Differentiable PAC-Bayes Objectives with Partially Aggregated Neural Networks

Entropy (Basel). 2021 Sep 29;23(10):1280. doi: 10.3390/e23101280.


We make two related contributions motivated by the challenge of training stochastic neural networks, particularly in a PAC-Bayesian setting: (1) we show how averaging over an ensemble of stochastic neural networks enables a new class of partially-aggregated estimators, proving that these lead to unbiased lower-variance output and gradient estimators; (2) we reformulate a PAC-Bayesian bound for signed-output networks to derive in combination with the above a directly optimisable, differentiable objective and a generalisation guarantee, without using a surrogate loss or loosening the bound. We show empirically that this leads to competitive generalisation guarantees and compares favourably to other methods for training such networks. Finally, we note that the above leads to a simpler PAC-Bayesian training scheme for sign-activation networks than previous work.

Keywords: PAC–Bayes theory; deep learning; statistical learning theory.