A Class of Near-Optimal Local Minima for Witsenhausen’s Problem

Abstract

This paper is the continuation of the research initiated in our previous work aiming to analytically support the existence of near-piecewise-linear local optima for Witsenhausen’s problem for a highly volatile state, which had been suspected of being the optimal solution in the past based on numerical evidence. The core idea of our previous work was to pose Witsenhausen’s problem as a leader-follower coordination game of asymmetric information, and carefully construct a class of near-piecewise-linear strategies for the leader that stays invariant under the best response operator. This invariance set of strategies was then used to show the existence of a near-piecewise-linear equilibrium strategy for the leader. One major contribution of the current work is to refine the set of strategies proposed in our previous work to include more efficient local minima. In fact, we explicitly specify a range of values for the number of segments in the proposed near-piecewise-linear strategies (as a function of the state variance), which yields local minima of the same order as of the global optimal solution. This has required major refinement of the bounds and proofs in the best response analysis. As another key contribution and to compare the performance of the proposed local optima with the global optimal solution, we use results from the asymptotic quantization theory to upper-bound the cost of the proposed local minima. Combining it with an asymptotic upper bound on the optimal cost obtained from existing results in the literature, we show that the cost of the proposed local minima are at most a constant factor away from the optimal cost.