Bayesian Deep Convolutional Networks with Many Channels are Gaussian Processes

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.

• Machine Intelligence