A Data-Driven Approach to Robust Hypothesis Testing Using Sinkhorn Uncertainty Sets

Wang, Jie; Xie, Youhua

doi:10.48550/arxiv.2202.04258

Cited by 1 publication

(1 citation statement)

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“…In [9], the minimax problem with the 0-1 loss, i.e., the exact probability of error, was considered, where a computationally tractable reformulation and the optimal robust test were characterized. In [10], the data-driven robust hypothesis testing problem with the Sinkhorn divergence, which is a variant of Wasserstein distance with entropic regularization, was studied. The original 0-1 loss, i.e., the error probability, was smoothed as in [8].…”

Section: Introductionmentioning

confidence: 99%

Kernel Robust Hypothesis Testing

Sun¹,

Zou²

2022

Preprint

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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.

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Section: Introductionmentioning

confidence: 99%