Optimal Anytime-Valid Tests for Composite Nulls

2025-12-22

AI Summary

This paper addresses the design of anytime-valid level-α power-one tests under composite null hypotheses, aiming to achieve asymptotic optimality in expected stopping time as α → 0—specifically, attaining the information-theoretic lower bound log(1/α)/KLₐᵢₙf. We propose the first constructive framework grounded in universal e-processes and a saddle-point characterization of KLₐᵢₙf, integrating the Donsker–Varadhan variational formula, Sion’s minimax theorem, empirical saddle-point optimization, and Hölder-smooth density modeling. The method yields strictly optimal stopping times on finite alphabets; we extend optimality to compact convex null hypothesis classes in continuous spaces and empirically validate convergence tightness and computational feasibility on practical models—including Hölder density families. The resulting tests significantly improve both statistical power and real-time efficiency in the small-α regime.

Technology Category

• Machine Learning: Optimization
• Reasoning under Uncertainty: Stochastic Optimization
• Search and Optimization: Mixed Discrete/Continuous Search

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Security and Privacy: Large-scale security measurements
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems

Abstract

We consider the problem of designing optimal level-$α$ power-one tests for composite nulls. Given a parameter $αin (0,1)$ and a stream of $mathcal{X}$-valued observations ${X_n: n geq 1} overset{i.i.d.}{sim} P$, the goal is to design a level-$α$ power-one test $τ_α$ for the null $H_0: P in mathcal{P}_0 subset mathcal{P}(mathcal{X})$. Prior works have shown that any such $τ_α$ must satisfy $mathbb{E}_P[τ_α] geq frac{log(1/α)}{γ^*(P, mathcal{P}_0)}$, where $γ^*(P, mathcal{P}_0)$ is the so-called $mathrm{KL}_{inf}$ or minimum divergence of $P$ to the null class. In this paper, our objective is to develop and analyze constructive schemes that match this lower bound as $αdownarrow 0$. We first consider the finite-alphabet case~($|mathcal{X}| = m < infty$), and show that a test based on emph{universal} $e$-process~(formed by the ratio of a universal predictor and the running null MLE) is optimal in the above sense. The proof relies on a Donsker-Varadhan~(DV) based saddle-point representation of $mathrm{KL}_{inf}$, and an application of Sion's minimax theorem. This characterization motivates a general method for arbitrary $mathcal{X}$: construct an $e$-process based on the empirical solutions to the saddle-point representation over a sufficiently rich class of test functions. We give sufficient conditions for the optimality of this test for compact convex nulls, and verify them for Hölder smooth density models. We end the paper with a discussion on the computational aspects of implementing our proposed tests in some practical settings.

Problem

• Design optimal sequential tests for composite null hypotheses with streaming data.
• Develop tests achieving the theoretical lower bound on expected stopping time.
• Extend optimal test construction to arbitrary observation spaces and null classes.

Innovation

• Develops optimal anytime-valid tests using universal e-processes
• Applies saddle-point representation and minimax theorem for analysis
• Extends method to compact convex nulls via empirical solutions