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Sequential Adversarial Hypothesis Testing

Eeshan Modak, Mayank Bakshi, Bikash Kumar Dey, Vinod M. Prabhakaran

TL;DR

This work studies sequential binary hypothesis testing under adversarial, adaptive distributions drawn from convex sets $\mathcal{P},\mathcal{Q}$. It proves that sequential testing strictly improves the error-exponent region, achieving the boundary characterized by $E_0E_1 \le D(q_1^*\|p_1^*)D(p_0^*\|q_0^*)$ and providing a feasible two-ratio SPRT construction to attain it. The authors also characterize the error-exponent regions under stopping-time constraints and under a probability-of-error constraint, showing rectangular regions with corners determined by the minimal KL divergences $D(p_0^*\|q_0^*)$ and $D(q_1^*\|p_1^*)$. The results connect adversarial hypothesis testing with sequential decision theory, offering exact first-order exponents and outlining directions for second-order analyses and broader adversarial models.

Abstract

We study the adversarial binary hypothesis testing problem in the sequential setting. Associated with each hypothesis is a closed, convex set of distributions. Given the hypothesis, each observation is generated according to a distribution chosen (from the set associated with the hypothesis) by an adversary who has access to past observations. In the sequential setting, the number of observations the detector uses to arrive at a decision is variable; this extra freedom improves the asymptotic performance of the test. We characterize the closure of the set of achievable pairs of error exponents. We also study the problem under constraints on the number of observations used and the probability of error incurred.

Sequential Adversarial Hypothesis Testing

TL;DR

This work studies sequential binary hypothesis testing under adversarial, adaptive distributions drawn from convex sets . It proves that sequential testing strictly improves the error-exponent region, achieving the boundary characterized by and providing a feasible two-ratio SPRT construction to attain it. The authors also characterize the error-exponent regions under stopping-time constraints and under a probability-of-error constraint, showing rectangular regions with corners determined by the minimal KL divergences and . The results connect adversarial hypothesis testing with sequential decision theory, offering exact first-order exponents and outlining directions for second-order analyses and broader adversarial models.

Abstract

We study the adversarial binary hypothesis testing problem in the sequential setting. Associated with each hypothesis is a closed, convex set of distributions. Given the hypothesis, each observation is generated according to a distribution chosen (from the set associated with the hypothesis) by an adversary who has access to past observations. In the sequential setting, the number of observations the detector uses to arrive at a decision is variable; this extra freedom improves the asymptotic performance of the test. We characterize the closure of the set of achievable pairs of error exponents. We also study the problem under constraints on the number of observations used and the probability of error incurred.

Paper Structure

This paper contains 19 sections, 10 theorems, 111 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Our work
  3. Notation
  4. Problem Setup
  5. Results
  6. Key Lemmas
  7. Proof of Theorem \ref{['thrm:bemv1']}
  8. Proof of achievability of Theorem \ref{['thrm:bemv1']}
  9. Proof of converse of Theorem \ref{['thrm:bemv1']}
  10. Probability constraint on the stopping time
  11. Proof of achievability in Theorem \ref{['thrm:bemv2']}
  12. Proof of converse in Theorem \ref{['thrm:bemv2']}
  13. Constraint on the probability of error
  14. Proof of achievability Theorem \ref{['thrm:bemv3']}
  15. Proof of converse of Theorem \ref{['thrm:bemv3']}
  16. ...and 4 more sections

Key Result

Theorem 1

For sets $\mathcal{P},\mathcal{Q} \subseteq \mathbb{R}^{\mathcal{X}}$ which satisfy the properties A1 and A2, the set of achievable pairs of exponents is Furthermore, all points on the curve $(E_0 E_1 = D(q_1^*\|p_1^*)D(p_0^*\|q_0^*))$ are achievable.

Figures (1)

  • Figure 1: The lighter curve shows the tradeoff between error exponents in the fixed length setting. The darker curve shows the tradeoff in the sequential setting. As we can see the pair $(E_1=D(p_1^*\|q_1^*),E_0=D(q_0^*\|p_0^*))$ can be achieved in the sequential case.

Theorems & Definitions (15)

  • Theorem 1
  • Lemma 1
  • proof
  • Lemma 2: Lemma 5, brandao2020adversarial
  • Lemma 3
  • proof
  • Lemma 4
  • proof
  • Theorem 2
  • Lemma 5: li2020second
  • ...and 5 more