# Bayesian Robustness: A Nonasymptotic Viewpoint

## TL;DR

This paper introduces Rob-ULA, a robust sampling algorithm for Bayesian inference that effectively handles adversarial outliers, providing finite-sample guarantees and demonstrating practical performance on various tasks.

## Contribution

The paper proposes Rob-ULA, a novel robust variant of ULA with finite-sample analysis, addressing adversarial contamination in Bayesian posterior sampling.

## Key findings

- Rob-ULA achieves accurate sampling within . .  iterations.
- Theoretical bounds relate sample complexity to data dimension and contamination level.
- Experimental results validate robustness and effectiveness on real-world datasets.

## 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 $T= \tilde{\mathcal{O}}(d/\varepsilon_{\textsf{acc}})$ 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.