Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Almost sure null bankruptcy of testing-by-betting strategies

Hongjian Wang, Shubhada Agrawal, Aaditya Ramdas

TL;DR

The paper analyzes testing-by-betting frameworks for bounded-mean problems, showing that a broad class of powerful strategies—including predictable plug-in, mixture, and hedging constructions—almost surely bankrupt under any nondegenerate null distribution. It introduces a sum-of-squares criterion that precisely characterizes null bankruptcy via the divergence of the bet-squared sequence, and proves that standard strategies like KT, GRAPA, and aGRAPA drive wealth to zero a.s. under the null, while mixture strategies converge to a cash component unless a cash reserve is present. The authors further show that some optimal-in-hindsight constructions (KL_inf related) admit a chi-square limit in their log-wealth, contrasting with the almost-sure bankruptcy of practical strategies. They also extend the analysis to sub-Gaussian and sub-psi settings and discuss the potential for improving non-bankrupt strategies on predictably safe paths, thereby deepening the understanding of wealth dynamics on almost all sample paths and informing the design of robust sequential testing procedures.

Abstract

The bounded mean betting procedure serves as a crucial interface between the domains of (1) sequential, anytime-valid statistical inference, and (2) online learning and portfolio selection algorithms. While recent work in both domains has established the exponential wealth growth of numerous betting strategies under any alternative distribution, the tightness of the inverted confidence sets, and the pathwise minimax regret bounds, little has been studied regarding the asymptotics of these strategies under the null hypothesis. Under the null, a strategy induces a wealth martingale converging to some random variable that can be zero (bankrupt) or non-zero (non-bankrupt, e.g. when it eventually stops betting). In this paper, we show the conceptually intuitive but technically nontrivial fact that these strategies (universal portfolio, Krichevsky-Trofimov, GRAPA, hedging, etc.) all go bankrupt with probability one, under any non-degenerate null distribution. Part of our analysis is based on the subtle almost sure divergence of various sums of $\sum O_p(n^{-1})$ type, a result of independent interest. We also demonstrate the necessity of null bankruptcy by showing that non-bankrupt strategies are all improvable in some sense. Our results significantly deepen our understanding of these betting strategies as they qualify their behavior on "almost all paths", whereas previous results are usually on "all paths" (e.g. regret bounds) or "most paths" (e.g. concentration inequalities and confidence sets).

Almost sure null bankruptcy of testing-by-betting strategies

TL;DR

The paper analyzes testing-by-betting frameworks for bounded-mean problems, showing that a broad class of powerful strategies—including predictable plug-in, mixture, and hedging constructions—almost surely bankrupt under any nondegenerate null distribution. It introduces a sum-of-squares criterion that precisely characterizes null bankruptcy via the divergence of the bet-squared sequence, and proves that standard strategies like KT, GRAPA, and aGRAPA drive wealth to zero a.s. under the null, while mixture strategies converge to a cash component unless a cash reserve is present. The authors further show that some optimal-in-hindsight constructions (KL_inf related) admit a chi-square limit in their log-wealth, contrasting with the almost-sure bankruptcy of practical strategies. They also extend the analysis to sub-Gaussian and sub-psi settings and discuss the potential for improving non-bankrupt strategies on predictably safe paths, thereby deepening the understanding of wealth dynamics on almost all sample paths and informing the design of robust sequential testing procedures.

Abstract

The bounded mean betting procedure serves as a crucial interface between the domains of (1) sequential, anytime-valid statistical inference, and (2) online learning and portfolio selection algorithms. While recent work in both domains has established the exponential wealth growth of numerous betting strategies under any alternative distribution, the tightness of the inverted confidence sets, and the pathwise minimax regret bounds, little has been studied regarding the asymptotics of these strategies under the null hypothesis. Under the null, a strategy induces a wealth martingale converging to some random variable that can be zero (bankrupt) or non-zero (non-bankrupt, e.g. when it eventually stops betting). In this paper, we show the conceptually intuitive but technically nontrivial fact that these strategies (universal portfolio, Krichevsky-Trofimov, GRAPA, hedging, etc.) all go bankrupt with probability one, under any non-degenerate null distribution. Part of our analysis is based on the subtle almost sure divergence of various sums of type, a result of independent interest. We also demonstrate the necessity of null bankruptcy by showing that non-bankrupt strategies are all improvable in some sense. Our results significantly deepen our understanding of these betting strategies as they qualify their behavior on "almost all paths", whereas previous results are usually on "all paths" (e.g. regret bounds) or "most paths" (e.g. concentration inequalities and confidence sets).
Paper Structure (30 sections, 17 theorems, 96 equations)

This paper contains 30 sections, 17 theorems, 96 equations.

Table of Contents

  1. Introduction
  2. Predictable plug-in.
  3. Mixture.
  4. Predictable hedging.
  5. Notation.
  6. Additional related work.
  7. Bankruptcy of predictable plug-in and hedging
  8. Necessary and sufficient condition for null bankruptcy
  9. Almost sure divergence of $\sum \Omega_p(n^{-1})$
  10. Null bankruptcy of KT, GRAPA, aGRAPA, etc.
  11. Bankruptcy of mixture strategies
  12. Do all good strategies go bankrupt?
  13. Further discussions
  14. Null asymptotics of $\operatorname{KL}_{\inf}$
  15. Null bankruptcy in subGaussian and sub-$\psi$ testing
  16. ...and 15 more sections

Key Result

Theorem 2.1

Let $P$ be a non-degenerate distribution on $[0,1]$ with mean $m$ and $(X_n)\overset{\mathrm{iid}}{\sim} P$. Let $\boldsymbol{\lambda} = (\lambda_n)$ be a predictable process taking values in $[-\frac{1}{1-m}, \frac{1}{m}]$. Then, the $\boldsymbol{\lambda}$-betting wealth process converges almost surely to a random variable $W_\infty$ satisfying and consequently

Theorems & Definitions (33)

  • Theorem 2.1: Sum-of-squares criterion
  • Theorem 2.2
  • Corollary 2.3
  • Corollary 2.4: $n^{-1/2}$ criterion
  • Proposition 2.5
  • Proposition 2.6
  • Proposition 2.7
  • Proposition 2.8
  • Theorem 3.1: No-cash criterion
  • Proposition 3.2
  • ...and 23 more