A Deterministic Annealing Approach to Witsenhausen's Counterexample

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.Comment: submitted to ISIT'1

A Generic Solver for Unconstrained Control Problems with Integral Functional Objectives

We present a generic solver for unconstrained control problems (UCPs) whose objectives take the form of an integral functional of the controllers. The solver generalizes and improves upon the algorithm in [1] for the Witsenhausen’s counterexample, which provides the best-known results. In essence, we show that minimizing the objective implies minimizing the marginal cost functions almost everywhere, and we perform the latter task pointwisely by the adaptive minimization technique, which speeds up the computation. We implement single-threaded and parallelized versions of the proposed algorithm. Our implementation runs 30× faster than the algorithm in [1] on the Witsenhausen’s counterexample, and we demonstrate the applicability of the solver and discuss the possible generalization to constrained problems and multidimensional controllers through three more examples

Slopey quantizers are locally optimal for Witsenhausen's counterexample

We study the perfect Bayesian equilibria of a leader-follower game of incomplete information. The follower makes a noisy observation of the leader's action (who moves first) and chooses an action minimizing her expected deviation from the leader's action. Knowing this, leader who observes the realization of the state, chooses an action that minimizes her distance to the state of the world and the ex-ante expected deviation from the follower's action. We show the existence of what we call “near piecewise-linear equilibria” when there is strong complementarity between the leader and the follower and the precision of the prior is poor. As a major consequence of this result, we prove local optimality of a class of slopey quantization strategies which had been suspected of being the optimal solution in the past, based on numerical evidence for Witsenhausen's counterexample

A Deterministic Annealing Framework for Global Optimization of Delay-Constrained Communication and Control Strategies

This dissertation is concerned with the problem of global optimization of delay constrained communication and control strategies. Specifically, the objective is to obtain optimal encoder and decoder functions that map between the source space and the channel space, to minimize a given cost functional. The cost surfaces associated with these problems are highly complex and riddled with local minima, rendering gradient descent based methods ineffective. This thesis proposes and develops a powerful non-convex optimization method based on the concept of deterministic annealing (DA) - which is derived from information theoretic principles with analogies to statistical physics, and was successfully employed in several problems including vector quantization, classification and regression. DA has several useful properties including reduced sensitivity to initialization and strong potential to avoid poor local minima. DA-based optimization methods are developed here for the following fundamental communication problems: the Wyner-Ziv setting where only a decoder has access to side information, the distributed setting where independent encoders transmit over independent channels to a central decoder, and analog multiple descriptions setting which is an extension of the well known source coding problem of multiple descriptions. Comparative numerical results are presented, which show strict superiority of the proposed method over gradient descent based optimization methods as well as prior approaches in literature. Detailed analysis of the highly non-trivial structure of obtained mappings is provided. The thesis further studies the related problem of global optimization of controller mappings in decentralized stochastic control problems, including Witsenhausen's celebrated 1968 counter-example. It is well-known that most decentralized control problems do not admit closed-form solutions and require numerical optimization. An optimization method is developed, based on DA, for a class of decentralized stochastic control problems. Comparative numerical results are presented for two test problems that show strict superiority of the proposed method over prior approaches in literature, and analyze the structure of obtained controller functions

J.S.: Learning approaches to the witsenhausen counterexample from a view of potential games

Abstract — Since Witsenhausen put forward his remarkable counterexample in 1968, there have been many attempts to develop efficient methods for solving this non-convex functional optimization problem. However there are few methods designed from game theoretic perspectives. In this paper, after discretizing the Witsenhausen counterexample and re-writing the formulation in analytical expressions, we use fading memory JSFP with inertia, one learning approach in games, to search for better controllers from a view of potential games. We achieve a better solution than the previously known best one. Moreover, we show that the learning approaches are simple and automated and they are easy to extend for solving general functional optimization problems. I