# Inference under Information Constraints II: Communication Constraints and Shared Randomness

## TL;DR

This paper investigates distributed statistical inference under communication constraints, demonstrating the optimality of certain protocols for distribution learning and testing, and highlighting the benefits of shared randomness.

## Contribution

It introduces a simulate-and-infer strategy for private-coin protocols and a novel public-coin protocol that outperforms it, achieving sample-optimality in distribution testing.

## Key findings

- Simulate-and-infer is sample-optimal for distribution learning.
- Public-coin protocol outperforms private-coin protocols in distribution testing.
- Shared randomness enhances the efficiency of distributed inference protocols.

## Abstract

A central server needs to perform statistical inference based on samples that are distributed over multiple users who can each send a message of limited length to the center. We study problems of distribution learning and identity testing in this distributed inference setting and examine the role of shared randomness as a resource. We propose a general-purpose simulate-and-infer strategy that uses only private-coin communication protocols and is sample-optimal for distribution learning. This general strategy turns out to be sample-optimal even for distribution testing among private-coin protocols. Interestingly, we propose a public-coin protocol that outperforms simulate-and-infer for distribution testing and is, in fact, sample-optimal. Underlying our public-coin protocol is a random hash that when applied to the samples minimally contracts the chi-squared distance of their distribution to the uniform distribution.

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Source: https://tomesphere.com/paper/1905.08302