Continuous Random Variable Estimation is not Optimal for the Witsenhausen Counterexample

Mael Le Treust (1) , Tobias J Oechtering (2)

1 ETIS - UMR 8051 - Equipes Traitement de l'Information et Systèmes
2 EECS - School of Electrical Engineering and Computer Science (KTH Royal Institute of Technology)

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 interim state, i.e. the result of the first decision maker action, 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 interim state that cannot be described by a continuous random variable which has a probability density function.

Domaines: Théorie de l'information [cs.IT]; Informatique et théorie des jeux [cs.GT]; Sciences de l'information et de la communication

Mael Le Treust, Tobias J Oechtering. Continuous Random Variable Estimation is not Optimal for the Witsenhausen Counterexample. IEEE ISIT 2021, Jul 2021, melbourne, Australia. ⟨hal-03223047v2⟩