Continuous Random Variable Estimation is not Optimal for the Witsenhausen Counterexample

TL;DR

This paper investigates the optimal control strategies for the Witsenhausen counterexample, revealing that assuming a continuous random variable for the internal state is generally suboptimal, thus challenging previous approaches.

Contribution

It characterizes the optimal control design in the vector setting and demonstrates the limitations of continuous random variable assumptions in the scalar problem.

Findings

Genie-aided outer bound established

Time-sharing strategies are suboptimal

Optimal scalar strategies involve non-continuous states

Abstract

Optimal design of distributed decision policies can be a difficult task, illustrated by the famous Witsenhausen counterexample. In this paper we characterize the optimal control designs for the vector-valued setting assuming that it results in an internal state that can be described by a continuous random variable which has a probability density function. More specifically, we provide a genie-aided outer bound that relies on our previous results for empirical coordination problems. This solution turns out to be not optimal in general, since it consists of a time-sharing strategy between two linear schemes of specific power. It follows that the optimal decision strategy for the original scalar Witsenhausen problem must lead to an internal state that cannot be described by a continuous random variable which has a probability density function.