The finite-dimensional Witsenhausen counterexample

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 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 "sphere-packing" philosophy, a lower bound to the optimal cost for this finite-length problem is derived that uses appropriate shadows of the infinite-length bound. We also introduce lattice-based quantization strategies for any finite length. Using the new finite-length lower bound, we show that good 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 -- the scalar case -- regular lattice-based strategies are observed to num...

• Type: Journal
• Year: 2010
• Where: CORR
• Authors: Pulkit Grover, Se Yong Park, Anant Sahai
• Keywords: CORR 2010, Education, Optimal Cost, Problem Parameters, Vector Length