Speaker: Aaditya Ramdas, Carnegie Mellon University

Title: Estimating means of bounded random variables by betting

Abstract: We derive confidence intervals (CI) and time-uniform confidence sequences (CS) for the classical problem of estimating an unknown mean from bounded observations. We present a general approach for deriving concentration bounds, that can be seen as a generalization (and improvement) of the celebrated Chernoff method. At its heart, it is based on deriving a new class of composite nonnegative martingales, with strong connections to Kelly betting and Robbins' method of mixtures. We show how to extend these ideas to sampling without replacement, another heavily studied problem. In all cases, our bounds are adaptive to the unknown variance, and empirically vastly outperform competing approaches based on Hoeffding or empirical Bernstein inequalities and their recent supermartingale generalizations. In short, we establish a new state-of-the-art for four fundamental problems: CSs and CIs for bounded means, with and without replacement. This work is joint with my student, Ian Waudby-Smith, a preprint is available here: https://arxiv.org/abs/2010.09686.