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Approximate Least-Favorable Distributions and Nearly Optimal Tests via Stochastic Mirror Descent

Andrés Aradillas Fernández, José Blanchet, José Luis Montiel Olea, Chen Qiu, Jörg Stoye, Lezhi Tan

TL;DR

The paper tackles hypothesis testing with a null consisting of $M$ distributions and a single alternative by recasting the search for the most powerful test of size $\alpha$ as a convex dual problem. It shows that stochastic mirror descent with a negative-entropy mirror map, using unbiased subgradient estimates, provably yields an approximate least-favorable distribution after finitely many iterations and, via averaging NP-type tests, a nearly optimal test with high-probability guarantees. A key contribution is the precise choice of iteration count $T$, step size $\eta$, and a practical, low-cost gradient estimator that can use as few as one Monte Carlo draw per density, along with a method to implement the nearly optimal test efficiently. The results extend to illustrative Gaussian-location problems and discuss confidence-region construction via test inversion, providing actionable guidance for implementing robust, near-optimal tests in high-dimensional composite-null settings.

Abstract

We consider a class of hypothesis testing problems where the null hypothesis postulates $M$ distributions for the observed data, and there is only one possible distribution under the alternative. We show that one can use a stochastic mirror descent routine for convex optimization to provably obtain - after finitely many iterations - both an approximate least-favorable distribution and a nearly optimal test, in a sense we make precise. Our theoretical results yield concrete recommendations about the algorithm's implementation, including its initial condition, its step size, and the number of iterations. Importantly, our suggested algorithm can be viewed as a slight variation of the algorithm suggested by Elliott, Müller, and Watson (2015), whose theoretical performance guarantees are unknown.

Approximate Least-Favorable Distributions and Nearly Optimal Tests via Stochastic Mirror Descent

TL;DR

The paper tackles hypothesis testing with a null consisting of distributions and a single alternative by recasting the search for the most powerful test of size as a convex dual problem. It shows that stochastic mirror descent with a negative-entropy mirror map, using unbiased subgradient estimates, provably yields an approximate least-favorable distribution after finitely many iterations and, via averaging NP-type tests, a nearly optimal test with high-probability guarantees. A key contribution is the precise choice of iteration count , step size , and a practical, low-cost gradient estimator that can use as few as one Monte Carlo draw per density, along with a method to implement the nearly optimal test efficiently. The results extend to illustrative Gaussian-location problems and discuss confidence-region construction via test inversion, providing actionable guidance for implementing robust, near-optimal tests in high-dimensional composite-null settings.

Abstract

We consider a class of hypothesis testing problems where the null hypothesis postulates distributions for the observed data, and there is only one possible distribution under the alternative. We show that one can use a stochastic mirror descent routine for convex optimization to provably obtain - after finitely many iterations - both an approximate least-favorable distribution and a nearly optimal test, in a sense we make precise. Our theoretical results yield concrete recommendations about the algorithm's implementation, including its initial condition, its step size, and the number of iterations. Importantly, our suggested algorithm can be viewed as a slight variation of the algorithm suggested by Elliott, Müller, and Watson (2015), whose theoretical performance guarantees are unknown.

Paper Structure

This paper contains 24 sections, 12 theorems, 165 equations, 2 figures, 1 table, 1 algorithm.

Table of Contents

  1. Introduction
  2. Notation and Statement of the Problem
  3. Statement of the Hypothesis Testing Problem
  4. Primal and Dual Problems in Hypothesis Testing
  5. Solving the dual problem
  6. Main Results
  7. Stochastic Mirror Descent
  8. Approximate Solutions to the Dual Problem
  9. Nearly Optimal Tests via Stochastic Mirror Descent
  10. Remarks on Confidence Regions
  11. Illustrative Example
  12. Approximately Unbiased Tests
  13. Conclusion
  14. Proofs of Main Results
  15. Proof of Lemma \ref{['lem:f_convex']}
  16. ...and 9 more sections

Key Result

Lemma 1

The function $f(\kappa)$ defined in Equation eqn:NP_oracle is convex. Furthermore, a subgradient of $f$ at $\kappa$ is given by where $\varphi_{\kappa}$ is defined as in eqn:test_NP_form.

Figures (2)

  • Figure 1: $\lambda_{T} \equiv \bar{\kappa}_{T} / \| \bar{\kappa}_{T}\|_1$ for $\alpha = 10\%$ and $\epsilon = 0.1$; where $\bar{\kappa}_{T}= (1/T) \sum_{t=1}^{T} \kappa_{t}.$
  • Figure 2: $\bar{\varphi}_T$ (red) alongside $\varphi_{\bar{\kappa}_T}$ (blue) for $\alpha=10\%$ and $\epsilon=.1$.

Theorems & Definitions (33)

  • Remark 1
  • Remark 2
  • Lemma 1
  • proof
  • Lemma 2
  • proof
  • Theorem 1
  • proof
  • Definition 1: $\epsilon$-least favorable distribution
  • Remark 3
  • ...and 23 more