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.

我们提出了一个新的框架来解决非凸鲁棒性假设检验问题，其中目标是寻找最佳探测器，最小化最坏情况下的第一型和第二型风险函数的最大值。我们构建了以 Sinkhorn 差异为基础的样本经验分布为中心的分布不确定性集合。通过引入问题的精确混合整数指数锥改进，我们可以在输入数据量适中的情况下得到全局最优解。然后，我们提出了一个凸近似，证明其优于当前最先进的方法学。此外，我们建立了鲁棒性假设检验和非鲁棒性风险函数的正则化表示之间的联系，提供了有深入见解的解释。我们的数值研究凸显了所提框架的令人满意的测试性能和计算效率。

使用 Sinkhorn 不确定集进行非凸健壮假设检验

Non-Convex Robust Hypothesis Testing using Sinkhorn Uncertainty Sets

We study statistical inference and distributionally robust solution methods
for stochastic optimization problems, focusing on confidence intervals for
optimal values and solutions that achieve exact coverage asymptotically. We
develop a generalized empirical likelihood framework---based on distributional
uncertainty sets constructed from nonparametric $f$-divergence balls---for
Hadamard differentiable functionals, and in particular, stochastic optimization
problems. As consequences of this theory, we provide a principled method for
choosing the size of distributional uncertainty regions to provide one- and
two-sided confidence intervals that achieve exact coverage. We also give an
asymptotic expansion for our distributionally robust formulation, showing how
robustification regularizes problems by their variance. Finally, we show that
optimizers of the distributionally robust formulations we study enjoy
(essentially) the same consistency properties as those in classical sample
average approximations. Our general approach applies to quickly mixing
stationary sequences, including geometrically ergodic Harris recurrent Markov
chains.

本文研究了基于经验似然和分布鲁棒解的方法进行随机优化问题的统计推断，特别关注最优值的置信区间和渐近达到精确覆盖的解决方案。我们提出了一个基于非参数 $f$- 分歧球构建的分布不确定性集合的广义经验似然框架，用于 Hadamard 可微函数和随机优化问题，从而提供了一个有原则的选择分布不确定性区域大小的方法，以实现达到精确覆盖的单侧和双侧置信区间。我们还给出了我们分布鲁棒的公式的渐近展开，表明如何通过方差来规范化问题。最后，我们证明了，我们研究的分布鲁棒公式的优化器具有与经典样本平均逼近中的优化器基本相同的一致性属性。我们的一般方法适用于快速混合的平稳序列，包括几何上遗传的 Harris 递归马尔科夫链。