The finite-dimensional Witsenhausen counterexample

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.

arXiv comment32 pages, 7 figures, 1 table. Presented at ConCom 2009, Seoul, Korea. Submitted to IEEE Transactions on Automatic Control

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• The finite-dimensional Witsenhausen counterexample
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Cited by in corpus (4)

• The finite-dimensional Witsenhausen counterexample
• Information embedding meets distributed control
• On the Inapproximability of the Discrete Witsenhausen Problem
• Iterative Source-Channel Coding Approach to Witsenhausen's Counterexample