Non-Convex Robust Hypothesis Testing using Sinkhorn Uncertainty Sets
Jie Wang Rui Gao Yao Xie
We present a new framework to address the non-convex robust hypothesis
testing problem, wherein the goal is to seek the optimal detector that
minimizes the maximum of worst-case type-I and type-II risk functions. The
distributional uncertainty sets are constructed to center around the empirical
distribution derived from samples based on Sinkhorn discrepancy. Given that the
objective involves non-convex, non-smooth probabilistic functions that are
often intractable to optimize, existing methods resort to approximations rather
than exact solutions. To tackle the challenge, we introduce an exact
mixed-integer exponential conic reformulation of the problem, which can be
solved into a global optimum with a moderate amount of input data.
Subsequently, we propose a convex approximation, demonstrating its superiority
over current state-of-the-art methodologies in literature. Furthermore, we
establish connections between robust hypothesis testing and regularized
formulations of non-robust risk functions, offering insightful interpretations.
Our numerical study highlights the satisfactory testing performance and
computational efficiency of the proposed framework.