Causal Coordination for Distributed Decision-Making

TL;DR

This work reframes the Witsenhausenn counterexample in distributed decision-making through empirical coordination, deriving a single-letter power-estimation-cost region for a vector-valued, causal-encoder/ noncausal-decoder setting. It introduces Zero Estimation Cost (ZEC) schemes that achieve S = 0 with dramatically reduced power, including a multi-level ZEC-k extension and a feedback-enabled ZEC-f variant that can attain zero power and zero estimation cost in the low-noise regime. A unified Gaussian-optimality insight shows time-sharing among linear policies suffices under Gaussianity, and the ZEC framework leverages discrete-continuous hybrids to tightly control the information constraint. Practically, the results demonstrate that modest coordination messages, coupled with structured quantization and feedback, can significantly improve the power-cost trade-off in decentralized control, providing a blueprint for cooperative guidance in distributed systems.

Abstract

In decentralized network control, communication plays a critical role by transforming local observations into shared knowledge, enabling agents to coordinate their actions. This paper investigates how communication facilitates cooperation behavior and therefore improves the overall performance in the vector-valued Witsenhausen counterexample, a canonical toy example in distributed decision-making. We consider setups where the encoder, i.e., the first decision-maker (DM) acts causally and the decoder, i.e., the second DM, operates noncausally, 1) without and 2) with access to channel feedback. Using a coordination coding framework, we characterize the achievable power-estimation cost regions in single-letter expressions for both scenarios. The first result is that, when restricted to Gaussian random variables, the cost is identical across all setups featuring at least one causal DM - regardless of the presence of feedback information. Next, building on the characterization of the power-estimation cost region, we propose a hybrid scheme that combines discrete quantization with a continuous Gaussian codebook - the Zero Estimation Cost (ZEC) scheme - which achieves an arbitrarily small estimation cost. This scheme uses coding tools that allow perfect reconstruction of the target symbols, leading to an asymptotic estimation cost equal to zero, while significantly reducing the asymptotic power consumption. Furthermore, when channel feedback is available at the first DM, we propose an analogous scheme that simultaneously achieves zero power and zero estimation cost in the low-noise regime.

Figures (11)

• Figure 1: Original scalar-valued Witsenhausen counterexample. $X_0\sim\mathcal{N}(0,Q)$ is the source state. The first DM $f: X_0\rightarrow U_1$. Then, $X_1=X_0+U_1$, and $Y_1 = X_1+Z_1$, where $Z_1\sim\mathcal{N}(0,N)$ is the Gaussian noise. At last, the second DM $g: Y_1\rightarrow U_2$ estimates the interim state $X_1$.
• Figure 2: Vector-valued Witsenhausen counterexample with causal encoder and noncausal decoder. 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 $Z_1^{n}\sim \mathcal{N}(0,N\mathbb{I})$. At each time instant $t\in\{1,\ldots,n\}$, the causal encoder takes the past observations $X_0^t$ and generates $U_{1,t}$. The noncausal decoder receives the whole sequence $Y_1^n$ and outputs $U_2^n$, which serves as the MMSE estimator of $X_1^n$.
• Figure 3: Illustration of the quantization step function $Q_k(\cdot)$ with $k=3$ (up) and $k=4$ (down): The 3-point quantization is parameterized by $0=a_1\leq a_2,0=B_1\leq B_2$, whereas the 4-point function is parameterized by $0< a_1< a_2,0=B_1\leq B_2$
• Figure 4: Witsenhausen counterexample for causal-encoding and noncausal-decoding model with channel feedback. At each time, the causal encoder observes the past sequence $X_0^t$ and the channel output $Y_1^{t-1}$ with one time-step delay, and generates $U_{1,t}$. In the end, the noncausal decoder takes the whole vector $Y_1^n$ and outputs $U_2^n$.
• Figure 5: Minimum required power cost $P^*$ as a function of the noise variance $N$ for the single-shot strategies, noncausal strategy that combines linear and DPC scheme, and our proposed ZEC-2, ZEC-3, ZEC-4, and ZEC-f schemes when $Q=1$.

Theorems & Definitions (27)

• Definition 2.1
• Definition 2.2
• Theorem 3.1
• Remark 3.2
• Theorem 3.3
• Corollary 3.4
• Proposition 4.1
• Theorem 4.2
• Definition 4.3: $k$-point quantization
• Theorem 4.4: $k$-point strategy