On Uniform, Bayesian, and PAC-Bayesian Deep Ensembles

TL;DR

This paper compares different ensemble methods for deep neural networks, showing that PAC-Bayesian weighted ensembles optimized with tandem loss outperform Bayesian and uniform ensembles in generalization and robustness.

Contribution

It introduces a PAC-Bayesian weighting method using tandem loss that improves ensemble generalization by accounting for model correlations, outperforming traditional Bayesian ensembles.

Findings

PAC-Bayesian weighted ensembles outperform Bayesian ensembles.

Tandem loss optimization enhances robustness against correlated models.

State-of-the-art Bayesian ensembles do not surpass simple uniform deep ensembles.

Abstract

It is common practice to combine deep neural networks into ensembles. These deep ensembles can benefit from the cancellation of errors effect: Errors by ensemble members may average out, leading to better generalization performance than each individual network. Bayesian neural networks learn a posterior distribution over model parameters, and sampling and weighting networks according to this posterior yields an ensemble model referred to as a Bayes ensemble. This study reviews the argument that neither the sampling nor the weighting in Bayes ensembles are particularly well suited for increasing generalization performance, as they do not support the cancellation of errors effect. In contrast, we show that a weighted average of models, where the weights are optimized by minimizing a second-order PAC-Bayesian generalization bound, can improve generalization. It is crucial that the optimization takes correlations between models into account. This can be achieved by minimizing the tandem loss, which requires hold-out data for estimating error correlations. The tandem loss based PAC-Bayesian weighting increases robustness against correlated models and models with lower performance in an ensemble. This allows us to safely add several models from the same learning process to an ensemble, instead of using early-stopping for selecting a single weight configuration. Our experiments provide further evidence that state-of-the-art intricate Bayes ensembles do not outperform simple uniformly weighted deep ensembles in terms of classification accuracy. Additionally, we show that these Bayes ensembles cannot match the performance of deep ensembles weighted by optimizing the tandem loss, which additionally provides nonvacuous rigorous generalization guarantees.