Data-dependent PAC-Bayes priors via differential privacy

Gintare Karolina Dziugaite, Daniel M. Roy

Published 2018-02-26Version 1

The Probably Approximately Correct (PAC) Bayes framework (McAllester, 1999) can incorporate knowledge about the learning algorithm and data distribution through the use of distribution-dependent priors, yielding tighter generalization bounds on data-dependent posteriors. Using this flexibility, however, is difficult, especially when the data distribution is presumed to be unknown. We show how an {\epsilon}-differentially private data-dependent prior yields a valid PAC-Bayes bound, and then show how non-private mechanisms for choosing priors obtain the same generalization bound provided they converge weakly to the private mechanism. As an application of this result, we show that a Gaussian prior mean chosen via stochastic gradient Langevin dynamics (SGLD; Welling and Teh, 2011) leads to a valid PAC-Bayes bound, despite SGLD only converging weakly to an {\epsilon}-differentially private mechanism. As the bounds are data-dependent, we study the bounds empirically on synthetic data and standard neural network benchmarks in order to illustrate the gains of data-dependent priors over existing distribution-dependent PAC-Bayes bound.