Bayesian Robustness: A Nonasymptotic Viewpoint

Kush Bhatia, Yi-An Ma, Anca D. Dragan, Peter L. Bartlett and Michael I. Jordan

Journal of the American Statistical Association, 2024, vol. 119, issue 546, 1112-1123

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/εacc) iterations, we can sample from pT such that dist(pT,p*)≤εacc+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.

Date: 2024
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DOI: 10.1080/01621459.2023.2174121

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