From Optimal Transport to Optimal Quantization: A Variational Study of the Witsenhausen Counterexample

RESUMO

We study the scalar Witsenhausen counterexample through optimal transport. Absorbing the first control into a transport map recasts the problem as the variational problem $J^=\inf_Q{k^2W_2^2(P,Q)+\operatorname{mmse}(Q)}$ over target laws $Q$. This formulation balances quadratic Wasserstein transport cost against minimum mean-square estimation error, with the optimal first controller recovered as the monotone rearrangement that pushes the prior $P$ to a minimizing law $Q^$. We establish existence and absolute continuity of $Q^$, derive an Euler-Lagrange condition, give an equivalent Fisher-information representation of the estimation cost, and obtain a semi-closed-form Gaussian benchmark with an explicit threshold for linear optimality. Restricting the target law to finite support yields an MMSE-regularized optimal quantization problem. Its stationarity conditions couple centroid levels with Voronoi decision cells and reduce to the classical Lloyd-Max conditions when control is expensive. We solve this nonconvex problem using a deterministic-annealing homotopy in the control penalty $k$. Finally, we derive the small- and large-$k$ asymptotics of $J^$ and identify the limiting controllers: a linear controller when control is expensive and a two-level signaling quantizer when it is cheap. Numerical results illustrate both regimes.