Universal Quantile Estimation with Feedback in the Communication-Constrained Setting

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 fusionbased 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 m sensors receives n observations, the mean-squared error of the optimal centralized protocol decays as O( 1 nm ). First, we describe a decentralized protocol based on k = m 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 O( 1 n √ m ), whereas for step sizes decaying as m−1/2 , it achiveves the same order of MSE—namely, O( ...

• Type: Journal
• Year: 2007
• Where: CORR
• Authors: Ram Rajagopal, Martin J. Wainwright
• Keywords: CORR 2007, Decentralized Protocol, Education, Single Bit, Standard Fusionbased Architecture