STaR-Bets: Sequential Target-Recalculating Bets for Tighter Confidence Intervals

TL;DR

This paper introduces STaR-Bets, a novel betting-based algorithm that constructs confidence intervals with optimal width guarantees, outperforming existing methods in fixed horizon settings for bounded random variables.

Contribution

It proposes a new betting strategy that guarantees near-optimal confidence interval widths with finite-time guarantees, improving upon prior heuristic and sub-optimal approaches.

Findings

Empirically outperforms existing confidence interval methods.

Achieves near-optimal width bounds up to a diminishing factor.

Provides finite-time guarantees for confidence interval construction.

Abstract

The construction of confidence intervals for the mean of a bounded random variable is a classical problem in statistics with numerous applications in machine learning and virtually all scientific fields. In particular, obtaining the tightest possible confidence intervals is vital every time the sampling of the random variables is expensive. The current state-of-the-art method to construct confidence intervals is by using betting algorithms. This is a very successful approach for deriving optimal confidence sequences, even matching the rate of law of iterated logarithms. However, in the fixed horizon setting, these approaches are either sub-optimal or based on heuristic solutions with strong empirical performance but without a finite-time guarantee. Hence, no betting-based algorithm guaranteeing the optimal $\mathcal{O}(\sqrt{\frac{\sigma^2\log\frac1\delta}{n}})$ width of the confidence intervals are known. This work bridges this gap. We propose a betting-based algorithm to compute confidence intervals that empirically outperforms the competitors. Our betting strategy uses the optimal strategy in every step (in a certain sense), whereas the standard betting methods choose a constant strategy in advance. Leveraging this fact results in strict improvements even for classical concentration inequalities, such as the ones of Hoeffding or Bernstein. Moreover, we also prove that the width of our confidence intervals is optimal up to an $1+o(1)$ factor diminishing with $n$. The code is available at https://github.com/vvoracek/STaR-bets-confidence-interval.