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On Stopping Times of Power-one Sequential Tests: Tight Lower and Upper Bounds

Shubhada Agrawal, Aaditya Ramdas

TL;DR

This paper develops a distribution-free theory for stopping times in alpha-correct power-one sequential tests between arbitrary composite nulls and alternatives. The authors establish two tight lower bounds on stopping times: in the small-error, fixed-gap regime as $eta o 0$, the essential term is $ rac{\, ext{log}(1/eta)}{ ext{KL}_{ ext{inf}}}$, and in the fixed-error, small-separation regime as $ ext{KL}_{ ext{inf}} o 0$, the optimal scale is $c rac{1}{ ext{KL}_{ ext{inf}}} ext{log log} rac{1}{ ext{KL}_{ ext{inf}}}$, with a universal constant $c>0$. The results hinge on the separation notion via $ ext{KL}_{ ext{inf}}(Q, rak{P})$ and are proved without reference measures or compactness assumptions. The paper further provides sufficient conditions under which existing or constructed sequential tests attain matching upper bounds, demonstrating tightness through various parametric and nonparametric examples, including mean testing for bounded distributions. The framework relies on e-processes, e-variables, and mixture strategies to build alpha-correct tests, and it unifies and extends classic results (e.g., Wald’s SPRT, Farrell’s LIL-type bounds) to broad hypothesis classes. The contributions thus offer a comprehensive, general theory for stopping-time performance of power-one sequential tests beyond parametric settings, with implications for adaptive testing and sequential decision-making.

Abstract

We prove two lower bounds for stopping times of sequential tests between general composite nulls and alternatives. The first lower bound is for the setting where the type-1 error level $α$ approaches zero, and equals $\log(1/α)$ divided by a certain infimum KL divergence, termed $\operatorname{KL_{inf}}$. The second lower bound applies to the setting where $α$ is fixed and $\operatorname{KL_{inf}}$ approaches 0 (meaning that the null and alternative sets are not separated) and equals $c \operatorname{KL_{inf}}^{-1} \log \log \operatorname{KL_{inf}}^{-1}$ for a universal constant $c > 0$. We also provide a sufficient condition for matching the upper bounds and show that this condition is met in several special cases. Given past work, these upper and lower bounds are unsurprising in their form; our main contribution is the generality in which they hold, for example, not requiring reference measures or compactness of the classes.

On Stopping Times of Power-one Sequential Tests: Tight Lower and Upper Bounds

TL;DR

This paper develops a distribution-free theory for stopping times in alpha-correct power-one sequential tests between arbitrary composite nulls and alternatives. The authors establish two tight lower bounds on stopping times: in the small-error, fixed-gap regime as , the essential term is , and in the fixed-error, small-separation regime as , the optimal scale is , with a universal constant . The results hinge on the separation notion via and are proved without reference measures or compactness assumptions. The paper further provides sufficient conditions under which existing or constructed sequential tests attain matching upper bounds, demonstrating tightness through various parametric and nonparametric examples, including mean testing for bounded distributions. The framework relies on e-processes, e-variables, and mixture strategies to build alpha-correct tests, and it unifies and extends classic results (e.g., Wald’s SPRT, Farrell’s LIL-type bounds) to broad hypothesis classes. The contributions thus offer a comprehensive, general theory for stopping-time performance of power-one sequential tests beyond parametric settings, with implications for adaptive testing and sequential decision-making.

Abstract

We prove two lower bounds for stopping times of sequential tests between general composite nulls and alternatives. The first lower bound is for the setting where the type-1 error level approaches zero, and equals divided by a certain infimum KL divergence, termed . The second lower bound applies to the setting where is fixed and approaches 0 (meaning that the null and alternative sets are not separated) and equals for a universal constant . We also provide a sufficient condition for matching the upper bounds and show that this condition is met in several special cases. Given past work, these upper and lower bounds are unsurprising in their form; our main contribution is the generality in which they hold, for example, not requiring reference measures or compactness of the classes.
Paper Structure (27 sections, 20 theorems, 161 equations)

This paper contains 27 sections, 20 theorems, 161 equations.

Table of Contents

  1. Introduction
  2. Contributions.
  3. Paper outline.
  4. Preliminaries: setup and background
  5. Instance-dependent vs worst-case bounds.
  6. KL-inf: the separation notion.
  7. E-variable, numeraire e-variable, and e-process.
  8. Literature
  9. Notation.
  10. Lower bounds for general hypothesis testing problems
  11. Small error, fixed gap regime
  12. Small gap, fixed error regime
  13. Proof of Theorem \ref{['th:lowerbound2']}
  14. Upper bounds: optimal sequential tests
  15. Non-parametric composite null, parametric point alternative
  16. ...and 12 more sections

Key Result

Theorem 3.1

For $Q \in \mathcal{Q}$ and $\alpha \in (0, 1)$, the stopping time $\tau_\alpha$ of any $\alpha$-correct power-one sequential test satisfies: and hence, where

Theorems & Definitions (44)

  • Theorem 3.1
  • proof
  • Remark 1
  • Theorem 3.2
  • Corollary 3.3
  • Remark 2
  • Lemma 3.4
  • proof
  • Lemma 3.5
  • proof
  • ...and 34 more