Instance Based Approximations to Profile Maximum Likelihood

TL;DR

This paper introduces a new efficient algorithm for approximating the profile maximum likelihood distribution, improving computational efficiency especially for instances with few distinct frequencies, and extends this to practical estimators for symmetric properties.

Contribution

The paper presents a novel sparsity-exploiting algorithm for approximate PML computation, along with the first provably efficient implementation of PseudoPML for broad symmetric property estimation.

Findings

Algorithm matches best known efficiency for approximate PML

First provable efficient implementation of PseudoPML

Empirical evaluation shows practical effectiveness

Abstract

In this paper we provide a new efficient algorithm for approximately computing the profile maximum likelihood (PML) distribution, a prominent quantity in symmetric property estimation. We provide an algorithm which matches the previous best known efficient algorithms for computing approximate PML distributions and improves when the number of distinct observed frequencies in the given instance is small. We achieve this result by exploiting new sparsity structure in approximate PML distributions and providing a new matrix rounding algorithm, of independent interest. Leveraging this result, we obtain the first provable computationally efficient implementation of PseudoPML, a general framework for estimating a broad class of symmetric properties. Additionally, we obtain efficient PML-based estimators for distributions with small profile entropy, a natural instance-based complexity measure. Further, we provide a simpler and more practical PseudoPML implementation that matches the best-known theoretical guarantees of such an estimator and evaluate this method empirically.