Causal Vector-valued Witsenhausen Counterexamples with Feedback

TL;DR

The paper studies the vector-valued Witsenhausen counterexample through empirical coordination and analyzes three setups with causal encoding/decoding and channel feedback. It shows that time-sharing is necessary to convexify the cost region in the causal-causal setting, and that feedback does not enlarge the region when both decision makers are causal, but does help when the decoder is noncausal, via a single-letter information constraint $I(W_1; Y_1) - I(U_2; X_0|W_1,Y_1) \ge 0$. The results provide precise single-letter characterizations of achievable cost regions and settle open questions about the role of feedback in vector-valued Witsenhausen problems, using genie-aided arguments and empirical coordination. Overall, the work clarifies when feedback is beneficial in distributed control with non-classical information patterns and highlights time-sharing as a key mechanism for convexifying performance regions in vector settings.

Abstract

We study the continuous vector-valued Witsenhausen counterexample through the lens of empirical coordination coding. We characterize the region of achievable pairs of costs in three scenarios: (i) causal encoding and causal decoding, (ii) causal encoding and causal decoding with channel feedback, and (iii) causal encoding and noncausal decoding with channel feedback. In these vector-valued versions of the problem, the optimal coding schemes must rely on a time-sharing strategy, since the region of achievable pairs of costs might not be convex in the scalar version of the problem. We examine the role of the channel feedback when the encoder is causal and the decoder is either causal or non-causal, and we show that feedback improves the performance, only when the decoder is non-causal.

Figures (6)

• Figure 1: Vector-valued Witsenhausen counterexample with causal encoder and causal decoder.
• Figure 2: Binary information source and binary symmetric stochastic encoder and decoder.
• Figure 3: The joint distribution induced by a binary system depicted in Fig. \ref{['fig: binary example1']}
• Figure 4: First row: joint distribution given by $(\alpha = 0,\beta = 0)$, second row: joint distribution given by $(\alpha = 1,\beta = 1)$, third row: a convex combination of the above two cases, but is not achievable by any single pair of $(\alpha,\beta)$.
• Figure 5: Witsenhausen counterexample for causal-encoding and causal-decoding with channel feedback $Y_1^{t-1}$ to the encoder. The dotted line describes the source feed-forward $X_0^{t-1}$ to the decoder, that is used for the genie-aided argument in the converse proof of Theorem \ref{['theorem: c-c w-f']} in Appendix B.

Theorems & Definitions (17)

• Definition 2.1
• Definition 2.2
• Theorem 2.3
• Remark 2.4
• Remark 2.5
• Example
• Definition 2.6
• Theorem 2.7
• Definition 2.8
• Theorem 2.9