A Deterministic Annealing Approach to Witsenhausen's Counterexample

Abstract

This paper proposes a numerical method, based on information theoretic ideas, to a class of distributed control problems. As a particular test case, the well-known and numerically "over-mined" problem of decentralized control and implicit communication, commonly referred to as Witsenhausen's counterexample, is considered. The method provides a small improvement over the best numerical result so far for this benchmark problem. The key idea is to randomize the zero-delay mappings. which become "soft", probabilistic mappings to be optimized in a deterministic annealing process, by incorporating a Shannon entropy constraint in the problem formulation. The entropy of the mapping is controlled and gradually lowered to zero to obtain deterministic mappings, while avoiding poor local minima. Proposed method obtains new mappings that shed light on the structure of the optimal solution, as well as achieving a small improvement in total cost over the state of the art in numerical approaches to this problem.