Local Optimality of Almost Piecewise-Linear Quantizers for Witsenhausen's Problem

TL;DR

This paper models Witsenhausen's problem as a leader-follower game with incomplete information, identifying near piecewise-linear equilibria and demonstrating the existence of local minima in the form of slopey quantizers that are close to optimal.

Contribution

It introduces a game-theoretic framework for Witsenhausen's problem and proves the existence of local minima represented by slopey quantizers that approximate the optimal cost.

Findings

Existence of near piecewise-linear equilibria in the game.

Identification of slopey quantizers as local minima.

Quantifiers are within a constant factor of the optimal cost.

Abstract

We pose Witsenhausen's problem as 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 study the perfect Bayesian equilibria of the game and identify a class of "near piecewise-linear equilibria" when leader cares much more about being close to the follower than the state, and the state is highly volatile. As a major consequence of this result, we prove the existence of a set of local minima for Witsenhausen's problem in form of slopey quantizers, which are at most a constant factor away from the optimal cost.