Power-Estimation Trade-off of Vector-valued Witsenhausen Counterexample with Causal Decoder

TL;DR

This paper explores the power-estimation trade-off in a vector-valued Witsenhausen counterexample with a causal decoder, extending finite alphabet results to Gaussian setups and demonstrating the superiority of mixed auxiliary variables.

Contribution

It introduces a novel single-letter characterization of the optimal trade-off in the Gaussian vector case, using a mixed auxiliary variable approach.

Findings

Discrete and continuous auxiliary variables outperform traditional strategies.

A single-letter expression characterizes the power-estimation trade-off.

The optimal auxiliary variable combination remains an open problem.

Abstract

The vector-valued extension of the famous Witsenhausen counterexample setup is studied where the encoder, i.e. the first decision maker, non-causally knows and encodes the i.i.d. state sequence and the decoder, i.e. the second decision maker, causally estimates the interim state. The coding scheme is transferred from the finite alphabet coordination problem for which it is proved to be optimal. The extension to the Gaussian setup is based on a non-standard weak typicality approach and requires a careful average estimation error analysis since the interim state is estimated by the decoder. We provide a single-letter expression that characterizes the optimal trade-off between the Witsenhausen power cost and estimation cost. The two auxiliary random variables improve the communication with the decoder, while performing the dual role of the channel input, which also controls the state of the system. Interestingly, we show that a pair of discrete and continuous auxiliary random variables, outperforms both Witsenhausen two point strategy and the best affine policies. The optimal choice of random variables remains unknown.