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Robust Multi-Hypothesis Testing with Moment Constrained Uncertainty Sets

Akshayaa Magesh, Zhongchang Sun, Venugopal V. Veeravalli, Shaofeng Zou

TL;DR

This work addresses robust multi-hypothesis testing when distributions under each hypothesis lie in moment-constrained uncertainty sets around empirical moments. It establishes a saddle-point existence, derives an exact optimal test for finite alphabets via dualization, and introduces a tractable discretization plus kernel-smoothing approach to extend the solution to infinite alphabets with convergence guarantees $\gamma_{N,\epsilon}\to\gamma$ and an error bound $|\gamma_{N,\epsilon}-\gamma|\le L_0\epsilon$. A 1-NN kernel-smoothed test is proposed with a bound on the worst-case excess risk, and experiments on synthetic and real data show superior performance over a heuristic benchmark and demonstrate exponential consistency for the smoothing-based tests. The framework offers a practical, data-driven method to perform robust hypothesis testing under moment uncertainty, with rigorous guarantees and scalable extensions to large alphabets.

Abstract

The problem of robust binary hypothesis testing is studied. Under both hypotheses, the data-generating distributions are assumed to belong to uncertainty sets constructed through moments; in particular, the sets contain distributions whose moments are centered around the empirical moments obtained from training samples. The goal is to design a test that performs well under all distributions in the uncertainty sets, i.e., minimize the worst-case error probability over the uncertainty sets. In the finite-alphabet case, the optimal test is obtained. In the infinite-alphabet case, a tractable approximation to the worst-case error is derived that converges to the optimal value using finite samples from the alphabet. A test is further constructed to generalize to the entire alphabet. An exponentially consistent test for testing batch samples is also proposed. Numerical results are provided to demonstrate the performance of the proposed robust tests.

Robust Multi-Hypothesis Testing with Moment Constrained Uncertainty Sets

TL;DR

This work addresses robust multi-hypothesis testing when distributions under each hypothesis lie in moment-constrained uncertainty sets around empirical moments. It establishes a saddle-point existence, derives an exact optimal test for finite alphabets via dualization, and introduces a tractable discretization plus kernel-smoothing approach to extend the solution to infinite alphabets with convergence guarantees and an error bound . A 1-NN kernel-smoothed test is proposed with a bound on the worst-case excess risk, and experiments on synthetic and real data show superior performance over a heuristic benchmark and demonstrate exponential consistency for the smoothing-based tests. The framework offers a practical, data-driven method to perform robust hypothesis testing under moment uncertainty, with rigorous guarantees and scalable extensions to large alphabets.

Abstract

The problem of robust binary hypothesis testing is studied. Under both hypotheses, the data-generating distributions are assumed to belong to uncertainty sets constructed through moments; in particular, the sets contain distributions whose moments are centered around the empirical moments obtained from training samples. The goal is to design a test that performs well under all distributions in the uncertainty sets, i.e., minimize the worst-case error probability over the uncertainty sets. In the finite-alphabet case, the optimal test is obtained. In the infinite-alphabet case, a tractable approximation to the worst-case error is derived that converges to the optimal value using finite samples from the alphabet. A test is further constructed to generalize to the entire alphabet. An exponentially consistent test for testing batch samples is also proposed. Numerical results are provided to demonstrate the performance of the proposed robust tests.
Paper Structure (12 sections, 6 theorems, 82 equations, 4 figures)

This paper contains 12 sections, 6 theorems, 82 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Our Contributions
  3. Problem Setup
  4. Insights from the Binary Case
  5. Robust Multi-Hypothesis Testing
  6. Finite Alphabet: Optimal Test
  7. Infinite Alphabet
  8. Experimental Results
  9. Synthetic Data
  10. Real Data
  11. Conclusion
  12. Exponential Consistency of the Heuristic Test

Key Result

Theorem 1

The minmax problem in eq:minmax_test_m_ary has a saddle-point solution $(\phi^*; P_1^*, \ldots, P_m^*)$, i.e.,

Figures (4)

  • Figure 1: Comparison of the Robust Test and the Heuristic Test for Three Hypothesis: Synthetic Data
  • Figure 2: Comparison of the Robust Test and the Heuristic Test for Four Hypothesis: Synthetic Data
  • Figure 3: Comparison of the Robust Test and the Heuristic Test for Three Hypothesis: Real Data
  • Figure 4: Comparison of the Robust Test and the Heuristic Test for Four Hypothesis: Real Data

Theorems & Definitions (17)

  • Remark 1
  • Claim 1
  • proof
  • Theorem 1
  • proof
  • Theorem 2
  • proof
  • Theorem 3
  • proof
  • Theorem 4
  • ...and 7 more