The finite-dimensional Witsenhausen counterexample

Recently, we considered a vector version of Witsenhausen´s counterexample and used a new lower bound to show that in that limit of infinite vector length, certain quantization-based 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 vector length being viewed as an additional problem parameter. By applying the ldquosphere-packingrdquo philosophy, a lower bound to the optimal cost for this finite-length problem is derived that uses appropriate shadows of the infinite-length bounds. We also introduce lattice-based quantization strategies for any finite length. Using the new finite-length lower bound, we show that the lattice-based 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 corresponds to the scalar case - lattice-based strategies attain within a factor of 8 of the optimal cost. Based on observations in the scalar case and the infinite-dimensional case, we also conjecture what the optimal strategies could be for any finite vector length.