On the near-optimality of betting confidence sets for bounded means

TL;DR

This paper demonstrates that betting confidence intervals and sequences for bounded means are nearly optimal, outperforming classical methods both asymptotically and in finite samples, with strong theoretical guarantees.

Contribution

The paper provides a theoretical comparison showing betting CIs are asymptotically narrower and nearly optimal, establishing lower bounds and matching them with betting methods.

Findings

Betting CIs have smaller asymptotic width than empirical Bernstein CIs.

Betting CIs and CSs match fundamental lower bounds up to constants.

Betting methods outperform classical approaches in finite-sample regimes.

Abstract

Constructing nonasymptotic confidence intervals (CIs) for the mean of a univariate distribution from independent and identically distributed (i.i.d.) observations is a fundamental task in statistics. For bounded observations, a classical nonparametric approach proceeds by inverting standard concentration bounds, such as Hoeffding's or Bernstein's inequalities. Recently, an alternative betting-based approach for defining CIs and their time-uniform variants called confidence sequences (CSs), has been shown to be empirically superior to the classical methods. In this paper, we provide theoretical justification for this improved empirical performance of betting CIs and CSs. Our main contributions are as follows: (i) We first compare CIs using the values of their first-order asymptotic widths (scaled by $\sqrt{n}$), and show that the betting CI of Waudby-Smith and Ramdas (2023) has a smaller limiting width than existing empirical Bernstein (EB)-CIs. (ii) Next, we establish two lower bounds that characterize the minimum width achievable by any method for constructing CIs/CSs in terms of certain inverse information projections. (iii) Finally, we show that the betting CI and CS match the fundamental limits, modulo an additive logarithmic term and a multiplicative constant. Overall these results imply that the betting CI~(and CS) admit stronger theoretical guarantees than the existing state-of-the-art EB-CI~(and CS); both in the asymptotic and finite-sample regimes.