Is Witsenhausen's counterexample a relevant toy?

Abstract

This paper answers a question raised by Doyle on the relevance of the Witsenhausen counterexample as a toy decentralized control problem. The question has two sides, the first of which focuses on the lack of an external channel in the counterexample. Using existing results, we argue that the core difficulty in the counterexample is retained even in the presence of such a channel. The second side questions the LQG formulation of the counterexample. We consider alternative formulations and show that the understanding developed for the LQG case guides the investigation for these other cases as well. Specifically, we consider 1) a variation on the original counterexample with general, but bounded, noise distributions, and 2) an adversarial extension with bounded disturbance and quadratic costs. For each of these formulations, we show that quantization-based nonlinear strategies outperform linear strategies by an arbitrarily large factor. Further, these nonlinear strategies also perform within a constant factor of the optimal, uniformly over all possible parameter choices (for fixed noise distributions in the Bayesian case). Fortuitously, the assumption of bounded noise results in a significant simplification of proofs as compared to those for the LQG formulation. Therefore, the results in this paper are also of pedagogical interest.