On the Efficient Implementation of High Accuracy Optimality of Profile Maximum Likelihood

TL;DR

This paper introduces an efficient, unified plug-in estimator based on profile-maximum-likelihood (PML) that achieves near-optimal accuracy for estimating symmetric properties of distributions, surpassing previous polynomial-time methods.

Contribution

The authors develop a sample-optimal PML-based estimator for symmetric properties, improving accuracy thresholds from  n^{-1/4} to  n^{-1/3}, and establish its theoretical optimality limits.

Findings

Achieves  n^{-1/3} accuracy threshold for symmetric property estimation.

Surpasses previous polynomial-time estimators with  n^{-1/4} threshold.

Reaches the theoretical limit for universal symmetric property estimation.

Abstract

We provide an efficient unified plug-in approach for estimating symmetric properties of distributions given $n$ independent samples. Our estimator is based on profile-maximum-likelihood (PML) and is sample optimal for estimating various symmetric properties when the estimation error $\epsilon \gg n^{-1/3}$. This result improves upon the previous best accuracy threshold of $\epsilon \gg n^{-1/4}$ achievable by polynomial time computable PML-based universal estimators [ACSS21, ACSS20]. Our estimator reaches a theoretical limit for universal symmetric property estimation as [Han21] shows that a broad class of universal estimators (containing many well known approaches including ours) cannot be sample optimal for every $1$-Lipschitz property when $\epsilon \ll n^{-1/3}$.