Asymptotics of representation learning in finite Bayesian neural networks

TL;DR

This paper investigates the asymptotic behavior of learned representations in finite Bayesian neural networks, revealing universal correction patterns and how task signals influence hidden layer features.

Contribution

It characterizes the leading finite-width corrections to feature kernels in Bayesian networks with linear readout, across multiple architectures, advancing understanding of finite versus infinite network representations.

Findings

Finite-width corrections have a universal form.

Learned features are shaped by task-relevant signals.

Results apply to deep linear, convolutional, and single nonlinear layer networks.

Abstract

Recent works have suggested that finite Bayesian neural networks may sometimes outperform their infinite cousins because finite networks can flexibly adapt their internal representations. However, our theoretical understanding of how the learned hidden layer representations of finite networks differ from the fixed representations of infinite networks remains incomplete. Perturbative finite-width corrections to the network prior and posterior have been studied, but the asymptotics of learned features have not been fully characterized. Here, we argue that the leading finite-width corrections to the average feature kernels for any Bayesian network with linear readout and Gaussian likelihood have a largely universal form. We illustrate this explicitly for three tractable network architectures: deep linear fully-connected and convolutional networks, and networks with a single nonlinear hidden layer. Our results begin to elucidate how task-relevant learning signals shape the hidden layer representations of wide Bayesian neural networks.