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.

翻译：我们提出了一种新的框架来解决非凸稳健假设检验问题，其目标是寻找最优检测器，以最小化最坏情况下的第一类与第二类风险函数的最大值。分布不确定集以基于Sinkhorn散度从样本导出的经验分布为中心。由于目标函数涉及非凸、非光滑的概率函数，通常难以优化，现有方法转而采用近似而非精确解。为应对这一挑战，我们引入了该问题的精确混合整数指数锥重构形式，可在适量输入数据下求解至全局最优。随后，我们提出了一种凸近似方法，并证明其优于当前文献中的顶尖方法。此外，我们建立了稳健假设检验与非稳健风险函数正则化公式之间的联系，提供了富有洞察力的解释。数值研究表明，所提出的框架具有令人满意的测试性能和计算效率。