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Universal Log-Optimality for General Classes of e-processes and Sequential Hypothesis Tests

Ian Waudby-Smith, Ricardo Sandoval, Michael I. Jordan

TL;DR

This work develops a general betting framework for sequential hypothesis testing with composite alternatives, showing that any e-process with sublinear portfolio regret achieves adaptive, asymptotic, and almost-sure log-optimality relative to a family of log-optimal portfolios.A primary novelty is the universal log-optimality result: the growth rate under any alternative Q is maximized by the log-optimal portfolio λ_Q^*, and the long-run log-wealth converges to ℓ_Q^* almost surely, matching any other admissible process within the class.The authors derive sharp, α→0⁺ bounds for the expected rejection time E_Q[τ_α], establishing both lower bounds for arbitrary strategies and achievable upper bounds under sublinear regret, with explicit nonasymptotic expressions and moment conditions.The theory is instantiated to common sequential testing problems (one- and two-sided bounded-mean tests, and difference-in-means tests for bounded tuples) and extended to a generalized regret framework with numeraire portfolios, yielding distribution-uniform log-optimality and uniform bounds on rejection times.Overall, the results provide a constructive, algorithm-agnostic toolkit for optimal sequential testing across a broad spectrum of nonparametric problems, and they introduce distribution-uniform notions of log-optimality that strengthen prior understandings.

Abstract

We consider the problem of sequential hypothesis testing by betting. For a general class of composite testing problems -- which include bounded mean testing, equal mean testing for bounded random tuples, and some key ingredients of two-sample and independence testing as special cases -- we show that any $e$-process satisfying a certain sublinear regret bound is adaptively, asymptotically, and almost surely log-optimal for a composite alternative. This is a strong notion of optimality that has not previously been established for the aforementioned problems and we provide explicit test supermartingales and $e$-processes satisfying this notion in the more general case. Furthermore, we derive matching lower and upper bounds on the expected rejection time for the resulting sequential tests in all of these cases. The proofs of these results make weak, algorithm-agnostic moment assumptions and rely on a general-purpose proof technique involving the aforementioned regret and a family of numeraire portfolios. Finally, we discuss how all of these theorems hold in a distribution-uniform sense, a notion of log-optimality that is stronger still and seems to be new to the literature.

Universal Log-Optimality for General Classes of e-processes and Sequential Hypothesis Tests

TL;DR

This work develops a general betting framework for sequential hypothesis testing with composite alternatives, showing that any e-process with sublinear portfolio regret achieves adaptive, asymptotic, and almost-sure log-optimality relative to a family of log-optimal portfolios.A primary novelty is the universal log-optimality result: the growth rate under any alternative Q is maximized by the log-optimal portfolio λ_Q^*, and the long-run log-wealth converges to ℓ_Q^* almost surely, matching any other admissible process within the class.The authors derive sharp, α→0⁺ bounds for the expected rejection time E_Q[τ_α], establishing both lower bounds for arbitrary strategies and achievable upper bounds under sublinear regret, with explicit nonasymptotic expressions and moment conditions.The theory is instantiated to common sequential testing problems (one- and two-sided bounded-mean tests, and difference-in-means tests for bounded tuples) and extended to a generalized regret framework with numeraire portfolios, yielding distribution-uniform log-optimality and uniform bounds on rejection times.Overall, the results provide a constructive, algorithm-agnostic toolkit for optimal sequential testing across a broad spectrum of nonparametric problems, and they introduce distribution-uniform notions of log-optimality that strengthen prior understandings.

Abstract

We consider the problem of sequential hypothesis testing by betting. For a general class of composite testing problems -- which include bounded mean testing, equal mean testing for bounded random tuples, and some key ingredients of two-sample and independence testing as special cases -- we show that any -process satisfying a certain sublinear regret bound is adaptively, asymptotically, and almost surely log-optimal for a composite alternative. This is a strong notion of optimality that has not previously been established for the aforementioned problems and we provide explicit test supermartingales and -processes satisfying this notion in the more general case. Furthermore, we derive matching lower and upper bounds on the expected rejection time for the resulting sequential tests in all of these cases. The proofs of these results make weak, algorithm-agnostic moment assumptions and rely on a general-purpose proof technique involving the aforementioned regret and a family of numeraire portfolios. Finally, we discuss how all of these theorems hold in a distribution-uniform sense, a notion of log-optimality that is stronger still and seems to be new to the literature.

Paper Structure

This paper contains 31 sections, 15 theorems, 139 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Preliminaries: notation and background
  3. 1. Notation and formalities.
  4. 2. $e$-processes.
  5. 3. Stopping times.
  6. 4. Deterministic portfolio regret.
  7. A concrete testing problem to keep in mind: bounded means
  8. Universal, asymptotic, almost-sure log-optimality
  9. Log-optimality via sublinear portfolio regret and universal portfolios
  10. Sharp bounds on the expected rejection time
  11. Optimal expected rejection times via sublinear portfolio regret
  12. Implications for some familiar sequential testing problems
  13. One-sided tests for the mean of a bounded random variable
  14. Two-sided tests for the mean of a bounded random variable
  15. Difference-in-means testing for bounded random tuples
  16. ...and 16 more sections

Key Result

Theorem 1

Let ${W} \equiv ({W}_n)_{{n}={1}}^{\infty}$ be a test $\mathcal{P}$-supermartingale taking the general form in eq:intro-general-test-sm. Let $\mathcal{Q}$ be any alternative hypothesis for which $(E_n^{(1)})_{{n}={1}}^{\infty}$ and $(E_n^{(2)})_{{n}={1}}^{\infty}$ are marginally i.i.d., such as If $(\lambda_n)_{{n}={1}}^{\infty}$ is chosen according to any algorithm with a sublinear portfolio reg

Figures (4)

  • Figure 1: Inclusions of testing problems considered in this paper. Those discussed in \ref{['section:corollaries']} use test supermartingales that are all special cases of that displayed in \ref{['eq:intro-general-test-sm']}. The properties of \ref{['eq:intro-general-test-sm']} are proven for more general test supermartingales discussed in \ref{['section:generalizations']}. We nevertheless focus our main discussions on sequential tests that fall under \ref{['eq:intro-general-test-sm']} as it strikes a balance between generality and concreteness.
  • Figure 2: Empirical growth rates (left) and distributions of rejection times (right) for various $e$-processes (see \ref{['section:preliminaries']} for a precise definition). As will be discussed in \ref{['corollary:adaptive optimality of UP and OJ']}, the $e$-processes labeled "Univ. Portfolio," "Regret-CO96," and "Regret-OJ23" all satisfy a sublinear portfolio regret bound. For this reason, they all have growth rates converging to $\ell_Q^\star$ and expected rejection times close to $\log(1/\alpha) / \ell_Q^\star$. In particular, Online Newton Step (ONS) is a commonly studied strategy in the literature for deriving growth rate and rejection time guarantees but it does not satisfy a portfolio regret bound, and can consequently be suboptimal from both of the perspectives discussed in the introduction. More details can be found in \ref{['section:asymptotic-equivalence', 'section:stopping time']}.
  • Figure 3: Empirical growth rates $\log ({W}_n) / n$ for various $e$-processes that test whether the mean of a bounded random variable is at most $0.3$ or $0.4$---for scenarios (a) and (b), respectively (the exact $e$-processes are discussed in \ref{['corollary:adaptive optimality of UP and OJ']} and \ref{['section:bounded one sided']}). In the left-hand and right-hand side plots, the true distributions of the bounded observations are Bernoulli(0.4) and Bernoulli(0.9) and hence the optimal log-wealth increments $\ell_Q^\star$ are 0.023 and 0.55, respectively. In particular, the maximizer $\lambda_Q^\star$ of the growth rate is within the implicit range $[1/4, 3/4]$ that is available in the regret bound for ONS in the first scenario while it is outside of that range in the second. Hence, in the latter case, only those $e$-processes with sublinear portfolio regret (Univ. Portfolio, Regret-CO96, Regret-OJ23) are asymptotically equivalent to the wealth of the log-optimal strategy $\lambda_Q^\star$. Due to numerical instabilities that arise for the Univ. Portfolio bets in the second scenario, we plot the conservative Regret-CO96-based empirical growth rate in place of Univ. Portfolio for very large $n$.
  • Figure 4: Distributions of the first rejection time for various $e$-processes under two scenarios for $\alpha = 0.01$. In particular, the log-optimal strategy lies inside the allowable range available to ONS in the scenario considered in the left-hand plot whereas it lies outside this range for that of the right-hand plot. Consequently, we see that the distribution of rejection times for $e$-processes with sublinear portfolio regret lie near the optimum $\log(1/\alpha) / \ell_Q^\star$ in both cases, whereas those of ONS only do so in the left-hand plot.

Theorems & Definitions (31)

  • Theorem : An informal summary of the main results of \ref{['section:asymptotic-equivalence', 'section:stopping time']}
  • Definition 1: Universal, asymptotic, and almost-sure log-optimality and equivalence
  • Remark 2.1
  • Theorem 2.1: Asymptotic log-optimality of $e$-processes with sublinear portfolio regret
  • Corollary 2.2
  • Proposition 3.1: Expected rejection times of constant rebalanced portfolios
  • Proposition 3.2: A lower bound on the expected rejection time of any betting strategy
  • Theorem 3.3: The expected rejection time of sublinear portfolio regret $e$-processes
  • Corollary 4.1: Log-optimality and expected rejection times for one-sided bounded mean testing
  • Corollary 4.2: Log-optimality and expected rejection times for two-sided bounded mean testing
  • ...and 21 more