Universal Quantile Estimation with Feedback in the Communication-Constrained Setting

TL;DR

This paper investigates decentralized quantile estimation under communication constraints, proposing protocols that match or approach the centralized optimal mean-squared error decay rates, with analysis and simulations demonstrating their effectiveness.

Contribution

It introduces new decentralized protocols with limited feedback that achieve near-optimal or optimal asymptotic mean-squared error decay rates in quantile estimation.

Findings

A protocol with  bits of feedback matches centralized asymptotic MSE.

A single-bit feedback protocol achieves   MSE decay with appropriate step sizes.

Simulations confirm the theoretical tradeoffs and performance of the proposed protocols.

Abstract

We consider the following problem of decentralized statistical inference: given i.i.d. samples from an unknown distribution, estimate an arbitrary quantile subject to limits on the number of bits exchanged. We analyze a standard fusion-based architecture, in which each of $m$ sensors transmits a single bit to the fusion center, which in turn is permitted to send some number $k$ bits of feedback. Supposing that each of $\nodenum$ sensors receives $n$ observations, the optimal centralized protocol yields mean-squared error decaying as $\order(1/[n m])$. We develop and analyze the performance of various decentralized protocols in comparison to this centralized gold-standard. First, we describe a decentralized protocol based on $k = \log(\nodenum)$ bits of feedback that is strongly consistent, and achieves the same asymptotic MSE as the centralized optimum. Second, we describe and analyze a decentralized protocol based on only a single bit ($k=1$) of feedback. For step sizes independent of $m$, it achieves an asymptotic MSE of order $\order[1/(n \sqrt{m})]$, whereas for step sizes decaying as $1/\sqrt{m}$, it achieves the same $\order(1/[n m])$ decay in MSE as the centralized optimum. Our theoretical results are complemented by simulations, illustrating the tradeoffs between these different protocols.