Entropy-SGD optimizes the prior of a PAC-Bayes bound: Data-dependent PAC-Bayes priors via differential privacy

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 show that an e-differentially private prior yields a valid PAC-Bayes bound, a straightforward consequence of results connecting generalization with differential privacy. Using stochastic gradient Langevin dynamics (SGLD) to approximate the well-known exponential release mechanism, we observe that generalization error on MNIST (measured on held out data) falls within the (empirically nonvacuous) bounds computed under the assumption that SGLD produces perfect samples. 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.