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Non-Convex Robust Hypothesis Testing using Sinkhorn Uncertainty Sets

Jie Wang, Rui Gao, Yao Xie

TL;DR

A convex approximation is proposed for robust hypothesis testing, demonstrating its superiority over current state-of-the-art methodologies in literature and establishing connections between robust hypothesis testing and regularized formulations of non-robust risk functions, offering insightful interpretations.

Abstract

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.

Non-Convex Robust Hypothesis Testing using Sinkhorn Uncertainty Sets

TL;DR

A convex approximation is proposed for robust hypothesis testing, demonstrating its superiority over current state-of-the-art methodologies in literature and establishing connections between robust hypothesis testing and regularized formulations of non-robust risk functions, offering insightful interpretations.

Abstract

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.
Paper Structure (12 sections, 7 theorems, 69 equations, 2 figures, 2 tables, 1 algorithm)

This paper contains 12 sections, 7 theorems, 69 equations, 2 figures, 2 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Problem Setup
  3. Random Feature Model
  4. Preliminaries on Sinkhorn DRO
  5. Optimization Methodology
  6. A Mixed-Integer Conic Formulation
  7. Convex Approximation
  8. Regularization Effects of Robust Testing
  9. Numerical Experiments
  10. Concluding Remarks
  11. Proofs of Technical Results
  12. Implementation Details

Key Result

Proposition 1

Fix the error probability $\delta\in(0,1)$. Suppose Assumption Assumption:detector holds and define the norm Then there exists a function $T$ in $\mathcal{F}_D$ such that with probability at least $1-\alpha$, it holds that

Figures (2)

  • Figure 1: Visualization of worst-case distributions for $H_1$ and $H_2$, respectively, on noisy Moon Dataset. Left: scatter plot for data points with testing size $n_{\text{Te}}=1000$; right: visualization of worst-case distributions based on training samples. The maximum of type-I/type-II error is $\mathsf{0.165}$, with computational time $264.1$s.
  • Figure 2: Testing performance (left plot) and computation time for various testing approaches on the HDGM dataset.

Theorems & Definitions (9)

  • Proposition 1: Direct Approximation Theorem
  • Proposition 2: Reformulation of Sinkhorn DRO
  • Remark 1: Recovery of Worst-case Distributions
  • Proposition 3: Consistency of $\widehat{\vartheta}^*$
  • Theorem 1: MIECP Reformulation of \ref{['Eq:CCP:testing:duality:app']}
  • Remark 2: Superior Performance of CVaR Approximation
  • Proposition 4
  • Proposition 5
  • Lemma 1