Some new numeric results concerning the Witsenhausen Counterexample

The Witsenhausen counterexample has been studied for decades as an elemental example of a challenging decentralized stochastic control problem. Its nonclassical information pattern induces an implicit need for controllers to ‘communicate’ (i.e., reduce uncertainty for another) via appropriate control actions. Nonlinear control strategies are required to do this well, but finding them is nontrivial since the problem is not convex. This paper uses contemporary neural-network toolkits to revisit this problem numerically and explore what makes it challenging. We find that without appropriately biasing the architecture to favor the kinds of solutions (”slopey quantizers”) that are believed to be nearly optimal, ample local minima make it hard to chance upon good solutions. Furthermore, inspired by earlier work that suggested that vector-versions of the counterexample should be easier to understand, we see numerically that indeed by going to even 2D, we can get better performance than the best known in 1D.