Bayesian Robustness: A Nonasymptotic Viewpoint

Abstract

We study the problem of robustly estimating the posterior distribution for the setting where observed data can be contaminated with potentially adversarial outliers. We propose Rob-ULA, a robust variant of the Unadjusted Langevin Algorithm (ULA), and provide a finite-sample analysis of its sampling distribution. In particular, we show that after T = O ˜ ( d / ε acc ) iterations, we can sample from p T such that dist ( p T , p * ) ≤ ε acc + O ˜ ( ϵ ) , where ϵ is the fraction of corruptions and dist represents the squared 2-Wasserstein distance metric. Our results for the class of posteriors p * which satisfy log-concavity and smoothness assumptions. We corroborate our theoretical analysis with experiments on both synthetic and real-world datasets for mean estimation, regression and binary classification. Supplementary materials for this article are available online.