On Uniform, Bayesian, and PAC-Bayesian Deep Ensembles

Nick Hauptvogel, Christian Igel

Keywords: deep learning, ensemble methods, Bayesian model average, PAC-Bayesian generalization bounds

TL;DR: Bayes ensembles cannot be expected to improve over uniformly weighted deep ensembles while weighting ensemble members by optimizing PAC-Bayesian generalization bounds can increase performance

Abstract: It is common practice to combine deep neural networks to ensembles. These deep ensembles can profit from the cancellation of errors effect: Errors by ensemble members may average out and the ensemble achieves 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 Bayes ensemble. In this study, we stress that neither the sampling nor the weighting in a Bayes ensemble are particularly well-suited for increasing generalization performance, as they do not support the cancellation of errors effect, which is evident in the limit from the Bernstein-von~Mises theorem for misspecified models. In contrast, a weighted average of models, where the weights are optimized by minimizing a PAC-Bayesian generalization bound, can improve generalization performance. This requires that the optimization takes correlations between models into account, which can be achieved by minimizing the tandem loss at the cost that hold-out data for estimating error correlations need to be available. The PAC-Bayesian weighting increases the 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 study presents empirical results supporting these conceptual considerations on four different classification datasets. We show that state-of-the-art Bayes ensembles from the literature, despite being computationally demanding, do not improve over simple uniformly weighted deep ensembles and cannot match the performance of deep ensembles weighted by optimizing the tandem loss, which additionally come with non-vacuous generalization guarantees.

Primary Area: other topics in machine learning (i.e., none of the above)

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Submission Number: 4344