Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Towards minimax optimal algorithms for Active Simple Hypothesis Testing

Sushant Vijayan

TL;DR

This work studies Active Simple Hypothesis Testing (ASHT) under a fixed budget by recasting minimax error-exponent bounds into a game-theoretic framework and linking them to a zero-sum differential game whose value equals R^{go}_{\infty}. It develops PDE-based numerical schemes to compute the minimax exponent for small numbers of hypotheses and introduces a Grid-based Optimal Action Policy (GOAP) with δ-optimal guarantees, while also proposing a scalable meta-strategy based on Blackwell Approachability that uniformly dominates static schemes and often matches the go-infinity exponent in practice. The results provide both theoretical connections between ASHT, differential games, and approachability, and practical algorithms with provable guarantees and favorable computational properties for small m and K. The approach opens avenues to extend to more complex hypothesis classes and to develop scalable, provably near-optimal strategies for fixed-budget active testing in broader settings.

Abstract

We study the Active Simple Hypothesis Testing (ASHT) problem, a simpler variant of the Fixed Budget Best Arm Identification problem. In this work, we provide novel game theoretic formulation of the upper bounds of the ASHT problem. This formulation allows us to leverage tools of differential games and Partial Differential Equations (PDEs) to propose an approximately optimal algorithm that is computationally tractable compared to prior work. However, the optimal algorithm still suffers from a curse of dimensionality and instead we use a novel link to Blackwell Approachability to propose an algorithm that is far more efficient computationally. We show that this new algorithm, although not proven to be optimal, is always better than static algorithms in all instances of ASHT and is numerically observed to attain the optimal exponent in various instances.

Towards minimax optimal algorithms for Active Simple Hypothesis Testing

TL;DR

This work studies Active Simple Hypothesis Testing (ASHT) under a fixed budget by recasting minimax error-exponent bounds into a game-theoretic framework and linking them to a zero-sum differential game whose value equals R^{go}_{\infty}. It develops PDE-based numerical schemes to compute the minimax exponent for small numbers of hypotheses and introduces a Grid-based Optimal Action Policy (GOAP) with δ-optimal guarantees, while also proposing a scalable meta-strategy based on Blackwell Approachability that uniformly dominates static schemes and often matches the go-infinity exponent in practice. The results provide both theoretical connections between ASHT, differential games, and approachability, and practical algorithms with provable guarantees and favorable computational properties for small m and K. The approach opens avenues to extend to more complex hypothesis classes and to develop scalable, provably near-optimal strategies for fixed-budget active testing in broader settings.

Abstract

We study the Active Simple Hypothesis Testing (ASHT) problem, a simpler variant of the Fixed Budget Best Arm Identification problem. In this work, we provide novel game theoretic formulation of the upper bounds of the ASHT problem. This formulation allows us to leverage tools of differential games and Partial Differential Equations (PDEs) to propose an approximately optimal algorithm that is computationally tractable compared to prior work. However, the optimal algorithm still suffers from a curse of dimensionality and instead we use a novel link to Blackwell Approachability to propose an algorithm that is far more efficient computationally. We show that this new algorithm, although not proven to be optimal, is always better than static algorithms in all instances of ASHT and is numerically observed to attain the optimal exponent in various instances.
Paper Structure (53 sections, 33 theorems, 256 equations, 2 tables, 4 algorithms)

This paper contains 53 sections, 33 theorems, 256 equations, 2 tables, 4 algorithms.

Table of Contents

  1. Introduction.
  2. Preliminaries.
  3. Assumptions.
  4. Problem Formulation.
  5. Previous relevant work.
  6. Recasting Information theoretic bounds.
  7. Consistency Results
  8. Zero sum differential game value interpretation of $R^{go}_\infty$.
  9. A Dynamic Programming Representation of the Value Function $V(x,t)$.
  10. Numerical schemes based on PDE formulation \ref{['HJI_pde']}.
  11. Finite Difference Explicit Upwind Scheme to approximate $R^{go}_{\infty}$.
  12. Computational effort in calculating $R^{go}_{\infty}$.
  13. Numerical calculations of $R^{go}_{\infty}$ and a conjecture of komiyama2022minimax.
  14. An optimal algorithm based on the PDE formulation \ref{['HJI_pde']}.
  15. Spatio-temporal grid construction.
  16. ...and 38 more sections

Key Result

Theorem 3.1

Let us define the following quantity between two distributions with a common support Suppose the following condition holds: (a) The second derivative of $\phi$ wrt $s$ is bounded for all arms and both pairs of hypothesis, that is, Then, if an algorithm $\mathbb{A}$ achieves an error exponent of atleast $r$ on the instance $\nu$, that is $e(\nu,\mathbb{A}) \geq r$, then we necessarily have:

Theorems & Definitions (75)

  • Remark 2.1
  • Definition 2.2
  • Definition 2.3: Minimax optimality
  • Theorem 3.1: Theorem 2 hayashi2009discrimination
  • Theorem 3.2: Theorem 1 nitinawarat2013controlled
  • Remark 3.3
  • Remark 3.4
  • Theorem 3.5: Theorem 2 komiyama2022minimax
  • Proposition 4.1
  • Remark 4.2
  • ...and 65 more