  This paper studies the problem of mapping optimization in decentralized
control problems. A global optimization algorithm is proposed based on the
ideas of ``deterministic annealing" - a powerful non-convex optimization
framework derived from information theoretic principles with analogies to
statistical physics. The key idea is to randomize the mappings and control the
Shannon entropy of the system during optimization. The entropy constraint is
gradually relaxed in a deterministic annealing process while tracking the
minimum, to obtain the ultimate deterministic mappings. Deterministic annealing
has been successfully employed in several problems including clustering, vector
quantization, regression, as well as the Witsenhausen's counterexample in our
recent work[1]. We extend our method to a more involved setting, a variation of
Witsenhausen's counterexample, where there is a side channel between the two
controllers. The problem can be viewed as a two stage cancellation problem. We
demonstrate that there exist complex strategies that can exploit the side
channel efficiently, obtaining significant gains over the best affine and known
non-linear strategies.
