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Optimal Gaussian Strategies for Vector-valued Witsenhausen Counterexample with Non-causal State Estimator

Mengyuan Zhao, Tobias J. Oechtering, Maël Le Treust

TL;DR

This paper studies the vector-valued Witsenhausen counterexample with a causal encoder (DM1) and a noncausal decoder (DM2) to quantify the trade-off between control power and estimation error. Under Gaussian assumptions, it derives a single-letter achievable Gaussian cost $S_G(P)$ and shows that a time-sharing between two affine strategies, i.e., the convex envelope of the linear cost, attains the optimum for $P$ in a certain range when $Q>4N$; specifically $S_G(P) = \frac{N(Q - N - P)}{Q}$ for $P$ in $[P_1,P_2]$, where $P_1 = \frac{1}{2}(Q - 2N - \sqrt{Q^2 - 4QN})$ and $P_2 = \frac{1}{2}(Q - 2N + \sqrt{Q^2 - 4QN})$. However, Witsenhausen's two-point strategy and Grover–Sahai's noncausal scheme can outperform the optimal Gaussian policy in some regimes, implying that block-coding gains require fully noncausal DMs. The analysis also shows that channel feedback yields no Gaussian gain and that the same optimal Gaussian cost appears across several setups with at least one causal controller, highlighting a separation between Gaussian achievability and non-Gaussian gains.

Abstract

In this study, we investigate a vector-valued Witsenhausen model where the second decision maker (DM) acquires a vector of observations before selecting a vector of estimations. Here, the first DM acts causally whereas the second DM estimates non-causally. When the vector length grows, we characterize, via a single-letter expression, the optimal trade-off between the power cost at the first DM and the estimation cost at the second DM. In this paper, we show that the best linear scheme is achieved by using the time-sharing method between two affine strategies, which coincides with the convex envelope of the solution of Witsenhausen in 1968. Here also, Witsenhausen's two-point strategy and the scheme of Grover and Sahai in 2010 where both devices operate non-causally, outperform our best linear scheme. Therefore, gains obtained with block-coding schemes are only attainable if all DMs operate non-causally.

Optimal Gaussian Strategies for Vector-valued Witsenhausen Counterexample with Non-causal State Estimator

TL;DR

This paper studies the vector-valued Witsenhausen counterexample with a causal encoder (DM1) and a noncausal decoder (DM2) to quantify the trade-off between control power and estimation error. Under Gaussian assumptions, it derives a single-letter achievable Gaussian cost and shows that a time-sharing between two affine strategies, i.e., the convex envelope of the linear cost, attains the optimum for in a certain range when ; specifically for in , where and . However, Witsenhausen's two-point strategy and Grover–Sahai's noncausal scheme can outperform the optimal Gaussian policy in some regimes, implying that block-coding gains require fully noncausal DMs. The analysis also shows that channel feedback yields no Gaussian gain and that the same optimal Gaussian cost appears across several setups with at least one causal controller, highlighting a separation between Gaussian achievability and non-Gaussian gains.

Abstract

In this study, we investigate a vector-valued Witsenhausen model where the second decision maker (DM) acquires a vector of observations before selecting a vector of estimations. Here, the first DM acts causally whereas the second DM estimates non-causally. When the vector length grows, we characterize, via a single-letter expression, the optimal trade-off between the power cost at the first DM and the estimation cost at the second DM. In this paper, we show that the best linear scheme is achieved by using the time-sharing method between two affine strategies, which coincides with the convex envelope of the solution of Witsenhausen in 1968. Here also, Witsenhausen's two-point strategy and the scheme of Grover and Sahai in 2010 where both devices operate non-causally, outperform our best linear scheme. Therefore, gains obtained with block-coding schemes are only attainable if all DMs operate non-causally.
Paper Structure (4 sections, 8 theorems, 46 equations, 2 figures)

This paper contains 4 sections, 8 theorems, 46 equations, 2 figures.

Table of Contents

  1. Introduction
  2. System Model
  3. Optimal Gaussian Cost
  4. Discussions

Key Result

Theorem II.3

The pair of Witsenhausen costs $(P,S)$ is achievable if and only if there exists a joint distribution over the random variables $(X_0, W_1, W_2, U_1, X_1, Y_1, U_2)$ that decomposes according to such that where $\mathcal{P}_{X_0}$ and $\mathcal{P}_{X_1, Y_1|X_0, U_1}$ are the given Gaussian distributions, $W_1,W_2$ are two auxiliary random variables, and where the mutual information $I(W_1, W_2;

Figures (2)

  • Figure 1: The i.i.d. state and the channel noise are drawn according to Gaussian distributions $X_0^{n}\sim \mathcal{N}(0,Q\mathbb{I})$ and $Z_1^{n}\sim \mathcal{N}(0,N\mathbb{I})$.
  • Figure 2: Comparison of the four cost functions $S_\ell(P)$, $S_{\mathsf{G}}(P)$, $S_2(P)$ and $S_{\mathsf{dpc}}(P)$. In this particular case, $S_2(P)$ outperforms $S_{\mathsf{G}}(P)$ and $S_\ell(P)$, and the cost induced by the non-causal strategy $S_{\mathsf{dpc}}(P)$ outperforms all other cost functions.

Theorems & Definitions (18)

  • Definition II.1
  • Definition II.2
  • Theorem II.3: zhao2024coordination
  • Remark II.4
  • Definition III.1
  • Proposition III.2
  • Lemma III.3: witsenhausen1968, Treust2024power
  • Theorem III.4: Main Result
  • Lemma III.5
  • Corollary III.6
  • ...and 8 more