Dimension-Free Empirical Entropy Estimation

TL;DR

This paper introduces a novel entropy estimator for discrete distributions with infinite alphabets, achieving near-optimal finite-sample accuracy bounds under a natural moment assumption, and extends classical entropy continuity results.

Contribution

It proposes the first finite-sample entropy estimates for infinite alphabets under a minimal moment assumption, improving bounds and resolving open problems in the field.

Findings

Achieves nearly minimax optimal rates for entropy estimation with infinite alphabets.

Provides sharper empirical bounds than existing methods across various distributions.

Introduces a dimension-free analogue of the Cover-Thomas entropy continuity result.

Abstract

We seek an entropy estimator for discrete distributions with fully empirical accuracy bounds. As stated, this goal is infeasible without some prior assumptions on the distribution. We discover that a certain information moment assumption renders the problem feasible. We argue that the moment assumption is natural and, in some sense, {\em minimalistic} -- weaker than finite support or tail decay conditions. Under the moment assumption, we provide the first finite-sample entropy estimates for infinite alphabets, nearly recovering the known minimax rates. Moreover, we demonstrate that our empirical bounds are significantly sharper than the state-of-the-art bounds, for various natural distributions and non-trivial sample regimes. Along the way, we give a dimension-free analogue of the Cover-Thomas result on entropy continuity (with respect to total variation distance) for finite alphabets, which may be of independent interest. Additionally, we resolve all of the open problems posed by J\"urgensen and Matthews, 2010.