The finite-dimensional Witsenhausen counterexample

TL;DR

This paper extends the analysis of Witsenhausen's counterexample to finite-dimensional vector cases, deriving bounds and demonstrating that lattice-based strategies perform near-optimally across all dimensions.

Contribution

It introduces a new finite-dimensional lower bound using sphere-packing concepts and shows lattice strategies are within a constant factor of optimal for all vector lengths.

Findings

Lattice strategies achieve near-optimality within a constant factor.

Derived a new lower bound for finite-dimensional Witsenhausen problem.

The gap for the scalar case is never more than a factor of 8.

Abstract

Recently, a vector version of Witsenhausen's counterexample was considered and it was shown that in that limit of infinite vector length, certain quantization-based control strategies are provably within a constant factor of the optimal cost for all possible problem parameters. In this paper, finite vector lengths are considered with the dimension being viewed as an additional problem parameter. By applying a large-deviation "sphere-packing" philosophy, a lower bound to the optimal cost for the finite dimensional case is derived that uses appropriate shadows of the infinite-length bound. Using the new lower bound, we show that good lattice-based control strategies achieve within a constant factor of the optimal cost uniformly over all possible problem parameters, including the vector length. For Witsenhausen's original problem -- the scalar case -- the gap between regular lattice-based strategies and the lower bound is numerically never more than a factor of 8.