Cold Posteriors through PAC-Bayes
This addresses a theoretical problem in Bayesian deep learning for researchers, but it is incremental as it builds on existing PAC-Bayes and variational inference frameworks.
The paper tackles the cold posterior effect by analyzing it through PAC-Bayes generalization bounds, showing that in non-asymptotic settings with small training samples, this interpretation captures the effect for regression and classification tasks using isotropic Laplace approximations.
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 $λ$ which is not restricted to be $λ=1$. 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.