Optimal Rates for Nonparametric Density Estimation under Communication Constraints

TL;DR

This paper develops a nearly optimal nonparametric density estimator under communication constraints, leveraging wavelet sparsity and adaptive techniques, with theoretical bounds supporting its near-optimality.

Contribution

It introduces a novel adaptive estimator exploiting wavelet sparsity under limited communication, and establishes minimax lower bounds demonstrating near rate-optimality.

Findings

Estimator is nearly rate-optimal for Besov spaces.

Wavelet-based method performs well under communication constraints.

Lower bounds confirm the estimator's near optimality.

Abstract

We consider density estimation for Besov spaces when each sample is quantized to only a limited number of bits. We provide a noninteractive adaptive estimator that exploits the sparsity of wavelet bases, along with a simulate-and-infer technique from parametric estimation under communication constraints. We show that our estimator is nearly rate-optimal by deriving minimax lower bounds that hold even when interactive protocols are allowed. Interestingly, while our wavelet-based estimator is almost rate-optimal for Sobolev spaces as well, it is unclear whether the standard Fourier basis, which arise naturally for those spaces, can be used to achieve the same performance.