Cold Posteriors through PAC-Bayes

TL;DR

This paper examines the cold posterior effect in Bayesian inference using PAC-Bayes bounds, highlighting how temperature adjustments in the PAC-Bayes framework can explain improved out-of-sample performance in small data regimes.

Contribution

It establishes a connection between the PAC-Bayes framework and the cold posterior effect, introducing a temperature parameter that explains the phenomenon in variational approximations.

Findings

PAC-Bayes bounds relate to the ELBO objective in variational inference.

Adjusting the temperature parameter $bb$ captures the cold posterior effect.

The interpretation applies to both regression and classification with Laplace approximations.

Abstract

We investigate the cold posterior effect through the lens of PAC-Bayes generalization bounds. We argue that in the non-asymptotic setting, when the number of training samples is (relatively) small, discussions of the cold posterior effect should take into account that approximate Bayesian inference does not readily provide guarantees of performance on out-of-sample data. Instead, out-of-sample error is better described through a generalization bound. In this context, we explore the connections between the ELBO objective from variational inference and the PAC-Bayes objectives. We note that, while the ELBO and PAC-Bayes objectives are similar, the latter objectives naturally contain a temperature parameter $\lambda$ which is not restricted to be $\lambda=1$. For both regression and classification tasks, in the case of isotropic Laplace approximations to the posterior, we show how this PAC-Bayesian interpretation of the temperature parameter captures the cold posterior effect.