n d G i u X D U f X W h U d h f W E E k F D T T h W q u U s f Y y L D U j n A z U T R G J M B t G W o S E W V H n Z J P g Q n R i l i J I m g k m q i / L z I s l E q F b z b H M d W N x b / X x f z L W w a W X s T B O N A J H O Q c K Q j N K F d Z m k R P P U E E w k M + E R W O J i T b l U r n w H i R + V n L P S V H M N U + T g A I h F F y g D L c Q g V q Q C C B Z i B V + v J e r N G v t d c m a e z D H g f P x U H n c I = < / l a t e x i t > z 0 < l a t e x i t s h a 1 _ b a s e 6 4 = " U 1 l 7 D t q 3 3 F g r R s 5 4 b T L L s j N b C y Q = " > A A A B + 3 i c b V C 7 S g N B F L 0 b X 3 F 9 R S 1 t B o N g F W a 1 0 E Y M 2 l h G N A 9 I 1 j A 7 m U 2 G z O w u M 7 N C X P I J t v o B 6 c T W b / A T x N o f c Z J Y a O K B C 4 d z 7 u V c T p A I r g 3 G n 0 5 u Y X F p e S W / 6 q 6 t b 2 x u F b Z 3 a j p O F W V V G o t Y N Q K i m e A R q x p u B G s k i h E Z C F Y P + p d j v 3 7 P l O Z x d G s G C f M l 6 U Y 8 5 J Q Y K 9 0 8 3 O F 2 o Y h L e A I 0 T 7 w f U j x / d 8 + S 0 Y d b a R e + W p 2 Y p p J F h g q i d d P D i f E z o g y n g g 3 d V q p Z Q m i f d F n T 0 o h I p v 1 s 8 u o Q H V i l g 8 J Y 2 Y k M m q i / L z I i t R 7 I w G 5 K Y n p 6 1 h u L / 3 p B I G e i T X j q Z z x K U s M i O k 0 O U 4 F M j M Z F o A 5 X j B o x s I R Q x e 3 z i P a I I t T Y u l z b i j f b w T y p H Z W 8 4 x K + x s X y B U y R h z 3 Y h 0 P w 4 A T K c A U V q A K F L j z C E z w 7 Q 2 f k v D i v 0 9 W c 8 3 O z C 3 / g v H 0 D m i 6 X 7 Q = = < / l a t e x i t > z 1 < l a t e x i t s h a 1 _ b a s e 6 4 = " V 7 A S t E 6 F L E t g S P N 8 7 a p V g z m L N D U = " > A A A B + 3 i c b V C 7 S g N B F L 0 b X 3 F 9 R S 1 t B o N g F X a 1 0 E Y M 2 l h G N A 9 I 1 j A 7 m U 2 G z M w u M 7 N C X P I J t v o B 6 c T W b / A T x N o f c Z J Y a O K B C 4 d z 7 u V c T p h w p o 3 n f T q 5 h c W l 5 Z X 8 q r u 2 v r G 5 V d j e q e k 4 V Y R W S c x j 1 Q i x p p x J W j X M c N p I F M U i 5 L Q e 9 i / H f v 2 e K s 1 i e W s G C Q 0 E 7 k o W M Y K N l W 4 e 7 v x 2 o e i V v A n Q P P F / S P H 8 3 T 1 L R h 9 u p V 3 4 a n V i k g o q D e F Y 6 6 b v J S b I s D K M c D p 0 W 6 m m C S Z 9 3 K V N S y U W V A f Z 5 N U h O r B K B 0 W x s i M N m q i / L z I s t B 6 I 0 G 4 K b H p 6 1 h u L / 3 p h K G a i T X Q a Z E w m q a G S T J O j l C M T o 3 E R q M M U J Y Y P L M F E M f s 8 I j 2 s M D G 2 L t e 2 4 s 9 2 M E 9 q R y X / u O R d e 8 X y B U y R h z 3 Y h 0 P w 4 Q T K c A U V q A K B L j z C E z w 7 Q 2 f k v D i v 0 9 W c 8 3 O z C 3 / g v H 0 D m 8 O X 7 g = = < / l a t e x i t > y 1 < l a t e x i t s h a 1 _ b a s e 6 4 = " 1 Y 9 j t I A c X c 6 W D 6 d G 1 4 l q r S Z U M k Y = " > A A A B + 3 i c b V C 7 S g N B F L 0 b X z G + o p a K D A b B K u x q o W X Q x j J B 8 4 B k D b O T S T J k Z n a Z m R W W J a W l r X 6 A n d g K + R V r S 3 / C y a P Q x A M X D u f c y 7 m c I O J M G 9 f 9 d D J L y y u r a 9 n 1 3 M b m 1 v Z O f n e v p s N Y E V o l I Q 9 V I 8 C a c i Z p 1 T D D a S N S F I u A 0 3 o w u B 7 7 9 Q e q N A v l n U k i 6 g v c k 6 z L C D Z W u k 3 u v X a + 4 B b d C d A i 8 W a k U D o c V b 4 f j 0 b l d v 6 r 1 Q l J L K g 0 h G O t m 5 4 b G T / F y j D C 6 T D X i j W N M B n g H m 1 a K r G g 2 k 8 n r w 7 R i V U 6 q B s q O 9 K g i f r 7 I s V C 6 0 Q E d l N g 0 9 f z 3 l j 8 1 w s C M R d t u p d + y m Q U G y r J N L k b c 2 R C N C 4 C d Z i i x P D E E k w U s 8 8 j 0 s c K E 2 P r y t l W v P k O F k n t r O i d F 9 2 K r e c K p s j C A R z D K X h w A S W 4 g T J U g U A P n u A Z X p y h 8 + q 8 O e / T 1 Y w z u 9 m H P 3 A + f g A u r p h X < / l a t e x i t > y 2 < l a t e x i t s h a 1 _ b a s e 6 4 = " M 1 N z n f h X n M F o s L H 4 H X e 5 0 7 v F G X w = " > A A A B + 3 i c b V C 7 S g N B F L 3 r M 8 Z X 1 F K R w S B Y h d 1 Y a B m 0 s U z Q P C B Z w + x k k g y Z m V 1 m Z o V l S W l p q x 9 g J 7 Z C f s X a 0 p 9 w 8 i g 0 8 c C F w z n 3 c i 4 n i D j T x n U / n a X l l d W 1 9 c x G d n N r e 2 c 3 t 7 d f 0 2 G s C K 2 S k I e q E W B N O Z O 0 a p j h t B E p i k X A a T 0 Y X I / 9 + g N V m o X y z i Q R 9 Q X u S d Z l B B s r 3 S b 3 x X Y u 7 x b c C d A i 8 W Y k X z o a V b 4 f j 0 f l d u 6 r 1 Q l J L K g 0 h G O t m 5 4 b G T / F y j D C 6 T D b i j W N M B n g H m 1 a K r G g 2 k 8 n r w 7 R q V U 6 q B s q O 9 K g i f r 7 I s V C 6 0 Q E d l N g 0 9 f z 3 l j 8 1 w s C M R d t u p d + y m Q U G y r J N L k b c 2 R C N C 4 C d Z i i x P D E E k w U s 8 8 j 0 s c K E 2 P r y t p W v P k O F k m t W P D O C 2 7 F 1 n M F U 2 T g E E 7 g D D y 4 g B L c Q B m q Q K A H T / A M L 8 7 Q e X X e n P f p 6 p I z u z m A P 3 A + f g A w Q 5 h Y < / l a t e x i t > z2 < l a t e x i t s h a 1 _ b a s e 6 4 = " 4 w e G o m N j p m J m W Z L 3 O 9 4 Y 1 O s Z s 9 o = " > A A A C A H i c b V C 7 S g N B F L 0 b X 3 F 9 R S 1 t B o N g F X Z j o Y 0 Y t L G M Y B 6 Q r G F m M p s M m Z 1 d Z m a F G N L 4 C b b a 2 d i J r T / g J 4 i 1 P + L k U W j i g Q u H c + 7 l X A 5 J B N f G 8 7 6 c z M L i 0 v J K d t V d W 9 / Y 3 M p t 7 1 R 1 n C r K K j Q W s a o T r J n g k l U M N 4 L V E 8 V w R A S r k d 7 F y K / d M q V 5 L K 9 N P 2 F B h D u S h 5 x i Y 6 V 6 k 2 C F 7 m 6 K r V z e K 3 h j o H n i T 0 n + 7 M M 9 T Z 4 / 3 X I r 9 9 1 s x z S N m D R U Y K 0 b v p e Y Y I C V 4 V S w o d t M N U s w 7 e E O a 1 g q c c R 0 M B j / O 0 Q H V m m j M F Z 2 p E F j 9 f f F A E d a 9 y N i N y N s u n r W G 4 n / e o R E M 9 E m P A k G X C a p Y Z J O k s N U I B O j U R u o z R W j R v Q t w V R x + z y i X a w w N b Y z 1 7 b i z 3 Y w T 6 r F g n 9 U 8 K 6 8 f O k c J s j C H u z D I f h w D C W 4 h D J U g I K A B 3 i E J + f e e X F e n b f J a s a Z 3 u z C H z j v P w x n m d I = < / l a t e x i t > A K 1 1 < l a t e x i t s h a 1 _ b a s e 6 4 = " q K h e j S f y k + 5 4 m P 4 3 K q Y 2 N v y x d M 8 = " > A A A C H H i c b V A 9 S w N B E N 2 L X z F + R S 1 t j g Q h a c K d F l p G b Q S b C O Y D c v H Y 2 + w l S / b 2 j t 0 5 I R 7 p 7 W y t L C 1 t 9 Q f Y i a 1 g 7 R 9 x L 0 m h i Q 8 G H u / N M D P P i z h T Y F l f R m Z h c W l 5 J b u a W 1 v f 2 N z K b + 8 0 V B h L Q u s k 5 K F s e V h R z g S t A w N O W 5 G k O P A 4 b X q D s 9 R v 3 l C p W C i u Y B j R T o B 7 g v m M Y N C S m y 8 4 A Y Y + w T w 5 G T m c + l C 6 u L Z d h w k f h o 5 k v T 6 U 3 X z R q l h j m P P E n p J i t X x / G 2 W f H m p u / t v p h i Q O q A D C s V J t 2 4 q g k 2 A J j H A 6 y j m x o h E m A 9 y j b U 0 F D q j q J O N f R u a + V r q m H 0 p d A s y x + n s i w Y F S w 8 D T n e n l a t Z L x X 8 9 z w t m V o N / 3 E m Y i G K g g k w 2 + z E 3 I T T T p M w u k 5 Q A H 2 q C i W T 6 e J P 0 s c Q E d J 4 5 n Y o 9 m 8 E 8 a R x U 7 M O K d a n j O U U T Z N E e K q A S s t E R q q J z V E N 1 R N A d e k Y v 6 N V 4 N N 6 M d + N j 0 p o x p j O 7 6 A + M z x 8 3 4 K U z < / l a t e x i t >

Figure 3 : Visualization of sample finite neural networks and their infinitely wide Bayesian counterparts (NN-GPs). Presented are networks with nonlinearity φ, L = 2 hidden layers that regress n3 = 10-dimensional outputs z2 (x) for each of the 4 (1, 2, 3, 4) inputs x from the dataset X . A hierarchical computation performed by a NN corresponds to a hierarchical transformation of the respective covariance matrix (resulting in K vec ∞ ⊗ I 10 ). Top two rows: a fully connected network (FCN, above) and the respective FCN-GP (below). Bottom two rows: a 1D CNN with no (valid) padding, and the respective CNN-GP. The analogy between the two models (NN and GP) is qualitatively similar to the case of FCN, modulo additional spatial dimensions in the internal activations and the respective additional entries in intermediary covariance matrices. Note that in general d = 10 pixels would induce a 10 × 10-fold increase in the number of covariance entries (in K 0 , K 1 ∞ , . . . ) compared to the respective FCN, yet without pooling only a small fraction of them (displayed) needs be computed to obtain the top-layer GP covariance K vec ∞ ⊗ I 10 (see §3.1).

3. TRANSFORMING A GP OVER SPATIAL LOCATIONS INTO A GP OVER:



CLASSES:

In §2.2 we showed that in the infinite channel limit a deep CNN is a GP indexed by input samples, output spatial locations, and output channel indices. Further, its covariance matrix K L ∞ can be computed in closed form. Here we show that transformations to obtain class predictions that are common in CNN classifiers can be represented as either vectorization or projection (as long as we treat classification as regression (see §G), identically to Lee et al. (2018) ). Both of these operations preserve the GP equivalence and allow the computation of the covariance matrix of the respective GP (now indexed by input samples and target classes) as a simple transformation of K L ∞ . In the following we will denote nL+2 as the number of target classes, zL+1 (x) ∈ R nL+2 as the outputs of the CNN-based classifier, and ωL+1 , bL+1 as the last (number L + 1) layer weights and biases. For each class i we will denote the empirical uncentered covariance of the outputs as

EQUATION

a random variable of shape |X | × |X |. We will show that in the scenarios considered below in §3.1 and §3.2, the outputs zL+1 i |y 0 will converge in distribution to a multivariate Gaussian (i.i.d.

for every class i) as min{n 1 , . . . , n L+1 } → ∞ with covariance denoted as

K ∞ ∈ R |X |×|X | , or, jointly, zL+1 |y 0 D → N (0, K ∞ ⊗ I nL+2 ).

As in §2.2, we postpone the formal derivation until §E.4 (Theorem E.6).

3.1. VECTORIZATION:

One common readout strategy is to vectorize (flatten) the output of the last convolutional layer z L (x) ∈ R n L+1 d into a vector4 vec z L (x) ∈ R n L+1 d and stack a fully connected layer on top:

EQUATION

where the weights ωL+1 ∈ R nL+2 ×n L+1 d and biases bL+1 ∈ R nL+2 are i.i.d. Gaussian,

EQUATION

The argument of §2.2 can then be extended to conclude that in the limit of infinite width zL+1 i |y 0 converges in distribution to a multivariate Gaussian (i.i.d. for each class i) with covariance

EQUATION

A sample 2D CNN using this readout strategy is depicted in Figure 2 , and a sample correspondence between a FCN, FCN-GP, CNN, and CNN-GP is depicted in Figure 3 .

Note that as observed in Xiao et al. (2018) , to compute the summands K L+1 ∞ α,α (x, x ) in Equation 13, one needs only the corresponding terms K L ∞ α,α (x, x ). Consequently, we only need to compute K l ∞ α,α (x, x ) : x, x ∈ X , α ∈ {1 . . . d} (or O (d) per covariance entry in an iterative or distributed setting). Note that this approach ignores pixel-pixel covariances and produces a GP corresponding to a locally connected network (see §5.1).

3.2. PROJECTION:

Another readout approach is a projection to collapse the spatial dimensions. Let h ∈ R d be a deterministic projection vector, ωL+1 ij ∼ N 0, σ 2 ω /n L+1 , and bL+1 be the same as in §3.1. Define the output to be

EQUATION

Examples of this approach include 1. Global average pooling: ) is consistently better than other models due to translation invariance of the kernel. CNN-GP with zero padding ( §3.1) outperforms an analogous CNN-GP without padding as depth increases. At depth 15 the spatial dimension of the output without padding is reduced to 1 × 1, making the CNN-GP without padding equivalent to the center pixel selection strategy ( §3.2.2) -which also performs worse than the CNN-GP (we conjecture, due to overfitting to centrally-located features) but approaches the latter (right) in the limit of large depth, as information becomes more uniformly spatially distributed (Xiao et al., 2018) . CNN-GPs generally outperform FCN-GP, presumably due to the local connectivity prior, but can fail to capture nonlinear interactions between spatially-distant pixels at shallow depths (left). Values are reported on a 2K/4K train/validation subset of CIFAR10. See §G.3 for experimental details.

EQUATION

This approach corresponds to applying global average pooling right after the last convolutional layer. 5 This approach takes all pixel-pixel covariances into consideration and makes the kernel translation invariant. However, it requires O |X | 2 d 2 memory to compute the sample-sample covariance of the GP (or O d 2 per covariance entry in an iterative or distributed setting). It is impractical to use this method to analytically evaluate the GP, and we propose to use a Monte Carlo approach (see §4).

2.. Subsampling one particular pixel: take:

EQUATION

This approach makes use of only one pixel-pixel covariance, and requires the same amount of memory as vectorization ( §3.1) to compute.

We compare the performance of presented strategies in Figure 4 . Note that all described strategies admit stacking additional FC layers on top while retaining the GP equivalence, using a derivation analogous to §2.2 (Lee et al., 2018; Matthews et al., 2018b) .

We introduce a Monte Carlo estimation method for NN-GP kernels which are computationally impractical to compute analytically, or for which we do not know the analytic form. Similar in spirit to traditional random feature methods (Rahimi & Recht, 2007) , the core idea is to instantiate many random finite width networks and use the empirical uncentered covariances of activations to estimate the Monte Carlo-GP (MC-GP) kernel, where θ consists of M draws of the weights and biases from their prior distribution, θ m ∼ p (θ), and n is the width or number of channels in hidden layers. The MC-GP kernel converges to the analytic kernel with increasing width, lim n→∞ K l n,M = K l ∞ in probability. For finite width networks, the uncertainty in Daniely et al. (2016, Theorems 2, 3) , we know that for well-behaved nonlinearities (that include ReLU and erf considered in our experiments)

EQUATION

K l n,M is Var K l n,M = Var θ K l n (θ) /M . From

Var θ K l n (θ) ∝ 1 n , which leads to Var θ [K l n,M ] ∝ 1 M n . For finite n, K l n,M is also a biased estimate of K l ∞ ,

where the bias depends solely on network width. We do not currently have an analytic form for this bias, but we can see in Figures 5 and 10 that for the hyperparameters we probe it is small relative to the variance. In particular,

K l n,M (θ) -K L ∞ 2

F is nearly constant for constant M n. We thus treat M n as the effective sample size for the Monte Carlo kernel estimate. Increasing M and reducing n can reduce memory cost, though potentially at the expense of increased compute time and bias.

In a non-distributed setting, the MC-GP reduces the memory requirements to compute GP pool from

O |X | 2 d 2 to O |X | 2 + n 2 +

nd , making the evaluation of CNN-GPs with pooling practical.

5. DISCUSSION:



5.1. BAYESIAN CNNS WITH MANY CHANNELS ARE IDENTICAL TO LOCALLY CONNECTED NETWORKS, IN THE ABSENCE OF POOLING:

Locally Connected Networks (LCNs) (Fukushima, 1975; Lecun, 1989) are CNNs without weight sharing between spatial locations. LCNs preserve the connectivity pattern, and thus topology, of a CNN. However, they do not possess the equivariance property of a CNN -if an input is translated, the latent representation in an LCN will be completely different, rather than also being translated.

The CNN-GP predictions without spatial pooling in §3.1 and §3.2.2 depend only on sample-sample covariances, and do not depend on pixel-pixel covariances. LCNs destroy pixel-pixel covariances:

K L ∞ LCN α,α (x,

x ) = 0, for α = α and all x, x ∈ X and L > 0. However, LCNs preserve the covariances between input examples at every pixel:

K L ∞ LCN α,α (x, x ) = K L ∞ CNN α,α (x, x ).

As a result, in the absence of pooling, LCN-GPs and CNN-GPs are identical. Moreover, LCN-GPs with pooling are identical to CNN-GPs with vectorization of the top layer (under suitable scaling of y L+1 ). We confirm these findings experimentally in trained networks in the limit of large width in Figures 1 and 6 (b), as well as by demonstrating convergence of MC-GPs of the respective architectures to the same CNN-GP (modulo scaling of y L+1 ) in Figures 5 and 10 .

5.2. POOLING LEVERAGES EQUIVARIANCE TO PROVIDE INVARIANCE:

The only kernel leveraging pixel-pixel covariances is that of the CNN-GP with pooling. This enables the predictions of this GP and the corresponding CNN to be invariant to translations (modulo edge effects) -a beneficial quality for an image classifier. We observe strong experimental evidence supporting the benefits of invariance throughout this work (Figures 1, 4, 5, 6 (b) ; Table 1 ), in both CNNs and CNN-GPs. In the absence of pooling, the benefits of equivariance and weight sharing are more challenging to explain in terms of Bayesian priors on class predictions (since without pooling equivariance is not a property of the outputs, but only of intermediary representations). Indeed, in this work we find that the performance of finite width SGD-trained CNNs often approaches that of their CNN-GP counterpart (Figure 6 , b, c), 6 suggesting that in those cases equivariance does not play a beneficial role in SGD-trained networks.

However, as can be seen in Figures 1, 6 (c), 7, and Table 1 , the best CNN overall outperforms the best CNN-GP by a significant margin -an observation specific to CNNs and not FCNs or LCNs. 7 We observe this gap in performance especially in the case of ReLU networks trained with a large learning rate. In Figure 1 we demonstrate this large gap in performance by evaluating different models with equivalent architecure and hyperparameter settings, chosen for good SGD-trained CNN performance.

We conjecture that equivariance, a property lacking in LCNs and the Bayesian treatment of the infinite channel CNN limit, contributes to the performance of SGD-trained finite channel CNNs with the correct settings of hyperparameters. Nonetheless, more work is needed to disentangle and quantify the separate contributions of stochastic optimization and finite width effects, 8 and differences in performance between CNNs with weight sharing and their corresponding CNN-GPs.

6. CONCLUSION:

In this work we have derived a Gaussian process that corresponds to fully Bayesian multi-layer CNNs with infinitely many channels. The covariance of this GP can be efficiently computed either in closed form or by using Monte Carlo sampling, depending on the architecture.

The CNN-GP achieves state of the art results for GPs without trainable kernels on CIFAR10. It can perform competitively with CNNs (that fit the training set) of equivalent architecture and weight priors, which makes it an appealing choice for small datasets, as it eliminates all training-related hyperparameters. However, we found that the best overall performance, at least in the absence of pooling, is achieved by finite SGD-trained CNNs and not by their infinite Bayesian counterparts. We hope our work stimulates future research towards understanding the distribution over functions induced by model architecture and training approach, and what aspects of this distribution are important for model performance.

Another natural extension of our work is the study of other deep learning architectures in the infinitely wide limit. After the publication of this paper, Yang (2019) devised a unifying framework proving the GP convergence for even more models (such as ones using batch normalization, (self-)attention, LSTM) with slightly different assumptions on the nonlinearity.

6 This observation is conditioned on the respective NN fitting the training set to 100%. Underfitting breaks the correspondance to an NN-GP, since train set predictions of such a network no longer correspond to the true training labels. Properly tuned underfitting often also leads to better generalization (Table 1 ). 7 Performing an analogous large-dataset comparison between CNNs and CNN-GPs with pooling was not computationally feasible. Their relative performance remains an interesting open question for future research. 8 We remark that concerns about GPs not being able to learn hierarchical representations have been raised in the literature (Matthews et al., 2018a, Section 7) , (Neal, 1995 , Chapter 5), (MacKay, 2003, Section 45.7) However, practical impact of these assertions have not been extensively investigated empirically or theoretically, and we hope that our work stimulates research in this direction. Each line corresponds to a particular choice of architecture and initialization hyperparameters, with best learning rate and weight decay selected independently for each number of channels (x-axis).

(b): SGD-trained CNNs often approach the performance of their corresponding CNN-GP with increasing number of channels. All models have the same architecture except for pooling and weight sharing, as well as training-related hyperparameters such as learning rate, weight decay and batch size, which are selected for each number of channels (x-axis) to maximize validation performance (y-axis) of a neural network. As the number of channels grows, best validation accuracy increases and approaches accuracy of the respective GP (solid horizontal line). (c): However, the best-performing SGD-trained CNNs can outperform their corresponding CNN-GPs. Each point corresponds to the test accuracy of: (y-axis) a specific CNN-GP; (x-axis) the best (on validation) CNN with the same architectural hyper-parameters selected among the 100%-accurate models on the full training CIFAR10 dataset with different learning rates, weight decay and number of channels. While CNN-GP appears competitive against 100%-accurate CNNs (above the diagonal), the best CNNs overall outperform CNN-GPs by a significant margin (below the diagonal, right means only models that achieved 100% training accuracy were considered). CNN-GP outperforms SGD-trained models optimized with a small learning rate to 100% train accuracy. When SGD optimization is allowed to underfit the training set, there is a significant improvement in generalization. Further, when ReLU nonlinearities are paired with large learning rates, the performance of SGD-trained models again improves relative to CNN-GPs, suggesting a beneficial interplay between ReLUs and fast SGD training. These differences in performance between CNNs and CNN-GPs are not observed between FCNs and FCN-GPs, or between LCNs and LCN-GPs (Figure 1 ), suggesting that equivariance is the underlying factor responsible for the improved performance of finite SGD-trained CNNs relative to infinite Bayesian CNNs without pooling. Further comparison between SGD-trained CNNs and CNN-GPs with pooling (omitted due to computational limitations), where both models do share the property of invariance, would be an interesting direction for future research. See 

#Channels:

Figure 9 : No underfitting in small models. Training loss (vertical axis) of best (in terms of validation loss) neural networks as the number of channels increases (horizontal axis). While perfect 0 loss is not achieved (but 100% accuracy is), we observe no consistent improvement when increasing the capacity of the network (left to right). This eliminates underfitting as a possible explanation for why small models perform worse in Figure 6 

B RELATIONSHIP TO DEEP SIGNAL PROPAGATION:

The recurrence relation linking the GP kernel at layer l+1 to that of layer l following from Equation 9(i.e.

K l+1 ∞ = (C • A) K l ∞

) is precisely the covariance map examined in a series of related papers on signal propagation (Xiao et al., 2018; Poole et al., 2016; Schoenholz et al., 2017; Lee et al., 2018) (modulo notational differences; denoted as F , C or e.g. A C in Xiao et al. (2018) ). In those works, the action of this map on hidden-state covariance matrices was interpreted as defining a dynamical system whose large-depth behavior informs aspects of trainability. In particular, as

l → ∞, K l+1 ∞ = (C • A) K l ∞ ≈ K l ∞ ≡ K * ∞ , i.e.

the covariance approaches a fixed point K * ∞ . The convergence to a fixed point is problematic for learning because the hidden states no longer contain information that can distinguish different pairs of inputs. It is similarly problematic for GPs, as the kernel becomes pathological as it approaches a fixed point. Precisely, in both chaotic and ordered regimes, outputs of the GP become asymptically identically correlated. Either of these scenarios captures no information about the training data in the kernel and makes learning infeasible. This problem can be ameliorated by judicious hyperparameter selection, which can reduce the rate of exponential convergence to the fixed point. For hyperpameters chosen on a critical line separating two untrainable phases, the convergence rates slow to polynomial, and very deep networks can be trained, and inference with deep NN-GP kernels can be performed -see Figure 11 .

C STRIDED CONVOLUTIONS, AVERAGE POOLING IN INTERMEDIATE LAYERS, HIGHER DIMENSIONS:

Our analysis in the main text can easily be extended to cover average pooling and strided convolutions (applied before the pointwise nonlinearity). Recall that conditioned on K l the pre-activation z l j (x) ∈ R d1 is a zero-mean multivariate Gaussian. Let B ∈ R d2×d1 denote a linear operator. Then Bz l j (x) ∈ R d2 is a zero-mean Gaussian, and the covariance is

EQUATION

One can easily see that Bz l j K l j are i.i.d. multivariate Gaussian as well. Strided convolution. Strided convolution is equivalent to a non-strided convolution composed with subsampling. Let s ∈ N denote size of the stride. Then the strided convolution is equivalent to choosing B as follows:

B ij = δ(is -j) for i ∈ {0, 1, . . . (d 2 -1)}.

Average pooling. Average pooling with stride s and window size ws is equivalent to choosing B ij = 1/ws for i = 0, 1, . . . (d 2 -1) and j = is, . . . , (is + ws -1).

ND convolutions. Note that our analysis in the main text (1D) easily extends to higher-dimensional convolutions by replacing integer pixel indices and sizes d, α, β with tuples (see also Figure 2 ). In Equation 2 

EQUATION

EQUATION

We use the shorthand K (X , X ) to denote the 

E EQUIVALENCE BETWEEN RANDOMLY INITIALIZED NNS AND GPS:

In this section, we present two different approaches, the sequential limit ( §E.3) and simultaneous limit ( §E.4), to illustrate the relationship between many-channels Bayesian CNNs and GPs.

Sequential limit ( §E.3) involves taking the infinite channel limit in hidden layers in a sequence, starting from bottom (closest to inputs) layers and going upwards (to the outputs), i.e. n 1 → ∞, . . . , n L → ∞. Note that this approach in fact only gives intuition into construction a GP using a NN architecture to define its covariance, and does not provide guarantees on actual convergence of large but finite Bayesian CNNs to GPs (which is of most practical interest), nor does it guarantee the existence of the specified GP on a given probability space. However, it has the following benefits:

1. Weak assumptions on the NN activation function φ and on the distribution of the NN parameters. 2. The arguments can be easily extended to more complicated network architectures, e.g. architectures with max pooling, dropout, etc. 3. A straightforward and intuitive way to compute the covariance of the Gaussian process without diving into mathematical details.

Simultaneous limit ( §E.4) considers growing the number of channels in hidden layers uniformly, i.e. min n 1 , . . . , n L → ∞. This approach establishes convergence of finite channel Bayesian CNNs to GPs and is thus a more practically relevant result. However, it makes stronger assumptions, and the proof is more involved.

We highlight that the GPs obtained by the two approaches are identical.

In both sections, we only provide the arguments for CNNs. It is straightforward (and in fact simpler) to extend them to LCNs and FCNs. Indeed, an FCN is a particular case of a CNN where the inputs and filters have singular spatial dimensions (d = 1, k = 0). For LCNs, the proof goes through in an identical fashion if we replace A with A LCN defined as

A LCN (K) α,α (x, x ) ≡ δ α,α [A (K)] α,α (x, x ).

E.1 SETUP Probability space. Let P be a collection of countably many mutually independent random variables (R.V.s) defined on a probability space (Ω, F, P ), where F is a product Borel σ-algebra and P is the probability measure. Here P ≡ W ∪ B ∪ H is the collection of parameters used to define neural networks: Inputs. We will consider a fixed X ⊆ R n 0 ×d to denote the inputs, with input channel count n 0 , number of pixels d. Assume x = 0 for all x ∈ X and |X |, the cardinality of the inputs, is finite.

(i) Weights. W = l∈N W l and W l = ω l ij,β : i, j ∈ N, β ∈ [±k] , where [±k] ≡ [-k, k] ∩ Z. We assume ω l ij,

However, our results can be straightforwardly extended to a countably-infinite input indexing spaces X for certain topologies via an argument presented in Matthews et al. (2018a, section 2.2), allowing to infer weak convergence on X from convergence on any finite subset (which is the case we consider in this text; see also Billingsley (1999, page 19 ) for details). For this reason, as in Matthews et al.

(2018a), weak convergence of countably-infinite stochastic processes will be considered with respect to the topology generated by the following metric:

ν (s, s ) ≡ ∞ k=1 2 -k min (1, |s k -s k |) , ∀s, s ∈ R N .

Notation, shapes, and indexing. We adopt the notation, shape, and indexing convention similar to §2.1, which the reader is encouraged to review. We emphasize that whenever an index is omitted, the variable is assumed to contain all possible entries along the respective dimension (e.g. whole X if x is omitted, or all n l channels if the channel i is omitted).

E.2 PRELIMINARY:

We will use the following well-known theorem. Theorem E.1. Let X, {X n } n∈N be R.V.s in R m . The following are equivalent:

EQUATION

(iii) (Lévy's Continuity Theorem) The characteristic functions of X n , i.e. E e it T Xn , converge to those of X pointwise, i.e. for all t ∈ R m ,

EQUATION

where i denotes the imaginary unit.

Using the equivalence between (i) and (iii), it is straightforward to show that Theorem E.2 (Cramer-Wold, (Billingsley (1995) , Theorem 29.4)).

EQUATION

In particular,

EQUATION

E.3 SEQUENTIAL LIMIT:

In this section, we give intuition into constructing a Gaussian process using an infinite channel CNN with the limits taken sequentially. Informally, our argument amounts to showing that if the inputs z l-1 to the layer l are a multivariate Gaussian with covariance A K l ⊗ I n l , then it's outputs z l converge in distribution to a multivariate Gaussian with covariance A K l+1 ⊗ I n l+1 as n l → ∞. This "allows" us to sequentially replace z 1 , z 2 , . . . , z L with their respective limiting Gaussian R.V.s as we take n 1 → ∞, n 2 → ∞, . . . , n L → ∞. However, since each convergence only holds in distribution and does not guarantee the existence of the necessary GPs on a given probability space, what follows merely gives intuition into understanding the relationship between wide Bayesian neural networks and GPs (contrary to §E.4, which presents a rigorous convergence proof for min n 1 , . . . , n L → ∞).

Let Ψ 1 denote the space of functions with a uniformly bounded second moment, i.e. having

EQUATION

Let N * = N\{0}. We construct Gaussian processes with the following iterative in l (from 0 to L) procedure:

(i) If l > 0, define random variables (a GP) z l,∞ i i∈N * in (Ω, F, P ) as i.i.d. Gaussian with mean zero and covariance A K l ∞ , so that they are also independent from any future events, i.e. independent from the σ-algebra generated by all R.V.s with layer index greater than l. Here we implicitly assume the probability space (Ω, F, P ) has enough capacity to fit in the R.V.s generated by the above procedure, if not we will extend the sample space Ω using product measures.

(ii) For n ∈ N * , define In what follows, we use the central limit theorem to prove (iii). Proof. We proceed by induction. This is obvious in the base case l = 0, since the weights and biases are assumed to be independent Gaussian. Now we assume the theorem holds for l -1. This implies u j + q.

EQUATION

z l,n i,α (x) ≡    1 √ n 0 j∈[n 0 ] β∈[±k] √ v β ω l ij,

(33)

It is not difficult to see that u j are i.i.d. and q is Gaussian. Then we can use the central limit theorem to conclude that the above converges in distribution to a Gaussian, once we verify the second moment of u j is finite. Using the fact that ω l ij,β i,β is a collection of independent R.V.s, and integrating over these R.V.s first, we get 

EQUATION

)

where by Ā we denote the linear transformation part of A from Equation 5, i.e. A without the translation term σ 2 b :

EQUATION

To prove finiteness of the second moment in Equation 36, it is sufficient to show that all the diagonal terms of K l ∞ are finite. This easily follows from the assumption φ ∈ Ψ 1 and the definition of

K l ∞ = (C • A) l K 0 .

Together with the distribution of v (whose covariance is straightforward to compute), the joint distribution of z l,n i i∈ [m] converges weakly to a mean zero Gaussian with covariance matrix A K l ∞ ⊗ I m by Theorem E.1 (Equation 26). Remark E.1. The results of Theorem E.3 can be strengthened / extended in many directions.

(1) The same result for a countably-infinite input index set X follows immediately according to the respective remark in §E.1.

(2) The same analysis carries over (and the covariance matrix can be computed without much extra effort) if we stack a channel-wise deterministic affine transform after the convolution operator. Note that average pooling (not max pooling, which is not affine), global average pooling and the convolutional striding are particular examples of such affine tranforms. Moreover, valid padding (i.e. no padding) convolution can be regarded as a subsampling operator (a linear projection) composed with the regular circular padding.

(3) The same analysis applies to max pooling, but computing the covariance may require non-trivial effort. Let m denote the max pooling operator and assume it is applied right after the activation function.

The assumption y l i i∈[n] = φ(z l-1,∞ i ) i∈ [n] are i.i.d. implies m y l i i∈[n] are also i.i.d. Then we can proceed exactly as above except for verifying the finiteness of second moment of m(y l i ) with the following trivial estimate:

EQUATION

)

where s is the window size of the max pooling.

In general, one can stack a channel-wise deterministic operator op on y l i so long as the second moment of op y l i is finite. One can also stack a stochastic operator (e.g. dropout), so long as the outputs are still channel-wisely i.i.d. and have finite second moments.

E.4 SIMULTANEOUS LIMIT:

In this section, we present a sufficient condition on the activation function φ so that the neural networks converge to a Gaussian process as all the widths approach infinity simultaneously. Precisely, let t ∈ N and for each l ≥ 0, let n l : N → N be the width function at layer l (by convention n 0 (t) = n 0 is constant). We are interested in the simultaneous limit n l = n l (t) → ∞ as t → ∞, i.e., for any fixed L ≥ 1 min n 1 (t), . . . , n L (t) ---→ t→∞ ∞.

(39)

Define a sequence of finite channel CNNs as follows:

y l,t i,α (x) ≡     

x i,α , l = 0 φ z l-1,t i,α (x) , l > 0 , √ v β ω l ij,β y l,t j,α+β (x) + b l i , l > 0

EQUATION

)

This network induces a sequence of covariance matrices K l t (which are R.V.s): for l ≥ 0 and t ≥ 0, for x, x ∈ X K l t α,α (x, x ) ≡ 1 n l (t) n l (t) i=1 y l,t i,α (x)y l,t i,α (x ).

We make an extra assumption on the parameters.

Assumption: all R.V.s in W are Gaussian distributed. Let Ψ denotes the space of measurable functions φ with the following properties:

1. Uniformly bounded second moment: for every R ≥ 1, C 2 (φ, R) < ∞.

2. Lipschitz continuity: for every R ≥ 1, there exists β = β (φ, R) > 0 such that for all Σ, Σ ∈ PSD 2 (R),

EQUATION

3. Uniform convergence in probability: for every R ≥ 1 and every ε > 0 there exists a positive sequence ρ n (φ, ε, R) with ρ n (φ, ε, R) → 0 as n → ∞ such that for every Σ ∈ PSD 2 (R) and any {(x i , y i )} n i=1 i.i.d. ∼ N (0, Σ)

EQUATION

)

We will also use Ψ 1 , Ψ 2 and Ψ 3 to denote the spaces of measurable functions φ satisfying properties 1, 2, and 3, respectively. It is not difficult to see that for every i, Ψ i is a vector space, and so is , the exponent in the above expression is just a linear combination of independent Gaussian R.V.s ω l ij,β , b l i , which is also a Gaussian. Integrating out these

Ψ = ∩ i Ψ i .

Using the fact Eω ij,α ω i j ,α = 0 unless (ij, α) = (i j , α ) and integrating out the R.V.s of the dense layer, the characteristic function is equal to We first note that the affine transform A is σ 2 ω -Lipschitz and property 2 of Ψ implies that the C operator is β-Lipschitz (both w.r.t. w.r.t. • ∞ ). Indeed, if we consider Σ ≡

E exp      - 1 2 i∈[m] x,x ∈X a i (x)a i (x )      1 n l (t)d

[K] α,α (x, x)

[K] α,α (x, x ) [K] α ,α (x , x) [K] α ,α (x , x ) ,

then [C (K)] α,α (x, x ) = T ∞ (Σ). Thus C • A is σ 2 ω β-Lipschitz. We now prove the theorem by induction. Assume K l t P → K l ∞ as t → ∞ (obvious for l = 0). We first remark that K l ∞ ∈ PSD |X |d , since PSD |X |d E K l t → K l ∞ and PSD |X |d is closed. Moreover, due to Equation 4, A necessarily preserves positive semi-definiteness, and therefore A K l ∞ ∈ PSD |X |d as well. Now let ε > 0 be sufficiently small so that the ε 2β -neighborhood of A(K l ∞ ) is contained in PSD |X |d (R), where we take R to be large enough for K l ∞ to be an interior point of PSD |X |d (R).9 Since

EQUATION

to prove K l+1 t P → K l+1 ∞ , it suffices to show that for every δ > 0, there is a t * such that for all t > t * ,

EQUATION

By our induction assumption, there is a t l such that for all t > t l

EQUATION

)

Remark E.4. In practice the 1/n decay bound obtained by Chebyshev's inequality in Equation 100is often too weak to be useful. However, if φ is linearly bounded, then one can obtain an exponential decay bound via the following concentration inequality: Lemma E.8. If |φ(x)| ≤ a + b|x| a.e., then there is an absolute constant c > 0 and a constant κ = κ(a, b, R) > 0 such that property 3 (Equation 48) holds with 

We can apply Bernstein-type inequality (Vershynin, 2010, Lemma 5.16 ) to conclude that there is a c > 0 such that for every Σ ∈ PSD 2 (R) and any {(x i , y i )} n i=1 i.i.d. ∼ N (0, Σ)

EQUATION

It remains to prove Claim E.9. For p ≥ 1, 

E (x,

We applied Cauchy-Schwarz' inequality in the first inequality, the triangle inequality in the second one, the fact Σ 11 , Σ 22 ≤ R in the third one, absolute moments estimate of standard Gaussian in the fourth one, where c is a constant such that

EQUATION

F GLOSSARY

We use the following shorthands in this work:

1. NN -neural network; 2. CNN -convolutional neural network; 3. LCN -locally connected network, a.k.a. convolutional network without weight sharing; 4. FCN -fully connected network, a.k.a. multilayer perceptron (MLP); 5. GP -Gaussian process; 6. X-GP -a GP equivalent to a Bayesian infinitely wide neural network of architecture X ( §2).

:

We will use Roman letters to index channels and Greek letters for spatial location. We use letters i, j, i , j , etc to denote channel indices, α, α , etc to denote spatial indices and β, β , etc for filter indices.

The weak law of large numbers allows convergence in probability of individual entries of K l . However, due to the finite dimensionality of K l , joint convergence in probability follows.

Note that since per our notation described in §2.1 z L (x) is already considered a 1D vector, here this operation simply amounts to re-indexing z L (x) with one channel index j instead of two (channel i and pixel α).

Spatially local average pooling in intermediary layers can be constructed in a similar fashion ( §C). We focus on global average pooling in this work to more effectively isolate the effects of pooling from other aspects of the model like local connectivity or equivariance.

Such R always exists, because the diagonal terms of K l ∞ are always non-zero. See(Long & Sedghi, 2019, Lemma 4) for proof.

Due to time and compute limitations certain large configurations could not be evaluated. We believe this did not impact the results of this work in a qualitative way.

Figures and Tables:

Figure 1: Disentangling the role of network topology, equivariance, and invariance on test performance, for SGD-trained and infinitely wide Bayesian networks (NN-GPs). Accuracy (%) on CIFAR10 of different models of the same depth, nonlinearity, and weight and bias variances. (a) Fully connected models -FCN (fully connected network) and FCN-GP (infinitely wide Bayesian FCN) underperform compared to (b) LCNs (locally connected network, a CNN without weight sharing) and CNN-GP (infinitely wide Bayesian CNN), which have a hierarchical local topology beneficial for image recognition. As derived in §5.1: (i) weight sharing has no effect in the Bayesian treatment of an infinite width CNN (CNN-GP performs similarly to an LCN), and (ii) pooling has no effect on generalization of an LCN model (LCN and LCN with pooling perform nearly identically). (c) Local connectivity combined with equivariance (CNN) is enabled by weight sharing in an

Figure1: Different dimensionality collapsing strategies described in §3. Validation accuracy of an MC-CNN-GP with pooling ( §3.2.1) is consistently better than other models due to translation invariance of the kernel. CNN-GP with zero padding ( §3.1) outperforms an analogous CNN-GP without padding as depth increases. At depth 15 the spatial dimension of the output without padding is reduced to 1 ⇥ 1, making the CNN-GP without padding equivalent to the center pixel selection strategy ( §3.2.2) -which also performs worse than the CNN-GP (we conjecture, due to overfitting to centrally-located features) but approaches the latter (right) in the limit of large depth, as information becomes more uniformly spatially distributed(Xiao et al., 2018). CNN-GPs generally outperform FCN-GP, presumably due to the local connectivity prior, but can fail to capture nonlinear interactions between spatially-distant pixels at shallow depths (left). Values are reported on a 2K/4K train/validation subsets of CIFAR10. See §A.7.3 for experimental details.

Figure 2: Validation accuracy (left) of an MC-CNN-GP increases with n ⇥ M (i.e. channel count times number of samples) and approaches that of the exact CNN-GP (not shown), while the distance (right) to the exact kernel decreases. The dark band in the left plot corresponds to ill-conditioning of K L+1n,M when the number of outer products contributing to K L+1 n,M approximately equals its rank. Values reported are for a 3-layer model applied to a 2K/4K train/validation subset of CIFAR10 downsampled to 8 ⇥ 8. See Figure7for similar results with other architectures and §A.7.2 for experimental details. random finite width networks and use the empirical uncentered covariances of activations to estimate the Monte Carlo-GP (MC-GP) kernel,

Figure1: Different dimensionality collapsing strategies described in §3. Validation accuracy of an MC-CNN-GP with pooling ( §3.2.1) is consistently better than other models due to translation invariance of the kernel. CNN-GP with zero padding ( §3.1) outperforms an analogous CNN-GP without padding as depth increases. At depth 15 the spatial dimension of the output without padding is reduced to 1 ⇥ 1, making the CNN-GP without padding equivalent to the center pixel selection strategy ( §3.2.2) -which also performs worse than the CNN-GP (we conjecture, due to overfitting to centrally-located features) but approaches the latter (right) in the limit of large depth, as information becomes more uniformly spatially distributed(Xiao et al., 2018). CNN-GPs generally outperform FCN-GP, presumably due to the local connectivity prior, but can fail to capture nonlinear interactions between spatially-distant pixels at shallow depths (left). Values are reported on a 2K/4K train/validation subsets of CIFAR10. See §A.7.3 for experimental details.

Figure 2: Validation accuracy (left) of an MC-CNN-GP increases with n ⇥ M (i.e. channel count times number of samples) and approaches that of the exact CNN-GP (not shown), while the distance (right) to the exact kernel decreases. The dark band in the left plot corresponds to ill-conditioning of K L+1n,M when the number of outer products contributing to K L+1 n,M approximately equals its rank. Values reported are for a 3-layer model applied to a 2K/4K train/validation subset of CIFAR10 downsampled to 8 ⇥ 8. See Figure7for similar results with other architectures and §A.7.2 for experimental details.

of post-nonlinearity activations given normal pre-activations)

e W P c e b C z 8 D 5 2 K p s G S n y y 5 s 3 e u M Z p 4 U U F q L o V x B e u 3 7 j 5 q 2 N 2 7 0 7 d + / d 3 + w / e H h s d

e W P c e b C z 8 D 5 2 K p s G S n y y 5 s 3 e u M Z p 4 U U F q L o V x B e u 3 7 j 5 q 2 N 2 7 0 7 d + / d 3 + w / e H h s d

e W P c e b C z 8 D 5 2 K p s G S n y y 5 s 3 e u M Z p 4 U U F q L o V x B e u 3 7 j 5 q 2 N 2 7 0 7 d + / d 3 + w / e H h s d

e W P c e b C z 8 D 5 2 K p s G S n y y 5 s 3 e u M Z p 4 U U F q L o V x B e u 3 7 j 5 q 2 N 2 7 0 7 d + / d 3 + w / e H h s d

r w B M / w 4 j w 4 r 8 6 b 8 z 5 d X X J m N w f w B 8 7 H D 0 B 7 m P Y = < / l a t e x i t > < l a t e x i t s h a 1 _ b a s e 6 4 = " S 9 7 h A c 1 X q 6

Figure4: Different dimensionality collapsing strategies described in §3. Validation accuracy of an MC-CNN-GP with pooling ( §3.2.1) is consistently better than other models due to translation invariance of the kernel. CNN-GP with zero padding ( §3.1) outperforms an analogous CNN-GP without padding as depth increases. At depth 15 the spatial dimension of the output without padding is reduced to 1 × 1, making the CNN-GP without padding equivalent to the center pixel selection strategy ( §3.2.2) -which also performs worse than the CNN-GP (we conjecture, due to overfitting to centrally-located features) but approaches the latter (right) in the limit of large depth, as information becomes more uniformly spatially distributed(Xiao et al., 2018). CNN-GPs generally outperform FCN-GP, presumably due to the local connectivity prior, but can fail to capture nonlinear interactions between spatially-distant pixels at shallow depths (left). Values are reported on a 2K/4K train/validation subset of CIFAR10. See §G.3 for experimental details.

Figure 5: Convergence of NN-GP Monte Carlo estimates. Validation accuracy (left) of an MC-CNN-GP increases with n × M (channel count × number of samples) and approaches that of the exact CNN-GP (not shown), while the distance (right) to the exact kernel decreases. The dark band in the left plot corresponds to ill-conditioning of K L+1 n,M when the number of outer products contributing to K L+1 n,M approximately equals its rank. Values reported are for a 3-layer model applied to a 2K/4K train/validation subset of CIFAR10 downsampled to 8 × 8. See Figure 10 for similar results with other architectures and §G.2 for experimental details.

FINITE CHANNEL SGD-TRAINED CNNS CAN OUTPERFORM INFINITE CHANNEL BAYESIAN CNNS, IN THE ABSENCE OF POOLING

Figure6: (a): SGD-trained CNNs often perform better with increasing number of channels. Each line corresponds to a particular choice of architecture and initialization hyperparameters, with best learning rate and weight decay selected independently for each number of channels (x-axis). (b): SGD-trained CNNs often approach the performance of their corresponding CNN-GP with increasing number of channels. All models have the same architecture except for pooling and weight sharing, as well as training-related hyperparameters such as learning rate, weight decay and batch size, which are selected for each number of channels (x-axis) to maximize validation performance (y-axis) of a neural network. As the number of channels grows, best validation accuracy increases and approaches accuracy of the respective GP (solid horizontal line). (c): However, the best-performing SGD-trained CNNs can outperform their corresponding CNN-GPs. Each point corresponds to the test accuracy of: (y-axis) a specific CNN-GP; (x-axis) the best (on validation) CNN with the same architectural hyper-parameters selected among the 100%-accurate models on the full training CIFAR10 dataset with different learning rates, weight decay and number of channels. While CNN-GP appears competitive against 100%-accurate CNNs (above the diagonal), the best CNNs overall outperform CNN-GPs by a significant margin (below the diagonal, right). For further analysis of factors leading to similar or diverging behavior between SGD-trained finite CNNs and infinite Bayesian CNNs see Figures1, 7, and Table1. Experimental details: all networks have reached 100% training accuracy on CIFAR10. Values in (b) are reported on an 0.5K/4K train/validation subset downsampled to 8 × 8 for computational reasons. See §G.5 and §G.1 for full experimental details of (a, c) and (b) plots respectively.

Figure 7: Aspects of architecture and inference influencing test performance. Test accuracy (vertical axis, %) for the best model within each model family (horizontal axis), maximizing validation accuracy over depth, width, and training and initialization hyperpameters ("No underfitting"means only models that achieved 100% training accuracy were considered). CNN-GP outperforms SGD-trained models optimized with a small learning rate to 100% train accuracy. When SGD optimization is allowed to underfit the training set, there is a significant improvement in generalization. Further, when ReLU nonlinearities are paired with large learning rates, the performance of SGD-trained models again improves relative to CNN-GPs, suggesting a beneficial interplay between ReLUs and fast SGD training. These differences in performance between CNNs and CNN-GPs are not observed between FCNs and FCN-GPs, or between LCNs and LCN-GPs (Figure1), suggesting that equivariance is the underlying factor responsible for the improved performance of finite SGD-trained CNNs relative to infinite Bayesian CNNs without pooling. Further comparison between SGD-trained CNNs and CNN-GPs with pooling (omitted due to computational limitations), where both models do share the property of invariance, would be an interesting direction for future research. See Table1for further results on other datasets, as well as comparison to GP performance in prior literature. See §G.5 for experimental details.

Figure 8: Validation loss convergence. Best validation loss (vertical axis) of trained neural networks (dashed line) as the number of channels increases (horizontal axis) approaches that of a respective (MC-)CNN-GP (solid horizontal line). See Figure 6 (b) for validation accuracy, Figure 9 for training loss and §G.1 for experimental details.No PoolingGlobal Average Pooling

Figure 9:  No underfitting in small models. Training loss (vertical axis) of best (in terms of validation loss) neural networks as the number of channels increases (horizontal axis). While perfect 0 loss is not achieved (but 100% accuracy is), we observe no consistent improvement when increasing the capacity of the network (left to right). This eliminates underfitting as a possible explanation for why small models perform worse in Figure6(b). See Figure8for validation loss and §G.1 for experimental details.

Figure 10: Convergence of NN-GP Monte Carlo estimates. As in Figure5, validation accuracy (left) of MC-GPs increases with n × M (i.e. width times number of samples), while the distance (right) to the the respective exact GP kernel (or the best available estimate in the case of CNN-GP with pooling, top row) decreases. We remark that when using shared weights, convergence is slower as smaller number of independent random parameters are being used. See §G.2 for experimental details.

β values would have to span the hypercube [±k] N = {-k, . . . , k} N in the preactivation definition. Similarly, in §3 the normalizing factor d (d 2 ) should be the product (squared) of its entries, and summations over α, β should span the [d 0 ] × • • • × [d N ] hypercube as well. The definition of the kernel propagation operator A in Equation5will remain exactly the same, so long as β is summed over the hypercube, and the variance weights remain respectively normalizedβ v β = 1.D REVIEW OF EXACT BAYESIAN REGRESSION WITH GPSOur discussion in the paper has focused on model priors. A crucial benefit we derive by mapping to a GP is that Bayesian inference is straightforward to implement and can be done exactly for regression(Rasmussen & Williams, 2006, chapter 2), requiring only simple linear algebra. Let X denote training inputs x 1 , ..., x |X | , t T = t 1 , ..., t |X | training targets, and collectively D for the training set. The integral over the posterior can be evaluated analytically to give a posterior predictive distribution on a test point y * which is Gaussian, (z * |D, y * ) ∼ N µ * , σ 2 * , with

β are i.i.d. R.V.s with mean zero and finite variance 0 < σ 2 ω < ∞ (note the lack of scaling by input dimensionality, compensated for later in Equations 29 and 42; see also similar notation used in Matthews et al. (2018a)). When l = 0, we further assume they are Gaussian distributed. (ii) Biases. B = l∈N B l and B l = b l j : j ∈ N . We assume b l j are i.i.d. Gaussian with mean zero and variance 0 ≤ σ 2 b < ∞. (iii) Place-holder. H is a place-holder to store extra (if needed) R.V.s , e.g. parameters coming from the final dense layer.

≡ {1, . . . , n} and [±k] ≡ {-k, . . . , 0, . . . k} . (30) (iii) Prove that for any finite m ≥ 1, z l,n i i∈[m] ⊆ R |X |d converges in distribution to a multivariate normal with mean zero and covariance A K l ∞ ⊗ I m as n → ∞, where K l ∞ is defined identically to Equation 9. As a consequence, per our remark in §E.1, z l,n i i∈N * converges weakly to the GP z l,∞ i i∈N * .

If φ ∈ Ψ 1 , then for every l ≥ 0 and every m ≥ 1, z l,n i i∈[m] ⊆ R |X |d converges in distribution to a multivariate normal with mean zero and covariance A K l ∞ ⊗ I m .

i.i.d. random variables. Choose a vector a ≡ [a i (x)] T i∈[m], x∈X ∈ R m|X | . Then i∈

Let PSD m denote the set of m × m positive semi-definite matrices and for R ≥ 1, definePSD m (R) ≡ {Σ ∈ PSD m : 1/R ≤ Σ α,α ≤ R for 1 ≤ α ≤ m} . (44)Further let T ∞ : PSD 2 → R be a function given byT ∞ (Σ) ≡ E (x,y)∼N (0,Σ) [φ(x)φ(y)] ,(45)and C k (φ, R) (may equal ∞) denotes the uniform upper bound for the k-th momentC k (φ, R) ≡ sup 1/R≤r≤R E x∼N (0,r) |φ(x)| k .(46)

, we say that a function f : R → R is exponentially bounded if there exist a, b > 0 such that |f (x)| ≤ ae b|x| a.e. (almost everywhere) (49)We now prove our main result presented in §2.2.3 through the following three theorems.Theorem E.4. If φ is absolutely continuous and φ is exponentially bounded then φ ∈ Ψ.Theorem E.5. If φ ∈ Ψ, then for l ≥ 0, If for l ≥ 0, K l t P → K l ∞ and m ≥ 1, the joint distribution of z l,t j j∈[m]converges in distribution to a multivariate normal distribution with mean zero and covariance A K l ∞ ⊗ I m .The proofs of Theorems E.4, E.5, and E.6 can be found in §E.7, §E.6, and §E.5 respectively. The proof of Theorem E.5 is slightly more technical and we will borrow some ideas fromDaniely et al. (2016).E.5 PROOF OF THEOREM E.6Per Equation26, it suffices to prove that for any vector [ai (x)] i∈[m], x∈X ∈ R m|X | , l (t) j∈[n l (t)] β∈[±k]√ v β ω l ij,β y l,t j,α+β (x) + b l i that conditioned on y l,t j j∈[n l (t)]

tr denotes the mean trace operator acting on the pixel by pixel matrix, i.e. the functional computing the mean of the diagonal terms of the pixel by pixel matrix. Therefore[f i ] i∈[m]converges weakly to a mean zero Gaussian with covariance σ 2 ω tr K l ∞ (x, x ) + σ 2 b x,x ∈X ⊗ I m in the case of vectorization ( §3.1).E.6 PROOF OF THEOREM E.5Proof. We recall K L t and K L ∞ to be random matrices in R d|X |×d|X | , and we will prove convergence K to• ∞ , the pointwise ∞ -norm(i.e. K ∞ = max x,x ,α,α |K α,α (x, x )|).Note that due to finite dimensionality of K L , convergence w.r.t. all other norms follows.

We postpone the proof of the following claim.Claim E.9. Assume |φ(x)| ≤ a+b|x|. Then there is a κ = κ(a, b, R) such that for all Σ ∈ PSD 2 (R) and all p ≥ 1,E (x,y)∼N (0,Σ) |φ(x)φ(y)| p 1/p ≤ κp. (102)Claim E.9 and the triangle inequality imply E (x,y)∼N (0,Σ) |φ(x)φ(y) -Eφ(x)φ(y)| p 1/p ≤ 2κp.

y)∼N (0,Σ) |φ(x)φ(y)| p 1/p ≤ E x∼N (0,Σ11) |φ(x)| 2p 1/2p E y∼N (0,Σ22) |φ(y)| 2p 1/2p (105) ≤ a + b E|x| 2p 1/2p a + b E|y| 2p 1

Presented is a CNN with stride 1 and no (valid) padding, i.e. the spatial shape of the input shrinks as it propagates through it d

Table content: <table><tr><td>2 hypercube</td></tr><tr><td>(see  §2.2,  §3.1,  §C).</td></tr></table>

See Figure7for similar results with other architectures and §A.7.2 for experimental details.random finite width networks and use the empirical uncentered covariances of activations to estimate the Monte Carlo-GP (MC-GP) kernel,

Table content: <table><tr><td>(CNN w/o weight sharing)</td></tr><tr><td>0 p 9 z 6 F 6 y i w S Y = &lt; / l a t e x i t &gt;</td></tr><tr><td>(error functio</td></tr><tr><td>(Monte Carlo estimate)</td></tr><tr><td>(Fully-connected network)</td></tr><tr><td>Under review as a conference paper at ICLR 2019</td></tr><tr><td>MC-CNN-GP</td></tr><tr><td>Figure 2: Validation accuracy (left) of an MC-CNN-GP increases with n ⇥ M (i.e. channel count times number of samples) and approaches that of the exact CNN-GP (not shown), while the distance (right) to the exact kernel decreases. The dark band in the left plot corresponds to ill-conditioning of K L+1 n,M when the number of outer products contributing to K L+1 n,M approximately equals its rank. Values reported are for a 3-layer model applied to a 2K/4K train/validation subset of CIFAR10 downsampled to 8 ⇥ 8.</td></tr></table>



Table content: <table><tr><td>CNN</td></tr><tr><td>Figure 3: (a): SGD-trai Each line corresponds to best learning rate and we (b): SGD-trained CNN with increasing number and weight sharing, as w and batch size, which ar performance (y-axis) of a racy increases and approa the best-performing SG point corresponds to the tion) CNN with the same on the full training CIFAR nels. While CNN-GP ap best CNNs overall outpe further analysis of factors and infinite Bayesian CN 100% training accuracy o set downsampled to 8 ⇥ details of (a, c) and (b) pl</td></tr><tr><td>4• CNNs suffer from exploding or va</td></tr><tr><td>gradients at large depths unless t</td></tr><tr><td>initialized correctly. The phase bo</td></tr><tr><td>order and chaos (rightmost plot f</td></tr><tr><td>nonlinearity) is the region where l</td></tr><tr><td>training is feasible.</td></tr></table>



Table content: <table><tr><td>(b)</td><td>N</td></tr><tr><td>LCN</td><td/></tr><tr><td>CNN</td><td/></tr><tr><td colspan="2">Figure 3: (a): SGD-trai Each line corresponds to best learning rate and we (b): SGD-trained CNN with increasing number and weight sharing, as w and batch size, which ar performance (y-axis) of a racy increases and approa the best-performing SG point corresponds to the tion) CNN with the same on the full training CIFAR nels. While CNN-GP ap best CNNs overall outpe further analysis of factors and infinite Bayesian CN 100% training accuracy o set downsampled to 8 ⇥ details of (a, c) and (b) pl</td></tr><tr><td>4. Computing the NN-GP covariance</td><td/></tr><tr><td colspan="2">8. Large-dep</td></tr><tr><td colspan="2">• CNNs suffer from exploding or va</td></tr><tr><td colspan="2">gradients at large depths unless t</td></tr><tr><td colspan="2">initialized correctly. The phase bo</td></tr><tr><td colspan="2">order and chaos (rightmost plot f</td></tr><tr><td colspan="2">nonlinearity) is the region where la</td></tr><tr><td>training is feasible.</td><td/></tr></table>

). For further analysis of factors leading to similar or diverging behavior between SGD-trained finite CNNs and infinite Bayesian CNNs see Figures1, 7, and Table1. Experimental details: all networks have reached 100% training accuracy on CIFAR10. Values in (b) are reported on an 0.5K/4K train/validation subset downsampled to 8 × 8 for computational reasons. See §G.5 and §G.1 for full experimental details of (a, c) and (b) plots respectively.

Table content: <table/>

Table1for further results on other datasets, as well as comparison to GP performance in prior literature. See §G.5 for experimental details. Best achieved test error for different model classes (vertical axis) and datasets (horizontal axis). Test error (%) for the best model within each model family, maximizing validation accuracy over depth, width, and training and initialization hyperpameters. Except where indicated by parentheses, all models achieve 100% training accuracy. For SGD-trained CNNs, numbers in parentheses correspond to the same model family, but without restriction on training accuracy. CNN-GP achieves state of the art results on CIFAR10 for GPs without trainable kernels and outperforms SGD-trained models optimized with a small learning rate to 100% train accuracy. See Figure7for visualization and interpretation of these results. See §G.5 for experimental details.

Table content: <table><tr><td>Model</td><td>CIFAR10</td><td colspan="2">MNIST Fashion-MNIST</td></tr><tr><td colspan="3">CNN with pooling CNN with ReLU and large learning rate CNN-GP CNN with small learning rate CNN with erf (any learning rate) Convolutional GP (van der Wilk et al., 2017) ResNet GP (Garriga-Alonso et al., 2018) Residual CNN-GP (Garriga-Alonso et al., 2018) -14.85 (15.65) -24.76 (17.64) -32.86 0.88 33.31(22.89) -33.31(22.17) -35.40 1.17 -0.84 0.96 CNN-GP (Garriga-Alonso et al., 2018) -1.03 FCN-GP 41.06 1.22 FCN-GP (Lee et al., 2018) 44.34 1.21 FCN 45.52 (44.73) -</td><td>--7.40 ------8.22 --</td></tr></table>

|X | × |X | matrix formed by evaluating the GP covariance on the training inputs, and likewise K (x

Table content: <table/>

Peer review (iclr2019):
 This paper extends the recent results concerning GP equivalence of infinitely wide FC nets to the convolutional case. This paper is generally of a high quality (notwithstanding the lack of keys on figures) and provides insights to an important class of model. I recommend that this paper be accepted, but I think it could be improved in a few ways. Firstly, and rather mundanely: the figures. Fig 1 is not easy to read due to the density of plotting, and as there is no key it isn’t possible to tell what it shows. Figure 2 is rather is called a ‘graphical model’ but the variables (weights and biases) are not shown. It should be specified that this is the graphical model of the infinite limit, in which case the K variables should not be random. Also, the caption on this figure refers to variables that aren’t in the figure, and is grammatically incorrect (perhaps something like ‘the limit of an infinitely wide convolutional’ is missing?). Figure 3 has a caption which seems to be inconsistent with the coloring (for example green is center pixel in the text, but blue in the key). Figure 6 is also missing a key. In Figure 5, what does the tick symbol denote? Finally, the value some of Table 1 is questionable as so many entries are missing. For example, the Fashion-MNIST column has only two values, which seems to me of little use. [I would have given the paper a rating of 7 were it not for these issues] Regarding the presentation of the content, I found this paper generally easy to follow and the arguments sound. Here are few points: There is an important distinction between finite width Bayesian-CNNs and the infinite limit, and this distinction is indeed made in the paper but not clearly enough in my view. I would anticipate that some readers might come away after a cursory reading thinking that Bayesian-CNNs are fundamentally worse than their parametric counterparts, but this is emphatically not the message of the paper. It seems that the infinite limit that is the cause of two problems. The first problem (or perhaps benefit) is that the infinite limit gives Gaussian inner layers, just as in the fully connected case. The second problem (and I’d say this is definitely a problem this time) is that the infinite limit loses the covariance between the pixels, at least with a fully connected final layer. I would recall [Matthews 2018, long version] section 7, which discusses that point that taking the infinite limit in the fully connected is actually potentially undesirable. To quote Matthews 2018, “MacKay (2002, p. 547) famously reflected on what is lost when taking the Gaussian process limit of a single hidden layer network, remarking that Gaussian processes will not learn hidden features”. Some discussion of this would enhance the presented paper, in my view. The discussion of eq (7) could be made more clear. Eq (7) is only defined on K, and not in composition with A. It is important that the alpha dependency is preserved by the A operation, and while I suppose this is obvious I would welcome a bit more detail. It would help to demonstrate the application of the results of [Cho and Saul 2009] to the convolution case explicitly (i.e. for C o A), in my view. Regarding results, effort has clearly gone to keep the comparisons as fair as possible, but with these large datasets it is difficult to disentangle the many factors that might effect performance (as acknowledged on p9). It is a weakness of the paper that there is no toy example. An example demonstrating a situation which can only be solved with hierarchical features (e.g. features that are larger than the receptive field of a single layer) would be particularly interesting, as in this case I think the GP-CNN would fail, even with the average pooling, whereas the finite Bayesian-CNN would succeed (with a sufficiently accurate inference method). It would improve readability to stress the 1D notation in the main text rather than in a footnote. On first reading I missed this detail and was confused as I was trying to interpret everything as a 2D convolution. On reflection I think notation is used in the paper is good, but I think the generalization to 2D should be elevated to something more than the footnote. Perhaps a paragraph explaining how the 2D case works would be appropriate, especially as all the experiments are in 2D cases. Some further smaller points on specific [section, paragraph, line]s 1,2,4 I think ‘easily’ is a bit of an overstatement. In this work the kernel is itself defined via a recursive convolutional operation, which doesn’t seem to me much more interpretable than the parametric convolution. At least the filters can be examined in parametric case, which isn’t the case here. I do agree with the sentiment that a function prior is better than an implicit weight prior, however. 1,2,-1 This seems too vague to me, as at least to some extent, Matthew 2018 did indeed consider using NN-GPs to gain insight about equivalent NN models (e.g. section 5.3) 1.1,:,: I find it very surprising that there are no references to Cho and Saul 2009 in this section (one does appear in 2.2.2, however). 1.1,3,-2:-1 ‘Our work differs from all of these in that our GP corresponds exactly to a fully Bayesian CNN in the many channel limit’ I do not think this is completely true, as the deep convolution GP does correspond to an infinite limit of a Bayesian CNN, just not the same limit as the one taken in this paper. Similarly a DGP following the Danianou and Lawrence 2013 is an infinite limit of a NN, but one with bottlenecks between layers. It is important that readers appreciate that infinite limits can be taken in different ways, and the resulting models may be very different. This certain limit taken in this work has desirable computational properties, but arguably undesirable modelling implications. 1.1,-1,-2 It should be made more clear here that the SGD trained models are non-Bayesian. Figure 3 The MC-CNN-GP appears to have performance that is nearly independent of the depth, even including 1 layer. Could this be explained? 2.2,2,: The z^l variables are zero mean Gaussian with a fixed covariance, not delta functions, as I understand it. They are independent of each other due to the deterministic K^l, certainly, but they are not themselves deterministic. Could this be clarified?