Entropy-SGD optimizes the prior of a PAC-Bayes bound: Generalization properties of Entropy-SGD and data-dependent priors

TL;DR

This paper connects Entropy-SGD with PAC-Bayes bounds, showing how data-dependent priors via SGLD can provide valid generalization guarantees, despite Entropy-SGD's data-dependent nature.

Contribution

It demonstrates that Entropy-SGD optimizes a PAC-Bayes prior, and introduces a method using SGLD to obtain valid data-dependent bounds.

Findings

Entropy-SGD optimizes a PAC-Bayes prior.

Data-dependent priors via SGLD can yield valid bounds.

Test errors on MNIST and CIFAR10 are within risk bounds.

Abstract

We show that Entropy-SGD (Chaudhari et al., 2017), when viewed as a learning algorithm, optimizes a PAC-Bayes bound on the risk of a Gibbs (posterior) classifier, i.e., a randomized classifier obtained by a risk-sensitive perturbation of the weights of a learned classifier. Entropy-SGD works by optimizing the bound's prior, violating the hypothesis of the PAC-Bayes theorem that the prior is chosen independently of the data. Indeed, available implementations of Entropy-SGD rapidly obtain zero training error on random labels and the same holds of the Gibbs posterior. In order to obtain a valid generalization bound, we rely on a result showing that data-dependent priors obtained by stochastic gradient Langevin dynamics (SGLD) yield valid PAC-Bayes bounds provided the target distribution of SGLD is {\epsilon}-differentially private. We observe that test error on MNIST and CIFAR10 falls within the (empirically nonvacuous) risk bounds computed under the assumption that SGLD reaches stationarity. In particular, Entropy-SGLD can be configured to yield relatively tight generalization bounds and still fit real labels, although these same settings do not obtain state-of-the-art performance.