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 $T= tilde{mathcal{O}}(d/varepsilon_{textsf{acc}})$ iterations, we can sample from $p_T$ such that $text{dist}(p_T, p^*) leq varepsilon_{textsf{acc}} + tilde{mathcal{O}}(epsilon)$, where $epsilon$ 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.
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