Martingale Methods for Sequential Estimation of Convex Functionals and Divergences

TL;DR

This paper introduces a unified martingale-based approach for sequentially estimating convex divergences and functionals, enabling the creation of confidence sequences with minimal additional cost over fixed-time bounds.

Contribution

It develops a novel technique using reverse submartingales and maximal inequalities to convert offline concentration bounds into time-uniform confidence sequences for a wide range of convex divergences and functionals.

Findings

Sequential bounds have only an iterated logarithmic overhead compared to fixed-time bounds.

The method applies to various divergences including Wasserstein, Kullback-Leibler, and kernel maximum mean discrepancy.

The approach extends to general convex functionals and processes with leave-one-out properties.

Abstract

We present a unified technique for sequential estimation of convex divergences between distributions, including integral probability metrics like the kernel maximum mean discrepancy, $\varphi$-divergences like the Kullback-Leibler divergence, and optimal transport costs, such as powers of Wasserstein distances. This is achieved by observing that empirical convex divergences are (partially ordered) reverse submartingales with respect to the exchangeable filtration, coupled with maximal inequalities for such processes. These techniques appear to be complementary and powerful additions to the existing literature on both confidence sequences and convex divergences. We construct an offline-to-sequential device that converts a wide array of existing offline concentration inequalities into time-uniform confidence sequences that can be continuously monitored, providing valid tests or confidence intervals at arbitrary stopping times. The resulting sequential bounds pay only an iterated logarithmic price over the corresponding fixed-time bounds, retaining the same dependence on problem parameters (like dimension or alphabet size if applicable). These results are also applicable to more general convex functionals -- like the negative differential entropy, suprema of empirical processes, and V-Statistics -- and to more general processes satisfying a key leave-one-out property.