Approximately Optimal Solutions to the Finite-Dimensional Witsenhausen Counterexample

Recently, a vector version of Witsenhausen's counterexample was considered and it was shown that in the asymptotic limit of infinite vector length, certain vector-quantization-based control strategies are provably within a constant factor of the asymptotically optimal cost for all possible problem parameters. While suggestive, a constant factor result for the finite-dimensional problem has remained elusive. In this paper, we provide a resolution to this issue. By applying a large-deviation "sphere-packing" philosophy, we derive a lower bound to the optimal cost for the finite dimensional case that uses appropriate shadows of an existing vector lower bound that is the same for all dimensions. Using this new lower bound, we show that good lattice-quantization-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-which is the scalar case-the gap between regular lattice-quantization-based strategies and the lower bound is provably never more than a factor of 100, and computer calculations strongly suggest that the factor in fact may be no larger than 8. Finally, to obtain a numerical understanding of the possible room for improvement in costs using alternative strategies, we also include numerical comparison with strategies that are conjectured to be optimal. Using this comparison, we posit that there is more room for improvement in our lower bounds than in our upper bounds.