Zero Estimation Cost Strategy for Witsenhausen Counterexample with Causal Encoder

TL;DR

This work addresses the vector-valued Witsenhausen counterexample under a causal encoder and noncausal decoder by framing the problem in coordination coding to study the power-estimation tradeoff. It introduces a zero estimation cost (ZEC) scheme that uses a Gaussian and a discrete auxiliary variable to deterministically describe the state, enabling block coding to achieve $S=0$ for sufficiently large power $P$ and defines the minimum power $P^*$ needed for this zero-cost regime. An extension, the Non-ZEC scheme, adds a test channel with crossover $\gamma$ to trade off estimation accuracy against power, effectively implementing a time-sharing between the two-point strategy and ZEC. Numerical results show substantial power gains for zero-cost reconstruction and indicate that time-sharing between the Gaussian optimal strategy and ZEC can outperform Non-ZEC in the tested scenarios.

Abstract

We propose a zero estimation cost (ZEC) scheme for causal-encoding noncausal-decoding vector-valued Witsenhausen counterexample based on the coordination coding result. In contrast to source coding, our goal is to communicate a controlled system state. The introduced ZEC scheme is a joint control-communication approach that transforms the system state into a sequence that can be efficiently communicated using block coding. Numerical results show that our approach significantly reduces the power required for achieving zero-estimation-cost state reconstruction at the decoder. In the second part, we introduce a more general non-zero estimation cost (Non-ZEC) scheme. We observe numerically that the Non-ZEC scheme operates as a time-sharing mechanism between the two-point strategy and the ZEC scheme. Overall, by leveraging block-coding gain, our proposed methods substantially improve the power-estimation trade-off for Witsenhausen counterexample.

Figures (5)

• Figure 1: Causal-encoding noncausal-decoding vector-valued Witsenhausen counterexample.
• Figure 2: Comparison of the four cost functions $S_{\mathsf{ZEC}}(P)$, $S_2(P)$, $S_\ell(P)$, and $S_{\mathsf{G}}(P)$ at $Q=1,N=0.15$. Our proposed scheme strictly outperforms the other strategies and achieves a zero-estimation-cost state reconstruction when $P\geq P^*= 0.383$.
• Figure 3: Variation of $P^*$ as a function of the noise level $N$ when $Q=1$. $P^*=P_2^{\min}=0.363$ when $N$ is small, and increases to $P^*=Q=1$ for $N\geq 0.65$.
• Figure 4: Evolution of $F(a,\gamma,P)$ with different values of $\gamma=0,0.05, 0.1,0.5$. When $\gamma=0$, $F(a,0,P) = S_{\mathsf{ZEC}}(P)$ in \ref{['eq: cost function lossless']}, and when $\gamma=0.5$, the lower boundary of $F(a,0.5,P)$ recovers that of the two-point strategy (yellow dashed curve).
• Figure 5: Comparison of the four closed-form cost functions $S_{\mathsf{Non}\text{-}\mathsf{ZEC}}(P)$, $S_2(P)$, $S_\ell(P)$, $S_{\mathsf{G}}(P)$, and the induced time-sharing cost $S_{\mathsf{t}\text{-}\mathsf{s}}(P)$ at $Q=1,N=0.15$.

Theorems & Definitions (9)

• Definition 1
• Definition 2
• Theorem 3: zhao2024coordination
• Remark 1
• Lemma 4
• Theorem 5: witsenhausen1968
• Theorem 6
• Theorem 7
• proof : Full Derivation for Theorem \ref{['thm: coord_2_testch']}