Data-dependent PAC-Bayes priors via differential privacy
This work addresses a theoretical limitation in machine learning generalization theory for researchers, offering a practical method to derive tighter bounds, though it is incremental in extending existing PAC-Bayes techniques.
The paper tackles the challenge of using data-dependent priors in the PAC-Bayes framework when the data distribution is unknown, by showing that ε-differentially private priors yield valid generalization bounds, and demonstrates empirically that these bounds can be nonvacuous in cases where other distribution-dependent bounds fail.
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 ε-differentially private data-dependent prior yields a valid PAC-Bayes bound, and then show how non-private mechanisms for choosing priors can also yield generalization bounds. 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 given control of the 2-Wasserstein distance to an ε-differentially private stationary distribution. We study our data-dependent bounds empirically, and show that they can be nonvacuous even when other distribution-dependent bounds are vacuous.