On Confidence Sequences for Bounded Random Processes via Universal Gambling Strategies

TL;DR

This paper introduces a new confidence sequence algorithm for bounded random processes using a gambling approach inspired by Cover's universal portfolio, achieving efficient performance with constant time complexity.

Contribution

It develops a novel algorithm based on a mixture of lower bounds that closely mimics Cover's universal portfolio, with improved theoretical properties and computational efficiency.

Findings

The algorithm provides tight confidence sequences for bounded processes.

It achieves constant per-round computational complexity.

A new higher-order lower bound technique is introduced.

Abstract

This paper considers the problem of constructing a confidence sequence, which is a sequence of confidence intervals that hold uniformly over time, for estimating the mean of bounded real-valued random processes. This paper revisits the gambling-based approach established in the recent literature from a natural \emph{two-horse race} perspective, and demonstrates new properties of the resulting algorithm induced by Cover (1991)'s universal portfolio. The main result of this paper is a new algorithm based on a mixture of lower bounds, which closely approximates the performance of Cover's universal portfolio with constant per-round time complexity. A higher-order generalization of a lower bound on a logarithmic function in (Fan et al., 2015), which is developed as a key technique for the proposed algorithm, may be of independent interest.