Asymptotics of representation learning in finite Bayesian neural networks
This work provides incremental theoretical insights into representation learning for researchers in Bayesian deep learning.
The paper tackled the problem of understanding how finite Bayesian neural networks learn hidden layer representations differently from infinite networks, showing that leading finite-width corrections to feature kernels have a largely universal form across three tractable architectures.
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.