From Optimal Transport to Optimal Quantization: A Variational Study of the Witsenhausen Counterexample
Abstract
Witsenhausen's counterexample is a canonical problem in decentralized control: a two-stage linear-quadratic-Gaussian team problem in which nonlinear policies strictly outperform every affine one. This paper develops a variational and computational framework for the scalar counterexample based on optimal transport. Building on the transport formulation of Wu and Verdú, we absorb the first controller into a transport map and recast the problem as J⋆=infQ{k2W22(P,Q)+mmse(Q)}, which trades a quadratic Wasserstein transport cost against a minimum mean-square estimation error. The optimal first controller is the monotone rearrangement that pushes the prior P to a minimizer Q⋆. We establish the existence and absolute continuity of minimizers, derive an Euler--Lagrange optimality condition, and obtain a semi-closed-form Gaussian benchmark with an explicit second-variation stability threshold. We then show that restricting to finitely supported laws yields an MMSE-regularized optimal quantizer: its optimal values decrease to J⋆, its weak limit points are minimizers of the continuous problem, and its stationarity conditions pair centroid levels with power cells, recovering Lloyd--Max quantization as control becomes expensive. This structure leads to an estimation-aware Lloyd algorithm with signalling-based initialization. As k→∞, the optimal controllers converge to the classical affine pair; as k→0, J⋆(k)=σ2k2+o(k2) and the optimal laws collapse onto δ0, so signalling constellations are a moderate-k phenomenon. Numerical experiments corroborate the theory, and the framework extends to the vector counterexample, where the finite-level program becomes an MMSE-regularized vector quantizer.
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BibTeX
@misc{zhu2026from,
title = {{From Optimal Transport to Optimal Quantization: A Variational Study of the Witsenhausen Counterexample}},
author = {Quanyan Zhu},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2609.18992},
eprint = {2609.18992},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.18992},
note = {CC BY-NC-SA 4.0},
}
RIS
TY - UNPB
TI - From Optimal Transport to Optimal Quantization: A Variational Study of the Witsenhausen Counterexample
AU - Quanyan Zhu
PY - 2026
DA - 2026/10/08
PB - arXiv
DO - 10.48550/arxiv.2609.18992
UR - https://arxiv.org/abs/2609.18992
AB - Witsenhausen's counterexample is a canonical problem in decentralized control: a two-stage linear-quadratic-Gaussian team problem in which nonlinear policies strictly outperform every affine one. This paper develops a variational and computational framework for the scalar counterexample based on optimal transport. Building on the transport formulation of Wu and Verdú, we absorb the first controller into a transport map and recast the problem as J^⋆=inf_Q{k^2W_2^2(P,Q)+mmse(Q)}, which trades a quadratic Wasserstein transport cost against a minimum mean-square estimation error. The optimal first controller is the monotone rearrangement that pushes the prior P to a minimizer Q^⋆. We establish the existence and absolute continuity of minimizers, derive an Euler--Lagrange optimality condition, and obtain a semi-closed-form Gaussian benchmark with an explicit second-variation stability threshold. We then show that restricting to finitely supported laws yields an MMSE-regularized optimal quantizer: its optimal values decrease to J^⋆, its weak limit points are minimizers of the continuous problem, and its stationarity conditions pair centroid levels with power cells, recovering Lloyd--Max quantization as control becomes expensive. This structure leads to an estimation-aware Lloyd algorithm with signalling-based initialization. As k→∞, the optimal controllers converge to the classical affine pair; as k→0, J^⋆(k)=σ^2k^2+o(k^2) and the optimal laws collapse onto δ_0, so signalling constellations are a moderate-k phenomenon. Numerical experiments corroborate the theory, and the framework extends to the vector counterexample, where the finite-level program becomes an MMSE-regularized vector quantizer.
ER -