Kernel Robust Hypothesis Testing

TL;DR

This paper develops kernel-based robust hypothesis tests that account for distributional uncertainties using data-driven uncertainty sets in RKHS, providing optimal and asymptotically consistent solutions for Bayesian and Neyman-Pearson frameworks.

Contribution

It introduces a novel kernel method for constructing uncertainty sets in hypothesis testing, leading to new robust tests with proven optimality and consistency.

Findings

Proposed robust tests outperform non-robust methods in uncertain environments.

Kernel smoothing enhances generalization to unseen data.

The tests are proven to be exponentially consistent and asymptotically optimal.

Abstract

The problem of robust hypothesis testing is studied, where under the null and the alternative hypotheses, the data-generating distributions are assumed to be in some uncertainty sets, and the goal is to design a test that performs well under the worst-case distributions over the uncertainty sets. In this paper, uncertainty sets are constructed in a data-driven manner using kernel method, i.e., they are centered around empirical distributions of training samples from the null and alternative hypotheses, respectively; and are constrained via the distance between kernel mean embeddings of distributions in the reproducing kernel Hilbert space, i.e., maximum mean discrepancy (MMD). The Bayesian setting and the Neyman-Pearson setting are investigated. For the Bayesian setting where the goal is to minimize the worst-case error probability, an optimal test is firstly obtained when the alphabet is finite. When the alphabet is infinite, a tractable approximation is proposed to quantify the worst-case average error probability, and a kernel smoothing method is further applied to design test that generalizes to unseen samples. A direct robust kernel test is also proposed and proved to be exponentially consistent. For the Neyman-Pearson setting, where the goal is to minimize the worst-case probability of miss detection subject to a constraint on the worst-case probability of false alarm, an efficient robust kernel test is proposed and is shown to be asymptotically optimal. Numerical results are provided to demonstrate the performance of the proposed robust tests.