A lower confidence sequence for the changing mean of non-negative right heavy-tailed observations with bounded mean

TL;DR

This paper introduces a non-parametric, non-asymptotic lower confidence sequence for the changing mean of non-negative, heavy-tailed data with bounded mean, applicable in various settings including bounded rewards and importance weights.

Contribution

It develops a novel lower confidence sequence that adapts to heavy-tailed data and can be efficiently approximated, improving sequential inference methods.

Findings

Dominates empirical Bernstein supermartingale when variance is finite

Adapts to known or unknown higher moments in infinite variance cases

Can be converted into a closed-interval confidence sequence with shrinking width

Abstract

A confidence sequence (CS) is an anytime-valid sequential inference primitive which produces an adapted sequence of sets for a predictable parameter sequence with a time-uniform coverage guarantee. This work constructs a non-parametric non-asymptotic lower CS for the running average conditional expectation whose slack converges to zero given non-negative right heavy-tailed observations with bounded mean. Specifically, when the variance is finite the approach dominates the empirical Bernstein supermartingale of Howard et. al.; with infinite variance, can adapt to a known or unknown $(1 + \delta)$-th moment bound; and can be efficiently approximated using a sublinear number of sufficient statistics. In certain cases this lower CS can be converted into a closed-interval CS whose width converges to zero, e.g., any bounded realization, or post contextual-bandit inference with bounded rewards and unbounded importance weights. A reference implementation and example simulations demonstrate the technique.