Bayesian Robustness: A Nonasymptotic Viewpoint

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= tildeO(d/varepsilon_textsfacc) iterations, we can sample from pT such that dist(pT, p^*) ≤ varepsilon_textsfacc + tildeO(ε), where ε is the fraction of corruptions. We corroborate our theoretical analysis with experiments on both synthetic and real-world data sets for mean estimation, regression and binary classification.