Huber-Robust Confidence Sequences

TL;DR

This paper develops robust confidence sequences for the mean of a distribution, accommodating a known moment bound and a fraction of corrupt data, enabling reliable sequential inference in adversarial settings.

Contribution

It introduces new robust exponential supermartingales that achieve optimal width for confidence sequences under contamination, advancing sequential robust inference methods.

Findings

Achieves confidence sequences with optimal width under contamination.

Constant margin between sequential and nonsequential bounds.

Enables robust sequential testing and bandit algorithms.

Abstract

Confidence sequences are confidence intervals that can be sequentially tracked, and are valid at arbitrary data-dependent stopping times. This paper presents confidence sequences for a univariate mean of an unknown distribution with a known upper bound on the $p$-th central moment ($p$ > 1), but allowing for (at most) $\epsilon$ fraction of arbitrary distribution corruption, as in Huber's contamination model. We do this by designing new robust exponential supermartingales, and show that the resulting confidence sequences attain the optimal width achieved in the nonsequential setting. Perhaps surprisingly, the constant margin between our sequential result and the lower bound is smaller than even fixed-time robust confidence intervals based on the trimmed mean, for example. Since confidence sequences are a common tool used within A/B/n testing and bandits, these results open the door to sequential experimentation that is robust to outliers and adversarial corruptions.