Multi-Terminal Remote Generation and Estimation Over a Broadcast Channel With Correlated Priors

TL;DR

This paper investigates multi-terminal remote estimation over broadcast channels with correlated priors, proposing a hierarchical importance sampling strategy that leverages correlation to reduce communication costs.

Contribution

It introduces a hierarchical importance sampling method utilizing common information among priors to optimize remote generation over broadcast channels.

Findings

Hierarchical importance sampling reduces communication costs when priors are correlated.

Upper bounds on bias and estimation error for unicast transmission are established.

Splitting transmission into broadcast and unicast phases improves efficiency.

Abstract

We study the multi-terminal remote estimation problem under a rate constraint, in which the goal of the encoder is to help each decoder estimate a function over a certain distribution -- while the distribution is known only to the encoder, the function to be estimated is known only to the decoders, and can also be different for each decoder. The decoders can observe correlated samples from prior distributions, instantiated through shared randomness with the encoder. To achieve this, we employ remote generation, where the encoder helps decoders generate samples from the underlying distribution by using the samples from the prior through importance sampling. While methods such as minimal random coding can be used to efficiently transmit samples to each decoder individually using their importance scores, it is unknown if the correlation among the samples from the priors can reduce the communication cost using the availability of a broadcast link. We propose a hierarchical importance sampling strategy that facilitates, in the case of non-zero G\'acs-K\"orner common information among the priors of the decoders, a common sampling step leveraging the availability of a broadcast channel. This is followed by a refinement step for the individual decoders. We present upper bounds on the bias and the estimation error for unicast transmission, which is of independent interest. We then introduce a method that splits into two phases, dedicated to broadcast and unicast transmission, respectively, and show the reduction in communication cost.