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# Causal Coordination for Distributed Decision-Making

Mengyuan Zhao, Tobias Oechtering, Maël Le Treust

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Mengyuan Zhao, Tobias Oechtering, Maël Le Treust. Causal Coordination for Distributed Decision-Making. 2025. hal-05314441

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# Causal Coordination for Distributed Decision-Making

Mengyuan Zhao, Tobias J. Oechtering, Maël Le Treust,

## 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.

## Index Terms

decentralized control, distributed decision-making, Witsenhausen counterexample, causal state communication, coordination coding, zero estimation cost, channel feedback

## I. INTRODUCTION

Decentralized control studies systems where multiple controllers or decision-makers (DMs) act based on different, locally available information. Among its most well-known challenges is the Witsenhausen counterexample, a deceptively canonical two-stage linear-quadratic-Gaussian (LQG) problem introduced in 1968 [5]. Despite its simplest formulation of scalar dynamics and quadratic cost, Witsenhausen showed that the optimal strategy is nonlinear, contradicting the classical separation principle that guarantees linear optimality in fully observed LQG settings [6]. In fact, even a nonlinear binary strategy, i.e., Witsenhausen's two-point strategy, outperforms the best linear strategy. More than five decades later, this problem remains unsolved in closed form and continues to serve as a benchmark for understanding the interplay between control and communication in decentralized systems. It has inspired a broad body of work on stochastic teams and optimality of affine policies [7], [8], control design under SNR and rate constraints [9]–[13], and information-theoretic perspectives on coordination, communication, and security [14]–[18].

The key challenge of the Witsenhausen counterexample is the *nonclassical information structure* [19]–[21]: The second DM receives a noisy observation of the first DM’s action but does not know its input. This information structure turns the Witsenhausen optimization problem into a non-convex problem, making the search for an optimal strategy difficult [7], [22]. An important question is the *dual role of control* at the first DM: it must spend some power to simultaneously regulate the system state while embedding and transmitting relevant information that aids the second DM’s estimation. Seen in this way, affine strategies appear ineffective with respect to both objectives: they consume more power for control yet keep the complexity of the message, whereas the Witsenhausen’s two-point strategy communicates more clearly by forcing more distinguishable signals [23].

Inspired by this, over the past five decades, researchers have developed different approaches based on quantization and lattice-based control policies. To name a few, [24], [25] adopted the numerical optimization approach, where the latter one uncovered a sloped step-like structure in near-optimal policies. [26] considered a generalized *n*-step quantization strategy. A quantized team policies have shown to be asymptotically optimal in [27]. Using a game-theoretic approach, the slopey quantization approach has been shown to be locally optimal in [28]. From the optimal transport point of view, [29] showed that the optimal decision strategy is a strictly increasing unbounded piecewise real analytic function with a real analytic left inverse. More control strategies have also been proposed in [30]–[32].

This work is supported by Swedish Research Council (VR) under grant 2020-03884. The work of M. Le Treust is supported by the French National Agency for Research (ANR) via the project n°ANR-22-PEFT-0010 of the France 2030 program PEPR Futur Networks.

The results have been presented in part at the 2024 IEEE International Symposium on Information Theory (ISIT) [1], at the 2024 IEEE Conference on Decision and Control (CDC) [2], at the 2024 IEEE Information Theory Workshop (ITW) [3], and at the 2025 IEEE ISIT [4].

M. Zhao and T. J. Oechtering are with the Division of Information Science and Engineering, KTH Royal Institute of Technology, 100 44 Stockholm, Sweden {mzhao, oech}@kth.se

M. Le Treust is with CNRS, University of Rennes, Inria, IRISA UMR 6074, F-35000 Rennes, France mael.le-treust@cnrs.fr
Despite its control-theoretic aspect, Witsenhausen counterexample effectively shows a strong communication requirement: one DM needs to embed and share information to another through its action. As a result, this problem has increasingly been interpreted through an information-theoretic lens, as a one-shot joint source-channel coding problem. For example, [33] developed an algorithm using an iterative joint source-channel coding approach that achieves better performances. However, compared to Shannon's traditional point-to-point communication setup [34], the goal here is to communicate the controlled interim state that results from the encoder's control action, rather than the intact original state itself. As a result, the problem becomes more dynamic and target-adjustable ('make the plant dance' [35]), which requires additional communication. This perspective motivates the study of vector-valued extensions of the Witsenhausen problem [36], in which the scalar states are replaced by vectors and multiple transmissions in blocks are considered. This formulation facilitates the adoption of many block-coding methods [37], such as Costa's dirty-paper coding (DPC) [38], state amplification [39], and state masking [40], etc. For example, Grover and Sahai derived lower and upper bounds on the averaged Witsenhausen pair of costs and established the constant-factor optimality of quantization-based strategies in both the finite-dimensional [41], and infinite-dimensional [36] cases when both DMs are noncausal. The upper bound for the latter case was established using a combination of linear and DPC schemes. Using this developed coding structure, [42] further characterized the optimal power-distortion trade-off. Some additional results for the generalization to the Witsenhausen counterexample are established in [43]–[46], building on the vector-valued extension.

A central question we want to ask is: under such a nonclassical information structure, how much communication is required, or more precisely, what information is relevant to communicate, so that the two potential conflicting objectives in the Witsenhausen counterexample no longer work against each other, but instead cooperate to achieve improved performance. In [47], [48], the authors investigated the communication requirement by introducing an external communication channel connecting the two noncausal DMs. By transmitting a truncated binary expansion of the source state, thereby enforcing the target state to a quantization point, they showed that the resulting scheme can simplify the source, reduce the costs, and achieve an $\varepsilon$-optimal asymptotic performances. Related results have been obtained for scalar-valued counterparts as well in [49], [50].

In this paper, we adopt the empirical coordination coding framework, formulated in [51]–[53], where the “coordination capacity” determines the minimum communication rate needed to induce the cooperative behavior of agents. Based on local observations, the encoder and decoder choose their action, not only to ensure reliable message transmission, but also to achieve a broader common objective. More precisely, they implement a coding scheme that ensures that the empirical distribution of symbol sequences converges to a target probability distribution. The characterization of the set of achievable joint probability distributions relies on a single-letter information constraint that depends on the local information available in the system. In [15], the coordination of two agents with a common average payoff function is considered, the Witsenhausen cost function being a special case. In [54], a point-to-point scenario is studied where the encoder is noncausal and the decoder is strictly causal based on its past observation. This work also provides a literature review on coordination results. In [55], necessary and sufficient conditions are established for several causal coding cases with two-sided state information. More optimal conditions for causal and noncausal encoding and decoding configurations are accessible in [56].

Prior to this work, Le Treust and Oechtering in [57] extended the finite-alphabet coordination problem to the Gaussian setting using a non-standard weak typicality approach, and investigated the noncausal-encoding and causal-decoding Witsenhausen setup. They provided an optimal single-letter characterization for the achievable power and estimation costs, that is, the image of the asymptotically averaged power and estimation costs induced by all possible pairs of noncausal encoding and causal decoding functions. This single-letter characterization involves auxiliary random variables (RVs) that highlight the dual role of control. Furthermore, they proposed a continuous-discrete hybrid design of the auxiliary RVs, inspired by DPC, which was shown to outperform both Witsenhausen's two-point strategy and the best affine policy.

However, in real-time communication systems, it is generally impractical for the encoder to wait until it fully aggregates the entire input sequence before acting. In this work, we consider a more realistic and operationally practical scenario: causal encoding and noncausal decoding setup. In this setup, the encoder observes and responds to data instantaneously as it arrives, while the decoder has the freedom to wait until all observations are available before deciding. Some causal transmission setups have been previously studied in the classical communication regime: Besides the pioneering work of Shannon [58], [59] studied causal and strictly causal state communication through a state-dependent channel. The problem of causal state estimation with a helper is analyzed in [60]. See more related work of side information available casually and noncausally to the transmitter in [61].

Our first contribution is the optimal single-letter characterization of the achievable power-estimation cost region in the causal encoder and noncausal decoder setup. This characterization employs two auxiliary RVs that capture the dual role of control: regulating the system state and embedding information for the second DM. The achievability proof extends the Markov block-coding approach from [59] with a refined cost analysis that improves upon the noncausal-encoding framework in [57]. This refinement is essential because, in the causal setting, the interim state depends dynamically on the first DM's causal observations, instead of the static source sequence in the classical noncausal setup. This result fully characterizes the fundamental trade-off between two stage costs and provides a basis for better understanding of what information is relevant and what communication scheme should be used to transmit that information. Next, we show that, when all the random variables involved are restricted to be Gaussian, all vector-valued Witsenhausen setups with either causal encoder or causal decoder, regardless of the help of
feedback or feedforward information, share a unified optimal solution: a time-sharing strategy over linear policies.

Motivated by this key observation, we propose a novel coding scheme — the Zero Estimation Cost (ZEC) scheme — which leverages the causal block-coding structure that enables perfect state reconstruction at the decoder. Inspired by Witsenhausen’s two-point strategy, the ZEC scheme drives the achievable estimation cost to zero, while saving a significant amount of power. When combined with the best linear strategy through time-sharing, the ZEC scheme strictly outperforms both the classical two-point method and the best affine policies across all possible directions in the power-estimation cost trade-off space. Then, we extend this design to a more general *k*-point ZEC scheme, which further reduces the minimum required power cost as the quantization level *k* increases.

Feedback, often an intrinsic component of communication and control systems, offers the potential to enhance performance by enabling DMs to refine their actions based on previous outcomes, thus reducing uncertainty about the unknown state. While in the classical point-to-point communication setting feedback does not increase the channel capacity, seminal work by Schalkwijk and Kailath [62] introduced a simple coding scheme that achieves an exponential decay of error probability, and error exponents for variable-length block codes are further investigated in [63]. Contradict to the point-to-point cause, feedback has proven helpful in enlarging the channel capacity in many multi-terminal scenarios. In particular, feedback can enlarge the capacity region and assist communication in the multiple-access channel [64], [65] as well as the broadcast channel [66], [67]. Within the empirical coordination framework, feedback has been shown to simplify the coordination problem. Specifically, [68] demonstrated that the perfect channel feedback enables the decoder to align its outputs more directly with the source state, reducing the complexity of the information constraint and decreasing the number of auxiliary RVs required. A state leakage and coordination problem with causal encoder at the presence of noisy channel feedback is considered in [69]. Together, these results show that channel feedback effectively enlarges the set of achievable joint distributions and strengthens coordination.

In this work, we characterize the region of achievable pairs of power and estimation costs for causal-encoding and noncausal-decoding with perfect channel feedback at the encoder. Consistent with the prior observations, the presence of feedback allows the encoder to anticipate potential decoding errors so as to recommend the most efficient action to the decoder. When meeting the associated information constraint, this mechanism effectively circumvents the core difficulty of the Witsenhausen problem — the nonclassical information structure. Indeed, thanks to the feedback, the action that is recommended to the second DM is also available at the first DM, ensuring a stronger alignment between their actions.

For this setup, we develop a second novel scheme — the Zero Estimation Cost with feedback (ZEC-f) scheme — analogous to the one proposed when there is no feedback. We show that feedback strictly improves performance: not only does the feedback-enabled scheme dominate its no-feedback counterpart, but surprisingly, in the low noise regime, it can also achieve the extreme point of zero power and zero estimation costs. Under our definition of achievable costs, this means that with only a finite number of effective transmissions from the encoder, the long-run average estimation cost at the decoder can be made arbitrarily small. Furthermore, we demonstrate that this scheme even strictly outperforms the noncausal scheme developed in [36], which relies on a combination of the linear scheme and the DPC scheme.

To summarize, this work demonstrates that even a moderate communication requirement can remarkably enhance coordination among multiple DMs and improve the power-estimation cost trade-off in the practically relevant causal-encoding noncausal-decoding Witsenhausen counterexample. Exploiting the advantage of communication from the fundamental coordination coding results, we establish that asymptotically zero estimation cost is achievable with significantly reduced power, and that perfect feedback enables the simultaneous attainment of zero power and zero estimation cost. More broadly, Witsenhausen counterexample provides a canonical setting for exploring a central question in general distributed decision-making: what is relevant to be communicated to establish cooperation. Our framework demonstrates how control actions shape information flow among DMs while communication enables tighter cooperation, more reliable control, and ultimately superior performance. This perspective underscores the importance of integrating coordination-coding and control to tackle fundamental challenges in decentralized and distributed systems.

This paper is structured as follows: Section II formulates the original scalar-valued Witsenhausen counterexample, reviews some basic results, and subsequently formulates its vector-valued extension with causal encoder and noncausal decoder. In Section III, we present its optimal single-letter cost region characterization and derive a unified optimal solution for all causal setups when every RV is restricted to Gaussian. We propose the ZEC scheme in Section IV and its *k*-point generalization. Section V formulates the causal-encoding noncausal-decoding with channel feedback model and provides the optimal cost region characterization, also in single-letter. Then, we provide a unified ZEC-f n scheme based on this setup. A numerical study is given in Section VI. Lastly, the conclusion follows in Section VII.

## II. SYSTEM MODEL

First, we introduce the notations applied throughout this paper. Capital letters, e.g. $X_0$, denote random variables, calligraphic fonts, e.g. $\mathcal{X}_0$, stand for alphabets where random variables take values, whereas lowercase letters, e.g. $x_0$, denote realisations of random variables. Sequences of length $n \in \mathbb{N}^*$ of random variables and realisations are denoted respectively by $X_0^n = (X_{0,1}, \dots, X_{0,t}, \dots, X_{0,n})$ and $x_0^n = (x_{0,1}, \dots, x_{0,t}, \dots, x_{0,n})$ for all $t \in \{1, \dots, n\}$. Furthermore, incomplete sequences up to stage $t \in \{1, \dots, n\}$ are similarly denoted by $X_0^t = (X_{0,1}, \dots, X_{0,t})$ and $x_0^t = (x_{0,1}, \dots, x_{0,t})$. Besides, $\mathbb{P}(\cdot), \mathcal{P}(\cdot)$ and $\mathcal{P}(\cdot|\cdot)$
<figure><img src="image_3.png" /><figcaption>Fig. 1. Original scalar-valued Witsenhausen counterexample. $X_0 \sim \mathcal{N}(0, Q)$ is the source state. The first DM $f: X_0 \to U_1$. 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 $g: Y_1 \to U_2$ estimates the interim state $X_1$.</figcaption></figure>

denote the general probability, probability density functions (PDFs) and conditional PDFs respectively. $1_{\{\cdot\}}$ is the indicator function. $\Phi(\cdot), \phi(\cdot)$ is the Gaussian cumulative density function (CDF) and PDF.

A. Original Model of Witsenhausen

In [5], Witsenhausen proposed the model illustrated in Figure 1 where the random variables $X_0, U_1, X_1, Z_1, Y_1, U_2$ stand for the source state, channel input, interim system state, channel noise, channel output, receiver's output respectively, and all take values from the real line $\mathbb{R}$. The source state and the noise are generated independently according to Gaussian distributions $X_0 \sim \mathcal{N}(0, Q), Z_1 \sim \mathcal{N}(0, N)$, for some strictly positive values $Q, N$. Moreover, $X_1$ and $Y_1$ are generated by

$$X_1 = X_0 + U_1, \qquad (1)$$

$$Y_1 = X_1 + Z_1 = X_0 + U_1 + Z_1. \qquad (2)$$

Furthermore, the control design

$$f_{U_1|X_0}: \mathcal{X}_0 \to \mathcal{U}_1, \quad g_{U_2|Y_1}: \mathcal{Y}_1 \to \mathcal{U}_2,$$

introduces a joint probability distribution over all the system symbols

$$\mathcal{P}_{X_0} f_{U_1|X_0} \mathcal{P}_{X_1, Y_1|X_0, U_1} g_{U_2|Y_1}. \qquad (3)$$

We denote by $\mathcal{P}_{X_0} = \mathcal{N}(0, Q)$ the generative Gaussian probability distribution of the state, and by $\mathcal{P}_{X_1, Y_1|X_0, U_1}$ the channel probability distribution according to (1) and (2). These two distributions are given and fixed by the system.

For a fixed trade-off constant $\lambda^2 > 0$, the objective is to design optimal controllers $(f,g)$ that minimize the weighted sum of power and estimation costs:

$$\min_{f,g} (\lambda^2 \cdot P + S), \qquad (4)$$

with

$$\text{power cost:} \qquad P = \mathbb{E}[U_1^2], \qquad (5)$$

$$\text{estimation cost:} \qquad S = \mathbb{E}[(X_1 - U_1)^2], \qquad (6)$$

where the expectations are taken over the distribution (3). In [5], Witsenhausen investigated the trade-off between the power cost $P$ at the first DM, and the estimation cost $S$ of the interim state at the second DM. It is well known that for a given $f$, the optimal $g$ is given by the minimum mean-square error (MMSE) estimator of $X_1$ given $Y_1$, see [70]. As a result, the estimation cost $S$ becomes

$$S = \mathbb{E}[(X_1 - \mathbb{E}[X_1|Y_1])^2].$$

Thus, the problem (4) effectively reduces to optimizing over only the first DM $f_{U_1|X_0}$.

In [5, Lemma 11], Witsenhausen determines the best linear policy to be $U_1 = -\sqrt{\frac{P}{Q}}X_0$, if $P \le Q$, otherwise $U_1 = -X_0 + \sqrt{P-Q}$. This induces the estimation cost function

$$S_{\ell}(P) = \begin{cases} \frac{(\sqrt{Q} - \sqrt{P})^2 \cdot N}{(\sqrt{Q} - \sqrt{P})^2 + N} & \text{if } P \in [0, Q], \\ 0 & \text{otherwise.} \end{cases} \qquad (7)$$

Note that the function $P \mapsto S_{\ell}(P)$ is generally not convex. Witsenhausen proposes the two-point strategy defined by

$$U_1 = a \cdot \text{sign}(X_0) - X_0, \qquad (8)$$

for some parameter $a \ge 0$, see [5, Theorem 2]. The corresponding power and estimation costs are given by

$$P_2(a) = Q + a \left( a - 2\sqrt{\frac{2Q}{\pi}} \right), \qquad (9)$$
<figure><img src="image_4.png" /><figcaption>Fig. 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 $t \in \{1, \dots, n\}$, 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 $X_1^n$.</figcaption></figure>

$$S_2(a) = a^2 \sqrt{\frac{2\pi}{N}} \phi \left( \frac{a}{\sqrt{N}} \right) \int \frac{\phi \left( \frac{ay_1}{\sqrt{N}} \right)}{\cosh \left( \frac{ay_1}{N} \right)} dy_1, \quad (10)$$

and the optimal receiver's strategy in this case is given by $\mathbb{E}[X_1|Y_1 = y_1] = a \tanh(\frac{ay_1}{N})$.

For some $Q$ and $N$, Witsenhausen's two-point strategy outperforms the best linear policy [5, Sec. 6]. The idea is to transform the continuous random variable $X_0$ into a two-point random variable $X_1 = U_1 + X_0 = a \cdot \text{sign}(X_0)$, so as to facilitate its estimation by the second DM.

## B. Vector-valued Extension with Causal Encoder

Contrast to the single-shot model above, we now consider the vector-valued Witsenhausen counterexample with causal encoder and noncausal decoder depicted in Figure 2.

The source states and channel noises are drawn independently according to the i.i.d. Gaussian distributions $X_0^n \sim \mathcal{N}(0, Q\mathbb{I})$ and $Z_1^n \sim \mathcal{N}(0, N\mathbb{I})$, for some $Q > 0$, $N > 0$. We denote by $X_1^n$ the memoryless interim state and $Y_1^n$ the output of the memoryless additive channel, where each symbol is generated according to (1) and (2).

We define the control design for this setup as follows.

**Definition II.1.** For $n \in \mathbb{N}^* = \mathbb{N}\{0\}$, a “control design” with causal encoder and noncausal decoder is a tuple of functions $c = (\{f_{U_{1,t}|X_0^t}^{(t)}\}_{t=1}^n, g_{U_2^n|Y_1^n})$ defined by

$$f_{U_{1,t}|X_0^t}^{(t)} : \mathcal{X}_0^t \to \mathcal{U}_1, \quad g_{U_2^n|Y_1^n} : \mathcal{Y}_1^n \to \mathcal{U}_2^n, \quad (11)$$

which induces a probability distribution over sequences of symbols

$$\prod_{t=1}^{n} \mathcal{P}_{X_{0,t}} \prod_{t=1}^{n} f_{U_{1,t}|X_0^t}^{(t)} \prod_{t=1}^{n} \mathcal{P}_{X_{1,t}, Y_{1,t}|X_{0,t}, U_{1,t}} g_{U_2^n|Y_1^n}. \quad (12)$$

We denote by $\mathcal{C}(n)$ the set of control designs with causal encoder and non-causal decoder.

As in [57], the objective is to determine the limit of the average pair of costs when block length $n \in \mathbb{N}^*$ grows.

**Definition II.2.** We define the two long-run cost functions

$$c_P(u_1^n) = \frac{1}{n} \sum_{t=1}^{n} (u_{1,t})^2, \quad (13)$$

$$c_S(x_1^n, u_2^n) = \frac{1}{n} \sum_{t=1}^{n} (x_{1,t} - u_{2,t})^2, \quad (14)$$

averaged over $n \in \mathbb{N}^*$ transmissions. The pair of costs $(P, S) \in \mathbb{R}^2$ is said to be achievable, if for all $\varepsilon > 0$, there exists $\tilde{n} \in \mathbb{N}^*$ such that for all $n \ge \tilde{n}$, there exists a control design $c \in \mathcal{C}(n)$ such that

$$\mathbb{E}[|P - c_P(U_1^n)| + |S - c_S(X_1^n, U_2^n)|] \le \varepsilon. \quad (15)$$

We denote by $\mathcal{R} \subset \mathbb{R}^2$ the set of all achievable pairs of costs obtained by the control designs with causal encoder and noncausal decoder.

By construction, $\mathcal{R}$ is defined through asymptotic average costs as the block length $n \to \infty$. Consequently, $\mathcal{R}$ is a closed subset of $\mathbb{R}^2$, containing all limit points of achievable pairs of costs. Next, we characterize $\mathcal{R}$ via a single-letter expression.
### III. CHARACTERIZATION OF THE COST REGION

Our main coding result characterizes the cost region $\mathcal{R}$.

**Theorem III.1.** The pair of Witsenhausen costs $(P, S) \in \mathcal{R}$ 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

$$ \mathcal{P}_{X_0} \mathcal{P}_{W_1} \mathcal{P}_{W_2 | X_0, W_1} \mathcal{P}_{U_1 | X_0, W_1} \mathcal{P}_{X_1, Y_1 | X_0, U_1} \mathcal{P}_{U_2 | W_1, W_2, Y_1}, \quad (16) $$

such that

$$ I(W_1, W_2; Y_1) - I(W_2; X_0 | W_1) \geq 0, \quad (17) $$

$$ P = \mathbb{E}[U_1^2], \quad S = \mathbb{E}[(X_1 - U_2)^2], \quad (18) $$

where $\mathcal{P}_{X_0}$ and $\mathcal{P}_{X_1, Y_1 | X_0, U_1}$ are the given distributions, and where $W_1, W_2$ are two auxiliary random variables.

The achievability and converse proofs are provided in Appendices A and B. The coding construction relies on the block-Markov coding scheme of [59], extended to the coordination framework in [55].

In Theorem III.1, the source $\mathcal{P}_{X_0}$ and the channel $\mathcal{P}_{X_1, Y_1 | X_0, U_1}$ are fixed by the system, and the other four probability distributions in (16) are the degrees of freedom, that must satisfy the information constraint (17) and the cost constraints (18). Furthermore, the two auxiliary RVs can be interpreted as follows: $W_1$ represents the independent codeword designed for the state-dependent channel with state $X_0$, consistent with the Shannon strategy [58]. $W_2$ is correlated with both $X_0$ and $W_1$, acting as a description of these two symbols. Both $W_1$ and $W_2$ are made available to the noncausal decoder. This formulation explicitly captures the dual role of control in Witsenhausen counterexample: $W_1$ regulates the controlled state $U_1$, whereas $W_2$ facilitates communication to the second DM.

We can gain a few insights from Theorem III.1.

**Remark III.2.** The decomposition of the probability distribution (16) is equivalent to the following Markov chains:

$$ \left\{ \begin{array}{l} X_0 \text{ is independent of } W_1, \\ U_1 \leftarrow (X_0, W_1) \leftarrow W_2, \\ (X_1, Y_1) \leftarrow (X_0, U_1) \leftarrow (W_1, W_2), \\ U_2 \leftarrow (W_1, W_2, Y_1) \leftarrow (X_0, U_1, X_1). \end{array} \right. \quad (19) $$

The first two Markov chains are consequences of causal encoding. The third Markov chain is related to the processing order of the Gaussian channel. The last Markov chain comes from noncausal decoding and symbol-wise reconstruction.

In the spirit of Witsenhausen's work, we determine the cost region characterized in Theorem III.1 when all the RVs are restricted to be jointly Gaussian: Given a power cost parameter $P \ge 0$, the optimal estimation cost writes

$$ S_G(P) = \inf_{P \in \mathcal{P}_G(P)} \mathbb{E} \left[ \left( X_1 - \mathbb{E}[X_1 | W_1, W_2, Y_1] \right)^2 \right], \quad (20) $$

where the set

$$ \mathbb{P}_G(P) = \left\{ \mathcal{P}_{W_1}, \mathcal{P}_{W_2 | X_0, W_1}, \mathcal{P}_{U_1 | X_0, W_1} : \begin{array}{l} P = \mathbb{E}[U_1^2], \\ I(W_1, W_2; Y_1) - I(W_2; X_0 | W_1) \ge 0, \text{ and} \\ X_0, W_1, W_2, U_1, X_1, Y_1, U_2 \text{ are jointly Gaussian} \end{array} \right\}. \quad (21) $$

Similarly to [57, Theorem 2] for causal-encoding and noncausal-decoding, we show that the estimation cost for joint Gaussian RVs is given by the convex envelop of the best linear cost function $S_\ell$ in (7).

**Theorem III.3.** When the RVs $(X_0, W_1, W_2, U_1, X_1, Y_1, U_2)$ are restricted to be jointly Gaussian, the closed form of the estimation cost (20) is given by

$$ S_G(P) = \begin{cases} \frac{N \cdot (Q - N - P)}{Q} & \text{if } Q > 4N \text{ and } P \in [P_1, P_2], \\ S_\ell(P) & \text{otherwise,} \end{cases} \quad (22) $$

where

$$ P_1 = \frac{1}{2}(Q - 2N - \sqrt{Q^2 - 4QN}), \quad (23) $$

$$ P_2 = \frac{1}{2}(Q - 2N + \sqrt{Q^2 - 4QN}). \quad (24) $$
The proof of Theorem III.3 is presented in Appendix C. The estimation-cost function $S_G(P)$ is attainable with a simple time-sharing strategy between the two operation points $P_1$ and $P_2$, even when both DMs operate causally.

In fact, when all involved RVs are restricted to be Gaussian, the same cost $S_G(P)$ is optimal for several superior causal configurations – those that permit feedback or feedforward links. The following corollary unifies these cases.

**Corollary III.4.** Under the joint Gaussian constraint, $S_G(P)$ in (22) is optimal for the following setups:

1) causal encoding noncausal decoding with channel feedback,

2) noncausal encoding causal decoding with source feedforward<fnref n="1" />,

3) causal encoding causal decoding with or without channel feedback, with or without source feedforward.

We provide the proof of this corollary in Appendix D. This result is remarkable since although these architectures have different fundamental achievable cost regions, but these regions coincide when all random variables are restricted to be Gaussian.

The next section drops the joint Gaussian assumption. We introduce the ZEC scheme, a continuous-discrete hybrid code that achieves zero estimation cost with a substantially low power cost.

## IV. ZERO ESTIMATION COST STRATEGY

Our objective is to design a scheme that achieves an estimation cost equal to zero, while reducing the required power cost as much as possible. Now, we first begin with a helpful single-shot benchmark. The proof of this result is in Appendix E.

**Proposition IV.1.** For original single-shot Witsenhausen counterexample in Sec. II-A with $N > 0$, the minimum power cost (5) for achieving a zero estimation cost (6) performance is $P = Q$. This is obtained by taking $U_1 = -X_0$.

This proposition states that if the first DM can completely cancel the continuous source state, perfect estimation becomes possible—but only with enough power budget $P \ge Q$. By using the coordination coding result of Theorem III.1, we seek to lower the power cost needed to achieve zero estimation cost.

### A. ZEC-2 Scheme

Inspired by Witsenhausen's two-point approach in (8), we propose a new coding scheme that induces a zero estimation cost, that we call the ZEC-2 scheme. Note that under Definition II.2 of achievable costs, zero estimation cost means the averaged long-run estimation cost (14) can be made arbitrarily small. The idea is to design the RVs ($X_0, W_1, W_2, U_1, X_1, Y_1, U_2$) involved in Theorem III.1, knowing that satisfying conditions (16), (17), (18) is equivalent to the existence of coding schemes that achieve the pair of costs ($P, S$). The ZEC-2 scheme employs a pair of auxiliary RVs: $W_1$ is a continuous Gaussian codebook and $W_2$ is a discrete binary random variable that quantizes the source state. This resembles the hybrid analog-digital design investigated in [71]. Now, for given parameters $V_1 \ge 0, a \ge 0, b \in \mathbb{R}$, the ZEC-2 scheme is designed as:

$$ 
\begin{aligned}
& X_0 \sim \mathcal{N}(0, Q), \\
& W_1 \sim \mathcal{N}(0, V_1), \quad W_1 \perp X_0, \\
& W_2 = a \cdot \text{sign}(X_0), \\
& U_1 = W_1 + a \cdot \text{sign}(X_0) + b \cdot X_0 \\
& \qquad = W_1 + W_2 + b \cdot X_0, \\
& X_1 = U_1 + X_0 = W_1 + W_2 + (b+1)X_0, \\
& Y_1 = X_1 + Z_1 = W_1 + W_2 + (b+1)X_0 + Z_1.
\end{aligned} 
\quad (25) $$

Given that the RVs ($W_1, W_2, Y_1$) are available at the decoder, the estimation cost is given by

$$ 
\begin{aligned}
S &= \mathbb{E}[(X_1 - \mathbb{E}[X_1|W_1, W_2, Y_1])^2] \\
&= (b+1)^2 \mathbb{E}[(X_0 - \mathbb{E}[X_0|W_1, S, Y_1])^2].
\end{aligned} 
$$

Hence, $S = 0$ if and only if $b = -1$. Setting $b = -1$ removes the residual uncertainty about $X_0$ and makes $X_1 = W_1 + W_2$ a deterministic function of the auxiliary RVs that are revealed to the decoder. Thus, the decoder can reconstruct $X_1$ perfectly, achieving zero long-run estimation cost. For the remainder of this section, we assume $b = -1$. This design resembles Witsenhausen's two-point approach (8) in that the control action $U_1$ cancels the source signal $X_0$. However, the key difference is that due to block coding, both $W_1$ and $W_2$ can be reliably communicated to the decoder.

The power cost needed is given by

$$ P = \mathbb{E}[U_1^2] = \mathbb{E}[(W_1 + a \cdot \text{sign}(X_0) - X_0)^2] $$

<footnote n="1">¹Feedforward link in this setting means the source state sequence $X_0^{t-1}$ is available at the (causal) decoder.</footnote>
$$ 
\begin{aligned}
&= V_1 + \left( Q + a^2 - 2a\sqrt{\frac{2Q}{\pi}} \right) \\
&= V_1 + P_2(a)
\end{aligned} 
\qquad (26) $$

where $P_2(\cdot)$ is the power cost function for Witsenhausen's two-point strategy given by (9). The information constraint (17) becomes

$$ 
\begin{aligned}
& I(W_1, W_2; Y_1) - I(W_2; X_0|W_1) \\
&= h(Y_1) - h(Y_1|W_1, W_2) - h(W_2|W_1) + h(W_2|X_0, W_1) \\
&= h(Y_1) - \frac{1}{2}\log(2\pi e N) - 1 \geq 0,
\end{aligned} 
\qquad (27) $$

where $h(Y_1)$ is calculated with regard to the Gaussian mixture distribution of the following form

$$ f_{Y_1}(y) = \frac{1}{2\sqrt{(V_1 + N)}} \left[ \phi\left(\frac{y-a}{\sqrt{V_1 + N}}\right) + \phi\left(\frac{y+a}{\sqrt{V_1 + N}}\right) \right]. \qquad (28) $$

Since there is no closed form for calculating the entropy of Gaussian mixture distributions, methods discussed in [72], [73] can be employed for numerical simulation.

Combining the above analysis, we have the following theorem, which summarizes the cost function for the ZEC-2 scheme.

**Theorem IV.2.** The achievable long-run estimation cost for the ZEC-2 coding scheme (25) is given by

$$ S_{\text{ZEC-2}}(P) = 0, \text{ for } P \geq P_{\text{ZEC-2}}^*, \qquad (29) $$

where the value

$$ 
\begin{aligned}
P_{\text{ZEC-2}}^* &= \min_{V_1 \ge 0, a \ge 0} V_1 + P_2(a) \\
\text{s.t.} \quad & h(Y_1) - \frac{1}{2}\log(2\pi e N) - 1 \ge 0.
\end{aligned} 
\qquad (30) $$

Note that the differential entropy term $h(Y_1)$ is not always convex with respect to the parameters $(V_1, a)$, hence this is not a convex optimization problem.

From equation (30), we see that since $V_1$ is nonnegative, the minimum required power cost $P_{\text{ZEC-2}}^*$ for the ZEC-2 scheme must satisfy

$$ P_{\text{ZEC-2}}^* \geq \min_a P_2(a) = Q \left(1 - \frac{2}{\pi}\right) \triangleq P_2^{\min}, \qquad (31) $$

where $P_2^{\min}$ denotes the lowest possible power budget required for the single-shot two-point strategy (9). This indicates that the ZEC-2 scheme achieves an improved estimation performance, at the expense of a higher power consumption than the original two-point strategy.

We now extend the ZEC-2 scheme to the ZEC-k scheme by generalizing the discrete auxiliary RV $W_2$ from a two-point binary design to a more expressive k-point quantization.

## B. ZEC-k Scheme

The idea of the ZEC-k scheme is inspired by the hierarchical search approach that generalizes Witsenhausen's original two-point strategy to an k-point quantization strategy in [26]. We adopt a symmetric indexing convention where the quantization levels and grids are placed symmetrically around zero. Thus, we optimize only the positive half of the parameters. Now, we formally define the k-point quantization function.

**Definition IV.3 (k-point quantization).** Let $k \in \mathbb{N}^*$ and define $m = \lceil \frac{k}{2} \rceil$. The construction of the k-point quantization function depends on whether k is odd or even:

* If $k = 2m-1$, we take the center level to be zero and define the quantization levels and decision boundaries as symmetric sequences:

$$ 
\begin{aligned}
& -a_m < \dots < -a_1 = 0 = a_1 < \dots < a_m, \\
& -B_m < \dots < -B_1 = 0 = B_1 < \dots < B_m.
\end{aligned} 
\qquad (32) $$

* If $k = 2m$, we take $a_1 \neq 0$ and the quantization levels and boundaries are defined as:

$$ 
\begin{aligned}
& -a_m < \dots < -a_1 < 0 < a_1 < \dots < a_m, \\
& -B_m < \dots < -B_1 = 0 = B_1 < \dots < B_m.
\end{aligned} 
\qquad (33) $$
<figure><img src="image_5.png" /><figcaption>Fig. 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 $0 = a_1 \le a_2, 0 = B_1 \le B_2$, whereas the 4-point function is parameterized by $0 < a_1 < a_2, 0 = B_1 \le B_2$</figcaption></figure>

$$P_k(\mathbf{a}_1^m, \mathbf{B}_1^m) = Q - 4\sqrt{Q} \sum_{i=1}^{m} a_i \left( \phi\left(\frac{B_i}{\sqrt{Q}}\right) - \phi\left(\frac{B_{i+1}}{\sqrt{Q}}\right) \right) + 2 \sum_{i=1}^{m} a_i^2 p_i \quad (36)$$

$$S_k(\mathbf{a}_1^m, \mathbf{B}_1^m) = 2 \sum_{i=1}^{m} a_i^2 p_i - \int \frac{\left[ \sum_{i=1}^{m} a_i \cdot p_i \left( \phi\left(\frac{y-a_i}{\sqrt{N}}\right) - \phi\left(\frac{y+a_i}{\sqrt{N}}\right) \right) \right]^2}{\sqrt{N} \sum_{i=1}^{m} p_i \left( \phi\left(\frac{y-a_i}{\sqrt{N}}\right) + \phi\left(\frac{y+a_i}{\sqrt{N}}\right) \right)} dy \quad (37)$$

In both cases, we define the k-point quantization function $Q_k(x) : \mathbb{R} \to \{-a_m, \dots, -a_1, a_1, \dots, a_m\}$ by

$$Q_k(x) = \begin{cases} -a_i & \text{if } x \in (-B_{i+1}, -B_i], \\ +a_i & \text{if } x \in [B_i, B_{i+1}), \end{cases} \quad (34) \\ \qquad i = 1, \dots, m \\ = \sum_{i=1}^{m} a_i (\mathbf{1}_{x \in [B_i, B_{i+1})} - \mathbf{1}_{x \in (-B_{i+1}, -B_i]}), \quad \forall x \in \mathbb{R}.$$

For notation consistency, we set $B_{k+1} = \infty$.

Figure 3 illustrates an example of the 3-point and 4-point quantization functions.

Based on this k-point quantization, we first recall the Witsenhausen k-point control strategy proposed in [26]. Since the estimation cost function derived in the reference involves Fisher information, we provide a more explicit expression of the cost functions in Theorem IV.4. In order to do this, we define the probability mass of $X_0$ lying in each quantization interval by

$$\begin{aligned} p_i &\triangleq \mathbb{P}(X_0 \in (-B_{i+1}, -B_i]) = \mathbb{P}(X_0 \in [B_i, B_{i+1})) \\ &= \Phi\left(\frac{B_{i+1}}{\sqrt{Q}}\right) - \Phi\left(\frac{B_i}{\sqrt{Q}}\right), \quad \text{for } i = 1, \dots, m. \end{aligned} \quad (35)$$

**Theorem IV.4 (k-point strategy).** For $k \in \mathbb{N}^*$, $m = \lfloor \frac{k}{2} \rfloor$, and given $(\mathbf{a}_1^m = [a_1, \dots, a_m]^\top, \mathbf{B}_1^m = [B_1, \dots, B_m]^\top)$ satisfying either (32) or (33), the Witsenhausen k-point strategy is defined by

$$U_1 = Q_k(X_0) - X_0, \quad (38)$$
where $Q_k(\cdot)$ is the quantization function defined in (34) parameterized by $\mathbf{a}_1^m, \mathbf{B}_1^m$. The power and estimation costs induced by this strategy are given by (36) and (37), where the quantity $p_i$ is defined by (35).

This control strategy refines $X_1$ into a more refined quantization symbol, improving the trade-off between the power and estimation costs. We now integrate this idea to generalize our ZEC scheme: Given $V_1 \ge 0$ and $(\mathbf{a}_1^m = [a_1, ..., a_m]^\top, \mathbf{B}_1^m = [B_1, ..., B_m]^\top)$ as above satisfying either (32) or (33), the ZEC-k scheme is given by the follows:

$$ 
\left\{
\begin{array}{l}
X_0 \sim \mathcal{N}(0, Q), \\
W_1 \sim \mathcal{N}(0, V_1), \quad W_1 \perp X_0, \\
W_2 = Q_k(X_0), \\
U_1 = W_1 + Q_k(X_0) - X_0 = W_1 + W_2 - X_0, \\
X_1 = U_1 + X_0 = W_1 + W_2, \\
Y_1 = X_1 + Z_1 = W_1 + W_2 + Z_1, \quad Z_1 \sim \mathcal{N}(0, N),
\end{array}
\right.
\qquad (39) $$

Here, $Q_k(\cdot)$ is the k-point quantizer defined as in (34), and the parameter $b = -1$ is fixed, as in ZEC-2 to ensure zero estimation cost, i.e., $X_1$ is fully determined by the auxiliary RVs $(W_1, W_2)$.

Following the same derivation procedure in the last section, the total power cost in this setup becomes:

$$ P = \mathbb{E}[U_1^2] = V_1 + P_k(\mathbf{a}_1^m, \mathbf{B}_1^m), $$

where $P_k(\mathbf{a}_1^m, \mathbf{B}_1^m)$ is given in (36).

To ensure the coordination coding scheme is valid, the ZEC-k scheme must satisfy the information constraint

$$ 
\begin{aligned}
0 \le I(W_1, W_2; Y_1) - I(W_2; X_0|W_1) \\
&= h(Y_1) - \frac{1}{2} \log 2\pi e N - H(Q_k(X_0))
\end{aligned}
\qquad (40) $$

where the entropy term $H(Q_k(X_0))$ takes the following form:

$$ 
\left\{
\begin{array}{ll}
-2 \sum_{i=1}^{m} p_i \log p_i & \text{k even,} \\
-2 \left( p_1 \log(2 \cdot p_1) + \sum_{i=2}^{m} p_i \log p_i \right) & \text{k odd,}
\end{array}
\right.
$$

Moreover, the marginal density function of $Y_1$ is the following mixture of Gaussian distributions

$$ f_{Y_1}(y) = \sum_{i=1}^{m} \frac{p_i}{\sqrt{V_1 + N}} \left[ \phi\left(\frac{y - a_i}{\sqrt{V_1 + N}}\right) + \phi\left(\frac{y + a_i}{\sqrt{V_1 + N}}\right) \right], $$

according to which, we calculate numerically the differential entropy $h(Y_1)$.

Conclusively, we obtain the following result:

**Theorem IV.5.** For the given power cost $P \ge 0$, the estimation cost for the ZEC-k coding scheme (39) is given by

$$ S_{\text{ZEC-k}}(P) = 0, \text{ for } P \ge P_{\text{ZEC-k}}^*, \qquad (41) $$

where the value

$$ 
\begin{array}{l}
P_{\text{ZEC-k}}^* = \min \quad V_1 + P_k(\mathbf{a}_1^m, \mathbf{B}_1^m), \\
\qquad \text{s.t.} \quad h(Y_1) - \frac{1}{2} \log(2\pi e N) - H(Q_k(X_0)) \ge 0.
\end{array}
\qquad (42) $$

Again, from (42), we get that the power consumption needed for the ZEC-k scheme is more than the power required for the single-shot k-point strategy denoted by

$$ P_k^{\min} = \min_{\mathbf{a}_1^m, \mathbf{B}_1^m} P_k(\mathbf{a}_1^m, \mathbf{B}_1^m). \qquad (43) $$

## V. CAUSAL-ENCODING WITH CHANNEL FEEDBACK

We consider vector-valued version of the Witsenhausen counterexample with the presence of perfect channel feedback at the causal encoder, see Figure 4.
<figure><img src="image_6.png" /><figcaption>Fig. 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 $U_2^n$.</figcaption></figure>

## A. System Model with Feedback

We first define control designs for this setup as in the last section.

**Definition V.1.** For $n \in \mathbb{N}$, a “control design” with causal encoder and noncausal decoder with channel feedback is a tuple of functions $c = (\{f_{U_{1,t}|X_0^t, Y_1^{t-1}}^{(f,t)}\}_{t=1}^n, g_{U_2^n|Y_1^n})$ defined by

$$f_{U_{1,t}|X_0^t, Y_1^{t-1}}^{(f,t)} : \mathcal{X}_0^t \times \mathcal{Y}_1^{t-1} \to \mathcal{U}_1, \quad g_{U_2^n|Y_1^n} : \mathcal{Y}_1^n \to \mathcal{U}_2^n,$$

which induces a distribution over sequences of symbols:

$$\prod_{t=1}^{n} \mathcal{P}_{X_0, t} \prod_{t=1}^{n} f_{U_{1,t}|X_0^t, Y_1^{t-1}}^{(f,t)} \prod_{t=1}^{n} \mathcal{P}_{X_{1,t}, Y_{1,t}|X_{0,t}, U_{1,t}} g_{U_2^n|Y_1^n}, \quad (44)$$

where $Y_1^0 = \emptyset$. We denote by $\mathcal{C}_f(n)$ the set of control designs with causal encoder and noncausal decoder with channel feedback.

Analogous to the no-feedback setup, we define the achievable pair of costs $(P, S)$ as the limit point of the averaged long-run costs induced by any control design, as in Definition II.2, and we denote by $\mathcal{R}_f$ the region of achievable pairs of costs in this setting. For brevity, we omit restating the definition and proceed directly to characterize the region $\mathcal{R}_f$.

**Theorem V.2.** The pair of Witsenhausen costs $(P, S) \in \mathcal{R}_f$ if and only if there exists a joint distribution over the random variables $(X_0, W_1, U_1, X_1, Y_1, U_2)$ that decomposes according to

$$\mathcal{P}_{X_0} \mathcal{P}_{W_1} \mathcal{P}_{U_1|X_0, W_1} \mathcal{P}_{X_1, Y_1|X_0, U_1} \mathcal{P}_{U_2|X_0, W_1, Y_1}, \quad (45)$$

such that

$$I(W_1; Y_1) - I(U_2; X_0|W_1, Y_1) \geq 0, \quad (46)$$

$$P = \mathbb{E}[U_1^2], \quad S = \mathbb{E}[(X_1 - U_2)^2], \quad (47)$$

where $\mathcal{P}_{X_0}$ and $\mathcal{P}_{X_1, Y_1|X_0, U_1}$ are the given distributions, and $W_1$ is an auxiliary RV.

We provide the achievability proof and converse proof in Appendix F and G respectively. The converse proof is adapted from the converse proof of Theorem III.2 in [68, App. C.], which dealt with a state-independent channel. We extend the arguments from this reference to apply to the state-dependent channel in our setting, without affecting the outcome of the result. Furthermore, the coding scheme for the achievability proof for Theorem V.2 also extends the approach provided in [68, App. B.] by changing the channel from state-independent to state-dependent.

**Remark V.3.** The decomposition of the distribution (45) is equivalent to the following Markov chains

$$\left\{ \begin{array}{l} X_0 \text{ is independent of } W_1, \\ (X_1, Y_1) \leftarrow (X_0, U_1) \leftarrow W_1, \\ U_2 \leftarrow (X_0, Y_1, W_1) \leftarrow (U_1, X_1). \end{array} \right. \quad (48)$$

Note that, the information constraint (46) in Theorem V.2 could be directly obtained from the information constraint in Theorem III.1 if we identify the following RVs based on the feature that perfect channel feedback is available:

$$\tilde{X}_0 = (X_0, Y_1), \quad (49)$$

$$\tilde{W}_2 = U_2. \quad (50)$$

Hence we have

$$I(W_1, \tilde{W}_2; Y_1) - I(\tilde{W}_2; \tilde{X}_0|W_1)$$
$$
\begin{align*}
&= I(W_1, U_2; Y_1) - I(U_2; X_0, Y_1 | W_1) \\
&= I(W_1; Y_1) + I(U_2; Y_1 | W_1) - I(U_2; Y_1 | W_1) \\
&\quad - I(U_2; X_0 | W_1, Y_1) \\
&= I(W_1; Y_1) - I(U_2; X_0 | W_1, Y_1).
\end{align*}
$$

The identification (49) means that the channel feedback $Y_1$ plays the role of an additional source symbol. And (50) means that it is optimal for the encoder to recommend to the decoder the action $U_2$ to implement.

**Remark V.4.** Compared to Theorem III.1 for the setup without channel feedback, the presence of channel feedback enables the decoder to directly coordinate with the source state $X_0$ instead of its noisy representation $W_2$ as in (16). Therefore, feedback helps with better communication and also enlarges the set of achievable cost pairs, i.e., $\mathcal{R} \subseteq \mathcal{R}_f$. Similar observations have been also pointed out in [68], [69], [74]

B. Zero Estimation Cost Strategy with Feedback

In this section, we derive an analogous ZEC scheme, so-called ZEC-f, that is based on the coordination coding result of
Theorem V.2 that further improves the power-estimation cost trade-off performance.

Given parameters $V_1 \ge 0$, $a, b \in \mathbb{R}$, we consider the following design

$$
\begin{array}{l}
|X_0 \sim \mathcal{N}(0, Q), \\
W_1 \sim \mathcal{N}(0, V_1), \\
U_1 = W_1 + a \cdot \text{sign}(X_0) + b \cdot X_0, \hfill (51) \\
X_1 = U_1 + X_0 = W_1 + a \cdot \text{sign}(X_0) + (b+1)X_0, \\
Y_1 = W_1 + a \cdot \text{sign}(X_0) + (b+1)X_0 + Z_1.
\end{array}
$$

This strategy also follows the idea of designing the system state $X_1$ being a deterministic function of $(X_0, W_1)$, which is known to the decoder. Notably, due to channel feedback, the source state $X_0$ is transmitted to the decoder perfectly, hence, even if $b \neq -1$, we still could exactly reconstruct $X_1$ from its MMSE estimator. We have

$$
\begin{align}
U_2 &= \mathbb{E}[X_1 | X_0, W_1, Y_1] \nonumber \\
    &= W_1 + a \cdot \text{sign}(X_0) + (b+1)X_0 = X_1 \tag{52}
\end{align}
$$

which results in an achievable estimation cost $S = 0$.

Before stating our cost function result, we first define the skew-normal distribution.

**Definition V.5.** The random variable $X$ is distributed according to the skew-normal distribution $X \sim \mathcal{SN}(u, \sigma, \alpha)$ with location $u$, scale $\sigma > 0$ and skewness parameter $\alpha$, if the probability density function writes

$$
f(x) = \frac{2}{\sigma} \cdot \phi \left( \frac{x-u}{\sigma} \right) \Phi \left( \alpha \cdot \frac{x-u}{\sigma} \right). \qquad (53)
$$

**Theorem V.6.** For a given power budget $P \ge 0$, the achievable estimation cost for ZEC-f scheme is

$$
S_{\text{ZEC-f}}(P) = 0, \text{ for } P \geq P_{\text{ZEC-f}}^*, \qquad (54)
$$

where the minimum required power

$$
P_{\text{ZEC-f}}^* = \min_{V_1 \ge 0, a, b \in \mathbb{R}} V_1 + a^2 + b^2 Q + 2ab \sqrt{\frac{2Q}{\pi}}, \qquad (55)
$$

$$
\text{s.t.} \quad h(Y_1) - h(X) - \frac{1}{2} \log 2\pi e N \ge 0, \qquad (56)
$$

and the differential entropies are computed according to

$$
f_{Y_1}(y) = \frac{1}{2} \mathcal{S}N(a, \sigma, \alpha) + \frac{1}{2} \mathcal{S}N(-a, \sigma, -\alpha), \qquad (57)
$$

$$
f_X(x) = \frac{1}{\sqrt{\delta^2 Q}} \phi \left( \frac{x-a}{\sqrt{\delta^2 Q}} \right) \mathbf{1}_{\text{sign}(\delta) \cdot (x-a) \ge 0} \\
\qquad + \frac{1}{\sqrt{\delta^2 Q}} \phi \left( \frac{x+a}{\sqrt{\delta^2 Q}} \right) \mathbf{1}_{\text{sign}(\delta) \cdot (x+a) \le 0}. \qquad (58)
$$

Here, we take $\delta = b + 1$, $\sigma^2 = V_1 + \delta^2 Q + N$, $\alpha = \frac{\sqrt{Q\delta}}{\sqrt{V_1+N}}$.
The proof of Theorem V.6 is stated in Appendix H. The following corollary is an immediate consequence of Theorem V.6.

**Corollary V.7.** If the Gaussian variances $Q > 0, N > 0$ satisfy

$$2\pi e \cdot \frac{QN}{Q+N} \le 1, \qquad (59)$$

then, the ZEC-f scheme achieves zero-power zero-estimation cost performance.

The proof is provided in Appendix I. Importantly, Corollary V.7 also provides a converse result: under the condition (59), ZEC-f strategy is the optimal strategy to conduct. To clarify this result, we have the following remark.

**Remark V.8.** Achieving zero-power and zero-estimation cost in Definition II.2 means that the averaged long-run quantities (13) and (14) can be made arbitrarily close to zero as $n \to \infty$. This does not mean that the two quantities can be made strictly identical to zero. Using our coordination-coding results, we show the existence of a sequence of control designs whose induced cost pairs converge to zero. Usually, at the strictly zero boundary, a valid coding scheme is not guaranteed to exist. Analogous behavior appears in the wideband regime of communication channels [75], where Verdú showed that the minimum energy-per-bit is achieved only as spectral efficiency tends to zero, i.e., “low, but non-zero” rate.

This counterintuitive result naturally raises the question: what is communicated when the averaged long-run cost $P = 0$, i.e., when the first DM only effectively transmits a finite number of times? In the presence of perfect channel feedback, the first DM has access to all the information available at the second DM. As a result, when the noise level $N \ge 0$ is small, the encoder can easily predict the noise to refine its actions and correct past errors online, such as in the classical Schalkwijk-Kailath coding scheme [62].

## VI. NUMERICAL RESULTS

In this section, we examine the performance of the proposed ZEC-2, ZEC-k, ZEC-f coordination coding schemes and we compare them with other control strategies.

First, we revisit the coding scheme proposed by Grover and Sahai [36] when both encoder and decoder are noncausal. Given a block-length $n \in \mathbb{N}^*$, a control design is given by a tuple of stochastic function $c = (f, g)$ defined by

$$f: \mathcal{X}_0^n \to \mathcal{U}_1^n, \quad g: \mathcal{Y}_1^n \to \mathcal{U}_2^n. \qquad (60)$$

This model is superior to our model described in Section II-B since the encoder is non-causal.

Besides their DPC-based scheme introduced in [36, App. D.1-D.7], they further extend it to a combination between the linear scheme and the DPC-based scheme [36, App. D.8] summarized as follows:

Given $-\sqrt{P/Q} \le \beta \le \sqrt{P/Q}$, the power budget $P$ is divided into a linear part $U_{1,1} = -\beta X_0$ and the other part $U_{1,2}$, which is used to implement the DPC-based method against the state $(1 - \beta)X_0 \sim \mathcal{N}(0, (1 - \beta)^2Q)$ with power constraint $\mathbb{E}[U_{1,2}^2] \le P - \beta^2Q$. By taking $\beta = -\rho\sqrt{P/Q}$, we obtain the following correlation matrix of $(X_0, U_1)$

$$K = \begin{pmatrix} Q & \rho\sqrt{PQ} \\ \rho\sqrt{PQ} & P \end{pmatrix}$$

and the state-dependent channel given by

$$\tilde{S} = \frac{\sqrt{Q} + \rho\sqrt{P}}{\sqrt{Q}} \cdot X_0, \\ \tilde{X} \perp (\tilde{S}, X_0), \text{ with } \mathbb{E}[\tilde{X}^2] \le P(1 - \rho^2), \\ X_1 = X_0 + U_1 = \tilde{X} + \tilde{S}, \\ Y_1 = \tilde{X} + \tilde{S} + Z.$$

This leads to the estimation cost function (61). In particular, the minimum power required for enabling zero estimation cost denoted by $P_{\text{lin+dpc}}^*$ for this scheme is given by the unique positive root of equation

$$P^2(P + Q + N) = QN^2. \qquad (62)$$

This scheme has also been evaluated in [57] and is shown to outperform all other estimation costs, including their newly-proposed noncausal-causal coordination coding scheme in [57, Sec. IV-C].
<figure><img src="image_7.png" /><figcaption>Fig. 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 $Q = 1$.</figcaption></figure>

$$S_{\text{lin+dpc}}(P) = \min_{\rho \in [-1,1]} \frac{N \left( P(1-\rho^2)\sqrt{P+Q+2\rho\sqrt{PQ+N}} - N(\sqrt{Q}+\rho\sqrt{P}) \right)^2}{(P(1-\rho^2)+N)^2(P+Q+2\rho\sqrt{PQ+N})} \quad (61)$$

## A. Comparison of Minimum Power Cost for Zero Estimation Cost Performance

In this section, for fixed $Q = 1$, we compare the minimum required power cost for achieving zero estimation cost, given different noise levels $N \ge 0$ for the following strategies:

* Single-shot scalar control strategies: such as the best linear policy (7), the optimal joint Gaussian strategy (22), and the $k$-point strategy (36) and (37). As summarized in Prop. IV.1, all of the above-mentioned strategies achieve zero estimation cost with the minimum required power budget denoted by $P_{\text{scalar}}^* = Q = 1$.

* ZEC-$k$ schemes given in Theorem IV.2 and Theorem IV.5 denoted by $P_{\text{ZEC-$k$}}^*$ for $k = 2, 3, 4$.

* ZEC-$f$ scheme given in Theorem V.6 denoted by $P_{\text{ZEC-$f$}}^*$.

* The combination of linear and DPC-based scheme when both DMs are noncausal recalled earlier, where the power cost required for achieving zero estimation cost is given by $P_{\text{lin+dpc}}^*$, i.e., the unique positive root of (62).

As we can see from Figure 5, when the noise variance $N$ increases from zero, the minimum required power $P_{\text{ZEC-$k$}}^*$ grows from the corresponding values of the single-shot $k$-point strategies defined in (43). Specifically, with our chosen parameter,

$$P_{2}^{\min} = 0.363, \quad P_{3}^{\min} = 0.19, \quad P_{4}^{\min} = 0.12. \qquad (63)$$

This indicates that when noise level increases from zero, the power $V_1$ for the auxiliary RV $W_1$ also increases from zero, and zero-cost estimation can be achieved without requiring additional power beyond that of the original $k$-point strategy. However, when $N \ge 0.65$, $P_{\text{ZEC-2}}^* \ge Q = 1$, meaning that the ZEC-2 scheme can no longer offer a block-coding gain in the high-noise regimes.

Moving from ZEC-2 to ZEC-3 yields substantial improvements across both low- and high-noise regimes, and the power cost saturates as $N$ continuously increases. This highlights the critical role of including a zero quantization level, i.e., $Q_3(x) = 0$ for $-B_2 \le x \le B_2$, see again Figure 3.

The transition from ZEC-3 to ZEC-4 provides further improvement only at small noise levels. Interestingly, for $N \ge 0.24$, the ZEC-4 scheme effectively reduces to ZEC-3, since the optimizer becomes $a_1 = 0$. This occurs because at higher noise levels, using fewer quantization levels simplifies the decoder's task of distinguishing signals, while still ensuring zero estimation cost.
<table><caption>TABLE I
$P^*$ AND CORRESPONDING OPTIMIZED PARAMETERS FOR ZEC-2, ZEC-3, ZEC-4, AND ZEC-F SCHEMES ACROSS DIFFERENT $N$.</caption><thead><tr><th rowspan="2">N</th><th colspan="3">ZEC-2</th><th colspan="4">ZEC-3</th><th colspan="4">ZEC-4</th><th colspan="3">ZEC-f</th></tr><tr><th>P*</th><th>V<sub>1</sub>*</th><th>a*</th><th>P*</th><th>V<sub>1</sub>*</th><th>a<sub>2</sub>*</th><th>B<sub>2</sub>*</th><th>P*</th><th>V<sub>1</sub>*</th><th>a<sub>1</sub>*</th><th>a<sub>2</sub>*</th><th>B<sub>2</sub>*</th><th>P*</th><th>a*</th><th>b*</th></tr></thead><tbody><tr><td>0.1</td><td>0.367</td><td>0.003</td><td>0.814</td><td>0.216</td><td>0.017</td><td>1.389</td><td>0.696</td><td>0.184</td><td>0.027</td><td>0.660</td><td>2.040</td><td>1.359</td><td>0.027</td><td>0.263</td><td>-0.247</td></tr><tr><td>0.2</td><td>0.410</td><td>0.033</td><td>0.914</td><td>0.338</td><td>0.057</td><td>1.812</td><td>0.935</td><td>0.335</td><td>0.060</td><td>0.883</td><td>2.824</td><td>1.904</td><td>0.088</td><td>0.444</td><td>-0.484</td></tr><tr><td>0.3</td><td>0.501</td><td>0.081</td><td>1.035</td><td>0.468</td><td>0.084</td><td>2.150</td><td>1.139</td><td>0.468</td><td>0.084</td><td>0.000</td><td>2.150</td><td>1.139</td><td>0.115</td><td>0.496</td><td>-0.556</td></tr><tr><td>0.4</td><td>0.623</td><td>0.138</td><td>1.147</td><td>0.579</td><td>0.102</td><td>2.425</td><td>1.308</td><td>0.579</td><td>0.102</td><td>0.000</td><td>2.425</td><td>1.308</td><td>0.131</td><td>0.533</td><td>-0.593</td></tr><tr><td>0.5</td><td>0.768</td><td>0.201</td><td>1.249</td><td>0.670</td><td>0.111</td><td>2.661</td><td>1.455</td><td>0.670</td><td>0.111</td><td>0.000</td><td>2.661</td><td>1.455</td><td>0.142</td><td>0.548</td><td>-0.617</td></tr><tr><td>0.6</td><td>0.927</td><td>0.268</td><td>1.342</td><td>0.743</td><td>0.115</td><td>2.874</td><td>1.588</td><td>0.743</td><td>0.115</td><td>0.000</td><td>2.874</td><td>1.588</td><td>0.151</td><td>0.560</td><td>-0.639</td></tr><tr><td>0.7</td><td>1.099</td><td>0.339</td><td>1.428</td><td>0.801</td><td>0.113</td><td>3.073</td><td>1.712</td><td>0.801</td><td>0.113</td><td>0.000</td><td>3.073</td><td>1.712</td><td>0.158</td><td>0.568</td><td>-0.654</td></tr><tr><td>0.8</td><td>1.281</td><td>0.413</td><td>1.508</td><td>0.848</td><td>0.108</td><td>3.264</td><td>1.830</td><td>0.848</td><td>0.108</td><td>0.000</td><td>3.264</td><td>1.830</td><td>0.163</td><td>0.576</td><td>-0.665</td></tr></tbody></table>

<figure><img src="image_8.png" /><figcaption>Fig. 6. Evolution of optimized parameters for the ZEC-2 scheme (top row) and the ZEC-f scheme (bottom row). Top: $\log V_1^*$ versus noise $N$ (left), and $a^*$ versus $N$ (right). Bottom: $a^*$ versus $N$ (left), and $b^*$ versus $N$ (right).</figcaption></figure>

Finally, the ZEC-f scheme with feedback strictly outperforms all no-feedback schemes, confirming the observation in Remark V.4 that feedback enlarges the achievable cost region without feedback. Remarkably, it even effectively surpasses the noncausal linear and DPC-based scheme. Furthermore, as explained in Corollary V.7, it achieves the extreme operating point of zero power and zero estimation cost whenever $N \le \frac{Q}{2\pi eQ-1} \approx 0.062$, and for larger $N$ it smoothly converges to a limiting power cost of 0.17.

## B. Optimized Parameters

Table I records the minimum required power cost $P^*$ and the corresponding optimized parameters $V_1^*$, $(a_1^m)^*$, $(B_1^m)^*$ for the shown ZEC-$k$ and ZEC-f schemes across different noise levels. To better illustrate how these parameters evolve with the noise level, we also plot the curves of the optimized values $V_1^*$ and $a^*$ for ZEC-2, and $a^*$ and $b^*$ for ZEC-f in Figure 6.

The results show that for ZEC-2 and ZEC-3, the optimized signal levels $V_1^*$, $a_i^*$, $B_i^*$ generally increase with $N$, reflecting the higher power required to maintain zero estimation cost under noisier conditions. On the other hand, as also observed in Figure 5, when $N \ge 0.24$, the ZEC-4 scheme effectively collapses to ZEC-3, with $a_1^* = 0$, in order to remain the sanity of zero-cost estimation at the decoder.

As for ZEC-f, we notice that the optimized $V_1^*$ remains small but nonzero across all noise levels. This is because the power cost to be minimized (55) penalizes $V_1$ linearly while a vanishingly small $V_1$ suffices to satisfy the information constraint (56). This means, the random codebook induced by $W_1$ becomes nearly deterministic, and we are close to the boundary of the achievable cost region characterized in Theorem V.2. Therefore, here, only the optimized parameters $a^*$, $b^*$ are recorded. As we can see, both $a^*$ and $|b^*|$ grow steadily with $N$, implying that the slope of the piecewise linear mapping $X_1 = a \cdot \text{sign}(X_0) + (b+1) \cdot X_0$ (with $-1 < b < 0$) decreases, thereby converging toward an effective two-point quantization scheme
with slope zero. This trend is consistent with the analysis of the role of slope uncovered in [76], which shows that higher power cost leads to flatter (less sloped) piecewise linear policies, making the estimation at the second DM more efficient.

Moreover, our simulations show that if we manually enforce $b = -1$ in the ZEC-f formulation (51), i.e., we eliminate the continuous source state $X_0$ from $X_1$ as we did in the no-feedback ZEC schemes, the resulting performance $(P^*, V_1^*, a^*)$ exactly coincides with that of the ZEC-2 scheme without feedback. This observation further underscores the importance of feedback, as it enables direct communication of the original source signal $X_0$ to the decoder, which allows the minimum required power cost to be significantly reduced.

### C. Power-Estimation Performance Comparison

We now compare the performance of the proposed ZEC schemes with classical single-shot strategies in the power-estimation cost space.

In order to evaluate the single-shot k-point strategy given in Theorem IV.4, we normalize the original cost function (4) by dividing it by $\lambda^2 + 1$. This is equivalent to a weighted optimization problem parameterized by $\omega \triangleq \frac{\lambda^2}{\lambda^2+1} \in [0, 1]$, each of which corresponds to a supporting hyperplane in a two-dimensional power-estimation cost space. Specifically, for a chosen $\lambda^2 \in [0, \infty)$, a quantization level $k \in \mathbb{N}^*$ and $m = \lceil \frac{k}{2} \rceil$, we determine the optimal parameters $\mathbf{a}_1^m, \mathbf{B}_1^m$ satisfying Definition IV.3, such that

$$ (\mathbf{a}_{\omega}^{*}, \mathbf{B}_{\omega}^{*}) = \arg \min \frac{1}{\lambda^2+1} [\lambda^2 P_k(\mathbf{a}_1^m, \mathbf{B}_1^m) + S_k(\mathbf{a}_1^m, \mathbf{B}_1^m)] \\ = \arg \min [\omega P_k(\mathbf{a}_1^m, \mathbf{B}_1^m) + (1-\omega)S_k(\mathbf{a}_1^m, \mathbf{B}_1^m)], $$

which gives us the best configuration for the given parameter $\lambda^2$. Then, we plot the curve $(P_k(\mathbf{a}_{\omega}^*, \mathbf{B}_{\omega}^*), S_k(\mathbf{a}_{\omega}^*, \mathbf{B}_{\omega}^*)) \in \mathbb{R}_+^2$ for all $\omega = \frac{\lambda^2}{\lambda^2+1} \in [0, 1]$, which gives us the optimized estimation cost value at each specific power consumption. For simplicity, from now on we denote this optimized power-estimation cost function for the k-point strategy by $S_k(P)$.

<figure><img src="image_9.png" /><figcaption>Fig. 7. Performance comparison at $Q = 1$ and $N = 0.15$ between ZEC-k schemes $(S_{ZEC-k}(P))$, the corresponding k-point strategies $(S_k(P))$ for $k = 2, 3, 4$, the best linear strategy $(S_{\ell}(P))$, and the optimal joint Gaussian strategy $(S_G(P))$. The ZEC-2, ZEC-3, and ZEC-4 schemes achieve zero-estimation-cost reconstruction at $P_{ZEC-2}^* = 0.38$, $P_{ZEC-3}^* = 0.27$, and $P_{ZEC-4}^* = 0.26$, respectively, while the feedback-enabled ZEC-f scheme requires only $P_{ZEC-f}^* = 0.065$.</figcaption></figure>

Now, we compare the cost functions of ZEC-2, ZEC-3, ZEC-4, and ZEC-f, denoted $S_{ZEC-2}(P)$, $S_{ZEC-3}(P)$, $S_{ZEC-4}(P)$, and $S_{ZEC-f}(P)^2$, respectively, with those of the single-shot 2-, 3-, and 4-point strategies $S_2(P)$, $S_3(P)$, $S_4(P)$ (after the weighted-optimization procedure mentioned-above), the best affine strategy $S_{\ell}(P)$ (7), the optimal joint Gaussian strategy $S_G(P)$ (22), and the noncausal strategy combining linear and DPC-based schemes (61). All comparisons are carried out at a fixed $Q = 1$ and three different noise levels $N \in \{0.15, 0.3, 0.06\}$, as shown in Figures 7-9.

As illustrated in Figure 7, when $N = 0.15$, the ZEC-k schemes achieve zero-estimation-cost reconstruction with only a modest increase in power compared to their single-shot counterparts $P_k^{\min}$. Specifically, while the minimum power requirements for the single-shot strategies are given in (63), the ZEC-2, ZEC-3, and ZEC-4 schemes achieve zero-cost estimation once the power budget exceeds $P_{ZEC-2}^* = 0.38$, $P_{ZEC-3}^* = 0.27$, and $P_{ZEC-4}^* = 0.26$, respectively. Notably, the feedback-enabled ZEC-f scheme achieves zero-cost estimation at a significantly lower threshold of $P_{ZEC-f}^* = 0.065$ which is even lower than that of the noncausal linear+DPC-based scheme $P_{\text{lin+dpc}}^* = 0.12$. By contrast, all other single-shot strategies can achieve zero-estimation-cost reconstruction only at the much larger power level $P = Q = 1$.

<footnote n="2">²For brevity, in the figures we plot only the minimum required power for ZEC-f.</footnote>
<figure><img src="image_10.png" /><figcaption>Fig. 8. Comparison of cost functions $S_{ZEC-k}(P)$, $S_k(P)$, $S_{ZEC-f}(P)$, and $S_G(P)$ for $k = 2, 3, 4$ at $Q = 1$ and $N = 0.3$. In this regime, the two-point strategy is outperformed by the optimal joint Gaussian (and equivalently, linear) strategy. Nevertheless, the ZEC-2 scheme achieves zero-estimation-cost performance once the power exceeds $P_{ZEC-2}^* = 0.5$. The ZEC-4 scheme collapses to ZEC-3, which attains zero-cost estimation at $P_{ZEC-3}^* = 0.46$. The feedback-enabled ZEC-f scheme requires only $P_{ZEC-f}^* = 0.12$ to achieve zero-estimation cost.</figcaption></figure>

As shown in Figure 8, when $N = 0.3$, the two-point strategy no longer outperforms the optimal linear strategy, which in this case coincides with the joint Gaussian solution. Nevertheless, the proposed ZEC schemes still maintain a distinct advantage: ZEC-2 achieves zero-estimation-cost reconstruction with $P \ge P_{ZEC-2}^* = 0.50$, while ZEC-4 reduces to ZEC-3 and reaches zero-cost estimation at $P_{ZEC-3}^* = 0.47$. Most notably, the feedback-enabled ZEC-f scheme requires only $P_{ZEC-f}^* = 0.12$, even in this relatively high-noise regime.

<figure><img src="image_11.png" /><figcaption>Fig. 9. Performance comparison at $Q = 1$ and $N = 0.06$ between ZEC-k schemes ($S_{ZEC-k}(P)$), the corresponding k-point strategies ($S_k(P)$) for $k = 2, 3, 4$, the best linear strategy ($S_\ell(P)$), and the optimal joint Gaussian strategy ($S_G(P)$). In this case, the ZEC-f scheme achieves the extreme point of zero-power zero-estimation cost</figcaption></figure>

Figure 9 shows that at the low noise level of $N = 0.06$, the ZEC-f scheme simultaneously achieves the extreme point of zero-power zero-estimation cost performance.

## VII. CONCLUSION

In this paper, we used coordination coding to characterize the communication constraints underlying cooperation between two DMs in the causal-encoding, noncausal-decoding formulation of Witsenhausen's counterexample. The result is a single-letter achievable region with auxiliary RVs that capture the dual role of control — state regulation and implicit communication — and an information constraint specifying the required communication rate for reliable coordination. Building on this, we proposed
the ZEC-k scheme, which significantly lowers the power needed for zero estimation cost as the quantization level increases. With channel feedback, the scheme is further enhanced, attaining the extreme point of zero power and zero estimation cost in the low-noise regime.

Taken together, these results provide an insightful answer to the central question motivating this work: how much communication is needed, and what is relevant to be transmitted in Witsenhausen counterexample? Our findings show that when we have sufficient transmissions, effective cooperation between communication and control is established with carefully designed quantization indices together with an independent Gaussian codebook. These components jointly suffice to regulate the power-controlled state, deterministically define the reconstruction target, and simplify the information constraint. Developed in the canonical benchmark, our framework for Witsenhausen counterexample extends naturally to more general distributed decision-making settings where multiple agents must coordinate actions under asymmetric, decentralized knowledge. This perspective invites deeper integration of coordination-coding and control, and highlights the fundamental role of communication in shaping information flow in decentralized network control.

## APPENDIX A
### ACHIEVABILITY PROOF OF THEOREM III.1

Since our achievability proof involves both continuous and discrete RVs, our analysis must incorporate both the Lebesgue measure $\lambda$ (for continuous RVs) and the counting measure $\mu$ (for discrete RVs). The corresponding information-theoretic quantities are defined using the Radon-Nikodym derivative, which generalizes the concept of density with respect to a base measure. This approach enables us to define entropy for mixed discrete-continuous RVs in a way that preserves consistency with both discrete entropy and differential entropy; see [77] and [57, App. A] for further details. Similar to what the authors do in the latter reference, we extend the standard definition of (jointly) weak typicality to random vectors with either discrete or continuous components. We denote by $\mathcal{A}_{\varepsilon}^{(n)}(\mathcal{P}_{X,Y})$ the set of jointly typical sequences given by

$$ \mathcal{A}_{\varepsilon}^{(n)}(\mathcal{P}_{X,Y}) = \left\{ (x^n, y^n) \in \mathbb{R}^{n \times 2} : \left| -\frac{1}{n} \log \prod_{i=1}^{n} \mathcal{P}_{X,Y}(x_i, y_i) - H(X, Y) \right| < \varepsilon, \right. \\ \left. x^n \in \mathcal{A}_{\varepsilon}^{(n)}(\mathcal{P}_{X}), y^n \in \mathcal{A}_{\varepsilon}^{(n)}(\mathcal{P}_{Y}) \right\}. \quad (64) $$

Moreover, the authors in [57, App. A] verify that the joint asymptotic equipartition property (AEP), covering lemma, and packing lemma, see [37] [78], can be straightforwardly extended to the above definition of typicality.

We consider an arbitrary but fixed $\varepsilon > 0$ and assume the sequence $(X_0^n, W_1^n, W_2^n, U_1^n, X_1^n, Y_1^n, U_2^n)$ is generated i.i.d. according to a distribution that decomposes as (16), with $P = \mathbb{E}[U_1^n]$, $S = \mathbb{E}[(X_1 - U_2)^2]$. Let $\psi^{(n)}: \mathcal{X}_0^n \times \mathcal{W}_1^n \times \mathcal{W}_2^n \times \mathcal{U}_1^n \times \mathcal{X}_1^n \times \mathcal{Y}_1^n \times \mathcal{U}_2^n \to \{0,1\}$ denote an indicator function for sequences of length $n$ with

$$ \psi^{(n)}(x_0^n, w_1^n, w_2^n, u_1^n, x_1^n, y_1^n, u_2^n) = \begin{cases} 1 & \text{if } |c_S(x_1^n, u_2^n) - S| \ge \frac{1}{12}\varepsilon \\ 0 & \text{or } (w_1^n, w_2^n, y_1^n) \notin \mathcal{A}_{\varepsilon}^n(W_1, W_2, Y_1), \\ 0 & \text{otherwise.} \end{cases} \quad (65) $$

Using the weak law of large numbers (LLN) and the union bound we have

$$ \mathbb{E}[\psi^{(n)}(X_0^n, W_1^n, W_2^n, U_1^n, X_1^n, Y_1^n, U_2^n)] \le \delta_n \to 0, $$

as $n \to \infty$. And similarly, we define the set

$$ S_{\varepsilon}^{(n)} = \{(x_0^n, w_1^n, w_2^n) \mid \eta^{(n)}(x_0^n, w_1^n, w_2^n) \le \sqrt{\delta_n}\}, $$

where $\eta^{(n)}(x_0^n, w_1^n, w_2^n) = \mathbb{E}[\psi^{(n)}(x_0^n, w_1^n, w_2^n, U_1^n, X_1^n, Y_1^n, U_2^n) \mid X_0^n = x_0^n, W_1^n = w_1^n, W_2^n = w_2^n]$. We then take

$$ \mathcal{B}_{\varepsilon}^{(n)} = \mathcal{A}_{\varepsilon}^{(n)} \cap \mathcal{S}_{\varepsilon}^{(n)}. \quad (66) $$

Then, we can easily prove the subsequent lemma:

**Lemma A.1.** Let the sequence $(X_0^n, W_1^n, W_2^n)$ i.i.d. $\sim \mathcal{P}_{X_0, W_1, W_2}$, then

$$ \mathbb{P}((X_0^n, W_1^n, W_2^n) \in \mathcal{B}_{\varepsilon}^{(n)}) \xrightarrow{n \to \infty} 1. $$
<figure><img src="image_12.png" /><figcaption>Fig. 10. Block-Markov codes with causal encoder.</figcaption></figure>

*Proof of Lemma A.1.* From the Markov inequality we obtain

$$ \mathbb{P}((X_0^n, W_1^n, W_2^n) \notin S_{\varepsilon}^{(n)}) \leq \frac{\delta_n}{\sqrt{\delta_n}} = \sqrt{\delta_n}. $$

□

Lemma A.1 elaborates that the certainty of an outcome being jointly weakly typical and satisfying the distortion cost constraint is guaranteed, as per the AEP, if it is generated i.i.d.

Moreover, it is also necessary to quantize the output of the non-causal decoder, as done in [57], to ensure that the second cost constraint remains bounded in case an error happens. Given a joint distribution with $\mathbb{E}[(X_1 - U_2)^2] = S$, for any $\hat{\delta} > 0$ there exists a quantization $q_{U_2}: \mathcal{U}_2 \to \{\hat{u}_{2,k}\}_{k=1}^{N_{U_2}}$, as explained in [79], such that

$$ \hat{S} = \mathbb{E}[(X_1 - q_{U_2}(U_2))^2] \leq (1 + \hat{\delta})S, $$

in particular $\hat{\delta}S < \frac{1}{4}\varepsilon$.

Now, we provide a coding scheme by considering a probability distribution $\mathcal{P}_{X_0,W_1,W_2,U_1,X_1,Y_1,U_2}$ that decomposes as (16) such that there exists a $\delta > 0$, and a rate $R > 0$, satisfying

$$ I(W_2; X_0|W_1) + \delta \leq R \leq I(W_1; Y_1) + I(W_2; Y_1|W_1) - \delta. $$

We consider a block-Markov code $c \in \mathcal{C}(Bn)$ defined over $B \in \mathbb{N}^*$ blocks of length $n$, similar to the scheme in [59]. Figure 10 is an illustration of this coding scheme.

*Random codebook generation:* We generate $|\mathcal{M}| = 2^{nR}$ sequences $W_1^n(m)$ i.i.d. $\sim \mathcal{P}_{W_1}$ with index $m \in \mathcal{M}$. For each index $m \in \mathcal{M}$, we generate the same number $|\mathcal{M}| = 2^{nR}$ sequences $W_2^n(m, \hat{m})$ with index $\hat{m} \in \mathcal{M}$ i.i.d. $\sim \mathcal{P}_{W_2|W_1}$ depending on sequence $W_1^n(m)$.

*Encoding function:* Due to the causal feature, the encoder performs forward encoding, where the encoder outputs the information associated with the past block. Let $m_b$ denote the message generated during block $b \in [1: B]$. During the first block, without loss of generality, the encoder takes $m_1 = 1$ and returns $W_1^n(m_1)$. At the beginning of block $b \in [2: B]$, the encoder has observed the full sequence of source symbols $X_{0,b-1} \in \mathcal{X}^n$ of the previous block. It recalls the index $m_{b-1} \in \mathcal{M}$ of the sequence $W_1^n(m_{b-1})$ used for block $b-1$. It finds an index $m_b$ such that sequences

$$ (X_{0,b-1}^n, W_1^n(m_{b-1}), W_2^n(m_{b-1}, m_b)) \in \mathcal{B}_{\varepsilon}^{(n)}(\mathcal{P}_{X_0,W_1,W_2}) $$

are jointly typical. We denote by $W_{2,b-1}^n = W_2^n(m_{b-1}, m_b)$ corresponding to the past block. Then, it returns $W_{1,b}^n = W_1^n(m_b)$ corresponding to the current block $b$, and for each time $t \in [1: n]$, it sends the symbol $U_{1,t,b}$ i.i.d. $\sim \mathcal{P}_{U_{1}|X_0,W_1}$ depending on $W_{1,t}(m_b)$ and $X_{0,t,b}$ causally observed in the current block $b \in [1: B]$.
**Decoding function:** The decoder first returns $\tilde{m}_1 = 1$. During block $b \in [2: B]$, the decoder recalls the past sequence $Y_{1,b-1}^n$ and the index $\tilde{m}_{b-1}$ that corresponds to the sequence $\tilde{W}_{1,b-1}^n = W_1^n(\tilde{m}_{b-1})$. It observes the channel output $Y_{1,b}^n$ and finds the unique index $\tilde{m}_b$ such that

$$ (Y_{1,b}^n, W_1^n(\tilde{m}_b)) \in \mathcal{A}_{\varepsilon}^{(n)}(\mathcal{P}_{Y_1, W_1}), \\ (Y_{1,b-1}^n, W_1^n(\tilde{m}_{b-1}), W_2^n(\tilde{m}_{b-1}), \tilde{m}_b) \in \mathcal{A}_{\varepsilon}^{(n)}(\mathcal{P}_{Y_1, W_1, W_2}). $$

We denote by $\tilde{W}_{1,b}^n = W_1^n(\tilde{m}_b)$ and $\tilde{W}_{2,b-1}^n = W_2^n(\tilde{m}_{b-1}, \tilde{m}_b)$ as our choice.

**Forward transmission of the decoder:** Upon receiving all sequences for all first ($B-1$) blocks, the decoder non-causally generates $U_{2,b}^n \sim \mathcal{P}_{U_2|Y_1, W_1, W_2}^\otimes$ depending on sequences $(Y_{1,b}^n, \tilde{W}_{1,b}^n, \tilde{W}_{2,b}^n)$ for $b \in [1: B-1]$. Finally, the decoder outputs the quantized sequence $\tilde{U}_{2,b}^n$ as the reconstruction for the interim state $X_{1,b}^n$.

**Termination block B:** As for the last block, the decoder simply outputs an all zero sequence, i.e., $\hat{U}_{2,B}^n = 0$. Usually, sequences are not jointly typical in the last block. Therefore we omit its error analysis in the subsequent part.

**Error analysis per block:** We first focus on the encoding error $\mathcal{E}^e$. For $b \in [2: B]$, let $\mathcal{E}_b^e(m_{b-1})$ denote the event of a failed encoding process during block $b$ given the knowledge of $m_{b-1}$, i.e., $\mathcal{E}_b^e(m_{b-1}) = \{\forall m_b \in \mathcal{M}, (X_{0,b-1}^n, W_1^n(m_{b-1}), W_2^n(m_{b-1}, m_b)) \notin \mathcal{B}_\varepsilon^{(n)}(\mathcal{P}_{X_0, W_1, W_2})\}$. Due to the independence between the codewords and Markov blocks, the probability of an encoding error in block $b$ given no encoding errors in the previous blocks is

$$ \mathbb{P}(\mathcal{E}_b^e(M_{b-1}) | \cap_{\beta=2}^{b-1} \bar{\mathcal{E}}_\beta^e(M_{\beta-1})) = \mathbb{P}(\mathcal{E}_b^e(M_{b-1})). $$

If $R \ge I(W_2; X_0|W_1) + \delta$, following the Covering Lemma [78] and Lemma A.1, we have $\mathbb{P}(\mathcal{E}_b^e(M_{b-1})) \to 0$ as $n \to \infty$, $\forall b \in [2: B]$. Thus, state sequences for the first $B-1$ blocks are successfully encoded with probability $\to 1$ as $n \to \infty$.

Next, we analyze the decoding error $\mathcal{E}^d$. For $b \in [2: B]$, let $\mathcal{E}_b^d(m_{b-1})$ denote the event of a failed decoding process in block $b$ given the past estimated index $m_{b-1}$. Furthermore, let $\mathcal{E}_{b-1}^{d,1}(m_{b-1})$ denote the event that the sequence $Y_{1,b-1}^n$ given $m_{b-1}$ is not jointly typical, i.e., $\mathcal{E}_{b-1}^{d,1}(m_{b-1}) = \{(Y_{1,b-1}^n, W_{1,b-1}^n(m_{b-1}), W_2^n(m_{b-1}, M_b)) \notin \mathcal{A}_\varepsilon^{(n)}(\mathcal{P}_{Y_1, W_1, W_2})\}$ and let $\mathcal{E}_b^{d,2}$ denote that the current sequence $Y_{1,b}^n$ is not jointly typical, i.e., $\mathcal{E}_b^{d,2} = \{(Y_{1,b}^n, W_1^n(m_b)) \notin \mathcal{A}_\varepsilon^{(n)}(\mathcal{P}_{Y_1, W_1})\}$. Note that, if a sequence with fewer terms $(Y_{1,b}^n, W_1^n(m_b))$ is atypical, it implies that a sequence with more terms $(Y_{1,b}^n, W_1^n(m_b), W_2^n(m_b, m_{b+1}))$ is also atypical. Therefore, we have $\mathcal{E}_b^{d,2} \subset \mathcal{E}_b^{d,1}(M_b)$. Then, the decoding error probability given no past decoding error or encoding error $\mathbb{P}(\mathcal{E}_b^d(m_{b-1}) | \cap_{\beta=2}^{b-1} \bar{\mathcal{E}}_\beta^d(m_{\beta-1}) \cap \bar{\mathcal{E}}^e)$ can be upperbounded by

$$ \mathbb{P}(\mathcal{E}_b^d(m_{b-1}) | \cap_{\beta=2}^{b-1} \bar{\mathcal{E}}_\beta^d(m_{\beta-1}) \cap \bar{\mathcal{E}}^e \cap \bar{\mathcal{E}}_{b-1}^{d,1}(m_{b-1}) \cap \bar{\mathcal{E}}_b^{d,2}) \\ + \mathbb{P}(\mathcal{E}_{b-1}^{d,1}(m_{b-1}) \cup \mathcal{E}_b^{d,2} | \cap_{\beta=2}^{b-1} \bar{\mathcal{E}}_\beta^d(m_{\beta-1}) \cap \bar{\mathcal{E}}^e) \quad (67) $$

using the union bound. The first term of (67) can be upperbounded by

$$ \mathbb{P}(\mathcal{E}_b^d(m_{b-1}) | \cap_{\beta=2}^{b-1} \bar{\mathcal{E}}_\beta^d(m_{\beta-1}) \cap \bar{\mathcal{E}}^e \cap \bar{\mathcal{E}}_{b-1}^{d,1}(m_{b-1}) \cap \bar{\mathcal{E}}_b^{d,2}) \\ = \mathbb{P} \left( \exists m' \neq M_b, \text{ s.t. } \{W_1^n(m') \in \mathcal{A}_\varepsilon^{(n)}(W_1|Y_{1,b}^n)\} \cap \right. \\ \left. \{W_2^n(m_{b-1}, m') \in \mathcal{A}_\varepsilon^{(n)}(W_2^n|Y_{1,b-1}^n, W_{1,b-1}^n)\} \right) \\ \le 2^{-n\varepsilon}, $$

where (a) is due to the joint packing lemma in order to satisfy both conditions at the same time given that $R \le I(W_1; Y_1) + I(W_2; Y_1|W_1) - 7\varepsilon$. Moreover, as for the second term in (67), we have

$$ \begin{align*} & \mathbb{P}(\mathcal{E}_{b-1}^{d,1}(m_{b-1}) \cup \mathcal{E}_b^{d,2} | \cap_{\beta=2}^{b-1} \bar{\mathcal{E}}_\beta^d(m_{\beta-1}) \cap \bar{\mathcal{E}}^e) \\ & \le \mathbb{P}((Y_{1,b-1}^n, W_{1,b-1}^n, W_{2,b-1}^n) \notin \mathcal{A}_\varepsilon^{(n)} | \\ & \qquad (X_{0,b-1}^n, W_{1,b-1}^n, W_{2,b-1}^n) \in \mathcal{B}_\varepsilon^{(n)}) \\ & \qquad + \mathbb{P}((Y_{1,b}^n, W_{1,b}^n, W_{2,b}^n) \notin \mathcal{A}_\varepsilon^{(n)} | (X_{0,b}^n, W_{1,b}^n, W_{2,b}^n) \in \mathcal{B}_\varepsilon^{(n)}) \\ & \le 2 \cdot \max_{(x_0^n, w_1^n, w_2^n) \in \mathcal{B}_\varepsilon^{(n)}} \eta^{(n)}(x_0^n, w_1^n, w_2^n) \\ & \le 2\sqrt{\delta_n} \xrightarrow{n \to \infty} 0, \end{align*} $$

where (b) comes from the independence of each Markov block and the fact that $\{(Y_{1,b}^n, W_{1,b}^n) \notin \mathcal{A}_\varepsilon^{(n)}\} \subset \{(Y_{1,b}^n, W_{1,b}^n, W_{2,b}^n) \notin \mathcal{A}_\varepsilon^{(n)}\}$, and (c) comes from the definition of $\mathcal{B}_\varepsilon^{(n)}$.

Therefore, following the above arguments, the encoding error and the decoding error are both asymptotically zero.
Witsenhausen costs analysis: We first analyze the power cost. To implement the power constraint, we impose a mean-square transmission power constraint as done in [36], [59]:

$$ \sum_{t=1}^{n} \mathbb{E}[U_{1,t}^2] \le nP, \qquad (68) $$

where the expectation is taken over random sequences $X_0^n$ and $W_1^n$. Therefore, the generated sequence $U_1^n$ satisfies the condition of uniform integrability. Thus, for $b \in [1: B]$, since at each time $U_{1,t,b}$ is generated i.i.d. with finite second moment $\mathbb{E}[U_1^2] = P$, due to the LLN, the non-negative averaged power sequence $c_P(U_{1,b}^n) = \frac{1}{n} \sum_{t=1}^n (U_{1,t,b}^2)$ converges in probability to $P$. Moreover, since sequence $U_{1,b}^n$ is also uniformly integrable, it holds (stronger) that

$$ \mathbb{E}[|c_P(U_{1,b}^n) - P|] < \varepsilon, \quad b \in [1: B] \qquad (69) $$

for large $n$, more details in [80]. Therefore, for all $B$ blocks, we have

$$ \mathbb{E}[|c_P(U_1^{Bn}) - P|] \le \frac{1}{B} \sum_{b=1}^{B} \mathbb{E}[|c_P(U_{1,b}^n) - P|] \le \varepsilon, \qquad (70) $$

for $n$ sufficiently large.

Next, we examine the estimation cost. Define the indicator function $\rho_b = 0$ to denote the absence of coding error in block $b \in [1: B-1]$. Now, take $\phi_b = (1 - \psi_b^{(n)})(1 - \rho_b)$ to indicate the occurrence of desired sequences that satisfy both the estimation cost and the typicality constraints AND no coding error event in block $b$. From the LLN and the error analysis, we have $\mathbb{E}[\phi_b] \to 1$ as $n \to \infty$ for $b \in [1: B-1]$. In particular, if $\phi_b = 1$, we have $\mathbb{E}[|c_S(X_{1,b}^n, \hat{U}_{2,b}^n) - S| | \phi_b = 1] < \frac{1}{12}\varepsilon$. Therefore, from the law of total expectation, we have

$$ \begin{aligned} & \mathbb{E}[|c_S(X_{1,b}^n, \hat{U}_{2,b}^n) - \hat{S}|] \\ &= \mathbb{P}(\phi_b = 1) \cdot \mathbb{E}[|c_S(X_{1,b}^n, \hat{U}_{2,b}^n) - \hat{S}| | \phi_b = 1] \\ & \quad + \mathbb{P}(\phi_b = 0) \cdot \mathbb{E}[|c_S(X_{1,b}^n, \hat{U}_{2,b}^n) - \hat{S}| | \phi_b = 0] \\ & \le \frac{1}{12}\varepsilon + \mathbb{P}(\phi_b = 0) \cdot \hat{S} \\ & \quad + \mathbb{P}(\phi_b = 0) \cdot \mathbb{E}[c_S(X_{1,b}^n, \hat{U}_{2,b}^n) | \phi_b = 0]. \end{aligned} $$

For $n$ sufficiently large, the second term $\mathbb{P}(\phi_b = 0) \cdot \hat{S} \le \frac{1}{12}\varepsilon$, since $\hat{S} < \infty$. Next, we evaluate the last term above. In particular, we need $c_S(X_{1,b}^n, \hat{U}_{2,b}^n)$ to be stochastically bounded when an unwanted error event happens. Since

$$ \begin{aligned} & \mathbb{P}(\phi_b = 0) \cdot \mathbb{E}[c_S(X_{1,b}^n, \hat{U}_{2,b}^n) | \phi_b = 0] \\ &= \mathbb{P}(\phi_b = 0) \cdot \mathbb{E}\left[\frac{1}{n} \sum_{t=1}^{n} |X_{0,t,b} + U_{1,t,b} - \hat{U}_{2,t,b}|^2 \mid \phi_b = 0\right], \end{aligned} \qquad (71) $$

by applying the inequality $(a+b)^2 \le 2a^2 + 2b^2$, $\forall a, b \in \mathbb{R}$, we have

$$ \begin{aligned} & \mathbb{P}(\phi_b = 0) \cdot \mathbb{E}[c_S(X_{1,b}^n, \hat{U}_{2,b}^n) | \phi_b = 0] \\ & \le 2\mathbb{P}(\phi_b = 0) \cdot \mathbb{E}[c_P(U_{1,b}^n) | \phi_b = 0] \\ & \quad + 2\mathbb{P}(\phi_b = 0) \cdot \mathbb{E}[c_S(X_{0,b}^n, \hat{U}_{2,b}^n) | \phi_b = 0] \\ & \le 2\mathbb{P}(\phi_b = 0) \cdot \mathbb{E}[|c_P(U_{1,b}^n) - P| + P | \phi_b = 0] \\ & \quad + 2\mathbb{P}(\phi_b = 0) \cdot \mathbb{E}[c_S(X_{0,b}^n, \hat{U}_{2,b}^n) | \phi_b = 0] \\ & = 2\mathbb{P}(\phi_b = 0) + 2\mathbb{P}(\phi_b = 0) \cdot \mathbb{E}[|c_P(U_{1,b}^n) - P| | \phi_b = 0] \\ & \quad + 2\mathbb{P}(\phi_b = 0) \cdot \mathbb{E}[c_S(X_{0,b}^n, \hat{U}_{2,b}^n) | \phi_b = 0]. \end{aligned} \qquad (72) $$

Here, the first term $2\mathbb{P}(\phi_b = 0) \le \frac{1}{36}\varepsilon$ since $P < \infty$. And the second term $2\mathbb{P}(\phi_b = 0) \cdot \mathbb{E}[|c_P(U_{1,b}^n) - P| | \phi_b = 0] \le 2\mathbb{E}[|c_P(U_{1,b}^n) - P|] \le \frac{1}{36}\varepsilon$ due to the nonnegativity of $|c_P(U_{1,b}^n) - P|$ and the power constraint (69).

Next, we analyse the last term in (72) with Wyner's trick [79] exploiting the discretization of $U_2$ as follows

$$ \mathbb{E}[c_S(X_{0,b}^n, \hat{U}_{2,b}^n) | \phi_b = 0] $$
$$ \leq \frac{1}{n} \sum_{t=1}^{n} \mathbb{E}[D(X_{0,t,b}) | \phi_b = 0], $$

with $D(X_{0,t,b}) = \max_{\hat{u}_{2,k}} c_S(X_{0,t,b}, \hat{u}_{2,k})$. The random variables $\{D(X_{0,t,b})\}_{t=1}^n$ are i.i.d. and integrable since $c_S(\cdot, \cdot)$ is a quadratic measure and $X_{0,t,b}$ is i.i.d. Gaussian distributed. Next, let $\chi_{\{D(X_{0,t,b})>d\}}$ denote an indicator function indicating if $D(X_{0,t,b}) > d$ for some $0 < d < \infty$. Then we have

$$ \begin{aligned} & \mathbb{P}(\phi_b = 0) \cdot \mathbb{E} \left[ c_S(X_{0,b}^n, \hat{U}_{2,b}^n) | \phi_b = 0 \right] \\ & \leq d \cdot \mathbb{P}(\phi_b = 0) + \frac{1}{n} \sum_{t=1}^{n} \mathbb{E} \left[ D(X_{0,t,b}) \chi_{\{D(X_{0,t,b})>d\}} \right]. \end{aligned} \quad (73) $$

Given that $X_{0,t,b}$ is generated i.i.d, and $D(X_{0,t,b})$ is integrable, for any $\varepsilon_d > 0$, there must exist a $d_0$ such that $\mathbb{E}[D(X_{0,t,b})\chi_{\{D(X_{0,t,b})>d\}}] < \varepsilon_d$ for all $d > d_0$, as per the Monotone Convergence Theorem. Thus, for a sufficiently small $\varepsilon_d$ and a sufficiently large $n$, term (73) can be upper bounded by $\frac{1}{2^2}\varepsilon$, so that (71) $\leq \frac{1}{12}\varepsilon$.

Thus, compared to the original cost constraint $S$, for block $b \in [1: B - 1]$ we have

$$ \begin{aligned} & \mathbb{E}[|c_S(X_{1,b}^n, \hat{U}_{2,b}^n) - S|] \\ & \leq |\hat{S} - S| + \mathbb{E}[|c_S(X_{1,b}^n, \hat{U}_{2,b}^n) - \hat{S}|] \\ & \leq \frac{1}{4}\varepsilon + \mathbb{E}[|c_S(X_{1,b}^n, \hat{U}_{2,b}^n) - \hat{S}|] \leq \frac{1}{2}\varepsilon. \end{aligned} \quad (74) $$

Lastly, we estimate the cost of the last block where the output $\hat{U}_{2,B}^n$ is set to be an all-zero sequence. In this case

$$ \begin{aligned} & \mathbb{E}[|c_S(X_{1,B}^n, \hat{U}_{2,B}^n) - \hat{S}|] \\ & \leq \mathbb{E}[|2c_P(X_{0,B}^n) + 2c_P(U_{1,B}^n) - \hat{S}|] \\ & \leq 2 \cdot \mathbb{E}[c_P(X_{0,B}^n)] + 2 \cdot \mathbb{E}[c_P(U_{1,B}^n)] + \hat{S} \\ & \stackrel{(d)}{\leq} 2 \cdot Q + 2 \cdot P + \hat{S} < \infty, \end{aligned} $$

where (d) comes from that $X_{0,B}^n$ is generated i.i.d. $\sim \mathcal{N}(0, Q\mathbb{I})$. Therefore, similar to (74), we have

$$ \mathbb{E}[|c_S(X_{1,B}^n, \hat{U}_{2,B}^n) - S|] \leq \frac{1}{4}\varepsilon + 2 \cdot Q + 2 \cdot P + \hat{S} \triangleq \tilde{S}. $$

For the average estimation cost, the impact of the last pathological block can be effectively diminished, since

$$ \begin{aligned} & \mathbb{E}[|c_S(X_1^{Bn}, \hat{U}_2^{Bn}) - S|] \\ & \leq \frac{1}{B} \sum_{b=1}^{B} \mathbb{E}[|c_S(X_{1,b}^n, \hat{U}_{2,b}^n) - S|] \\ & \leq \frac{1}{B} \tilde{S} + \frac{B-1}{B} \cdot \frac{1}{2} \varepsilon \leq \varepsilon, \end{aligned} \quad (75) $$

for $n$ and $B$ sufficiently large.

*Existence of the desired code:* With the aid of random coding arguments and applying the Selection Lemma [81] to both conditions (70) (75), we can obtain that there exists a control design $c \in \mathcal{C}(Bn)$ that simultaneously satisfies

$$ \mathbb{E}[|c_P(U_1^{Bn}) - P|] < \varepsilon, \quad \mathbb{E}[|c_S(X_1^{Bn}, \hat{U}_2^{Bn}) - S|] < \varepsilon. $$

*Closedness:* So far we have only shown under strict inequality of rate constraint (17) that the cost constraints hold. As for equality, we consider a sequence of probability distribution $\mathcal{P}^k \xrightarrow{k \to \infty} \mathcal{P}^*$, where each $\mathcal{P}^k$ satisfies the rate constraint with strict inequality, and $\mathcal{P}^*$ satisfies it with equality. Therefore,

$$ \begin{aligned} & \mathbb{E}_{\mathcal{P}^*} [ |c_S(X_1^n, U_2^n) - S| ] \\ & \leq \mathbb{E}_{\mathcal{P}_k} [ |c_S(X_1^n, U_2^n) - S| ] + \mathbb{E}_{\mathcal{P}^* - \mathcal{P}_k} [ |c_S(X_1^n, U_2^n) - S| ] \\ & \leq \frac{1}{2} \cdot \varepsilon + \frac{1}{2} \cdot \varepsilon \leq \varepsilon \end{aligned} $$

where the first $1/2 \cdot \varepsilon$ comes from the achievability proof with strict inequality as $n$ large enough, and the second $1/2 \cdot \varepsilon$ could similarly be derived repeating the estimation cost analysis steps above since the probability measure $\mathcal{P}^* - \mathcal{P}_k \to 0$ as $k \to 0$. Moreover, the power constraint for $\mathcal{P}^*$ also holds due to (68). This concludes our achievability proof. $\square$
APPENDIX B
CONVERSE PROOF OF THEOREM III.1

We consider a control design $c \in \mathcal{C}(n)$ of block length $n \in \mathbb{N}^*$ that achieves a pair of costs $(P, S)$. According to Csiszár sum identity, we have

$$
\begin{align*}
0 &= \sum_{t=1}^{n} I(X_0^{t-1}; Y_{1,t} | Y_{1,t+1}^n) - \sum_{t=1}^{n} I(Y_{1,t+1}^n; X_{0,t} | X_0^{t-1}) \\
&\le \sum_{t=1}^{n} I(X_0^{t-1}, Y_{1,t+1}^n; Y_{1,t}) - \sum_{t=1}^{n} I(Y_{1,t+1}^n; X_{0,t} | X_0^{t-1}) \\
&\stackrel{(a)}{=} \sum_{t=1}^{n} I(W_{1,t}, W_{2,t}; Y_{1,t}) - \sum_{t=1}^{n} I(W_{2,t}; X_{0,t} | W_{1,t}) \\
&\stackrel{(b)}{=} n \cdot (I(W_{1,T}, W_{2,T}; Y_{1,T} | T) - I(W_{2,T}; X_{0,T} | W_{1,T}, T)) \\
&\le n \cdot (I(W_{1,T}, T, W_{2,T}; Y_{1,T}) - I(W_{2,T}; X_{0,T} | W_{1,T}, T)) \\
&\stackrel{(c)}{=} n \cdot (I(W_1, W_2; Y_1) - I(W_2; X_0 | W_1)),
\end{align*}
$$

where (a) is from the identification of the auxiliary RVs $W_{1,t} = X_0^{t-1}$, $W_{2,t} = Y_{1,t+1}^n$, (b) is from introducing the uniform random variable $T \in \{1, ..., n\}$ and the corresponding mean auxiliary random variables $X_{0,T}, W_{1,T}, W_{2,T}, Y_{1,T}$, where $Y_{1,T}$ is distributed according to

$$ \mathbb{P}(Y_{1,T} = y_1) = \frac{1}{n} \sum_{t=1}^{n} \mathbb{P}(Y_{1,t} = y_1), \quad \forall y_1 \in \mathcal{Y}_1 $$

and (c) comes from the introduction of $X_0 \triangleq X_{0,T}, W_1 \triangleq (W_{1,T}, T), W_2 \triangleq W_{2,T}, Y_1 \triangleq Y_{1,T}$.

Next, we show that the introduced auxiliary RVs $W_{1,t} = X_0^{t-1}$, and $W_{2,t} = Y_{1,t+1}^n$ in equation (a) satisfy the Markov chain properties (19).

For $t \in \{1, ..., n\}$:

* $X_{0,t}$ and $W_{1,t} = X_0^{t-1}$ are independent because of the i.i.d. property of the source.

* $U_{1,t} \leftarrow (X_{0,t}, W_{1,t}) \leftarrow W_{2,t}$ is given by the causal encoding, that the current encoder's action $U_{1,t}$ depends on the past and current input states $(X_{0,t}, X_0^{t-1})$, not on the future sequence $W_{2,t}$.

* $(X_{1,t}, Y_{1,t}) \leftarrow (X_{0,t}, U_{1,t}) \leftarrow (W_{1,t}, W_{2,t})$ comes from the memoryless property of the channel.

* $U_{2,t} \leftarrow (W_{1,t}, W_{2,t}, Y_{1,t}) \leftarrow (X_{0,t}, U_{1,t}, X_{1,t})$ is shown as Lemma B.1 in Appendix B-A.

This implies that the auxiliary random variables $X_0 = X_{0,T}, W_1 = (W_{1,T}, T), W_2 = W_{2,T}, U_1 = U_{1,T}, X_1 = X_{1,T}, Y_1 = Y_{1,T}, U_2 = U_{2,T}$ satisfy the Markov chains (19). Equivalently, the distribution of auxiliary random variables decomposes as in (16).

Now, we reformulate n-stage costs with a control design $c \in \mathcal{C}(n)$. By defining distributions $\mathbb{P}(U_{1,T} = u_1) = \frac{1}{n} \sum_{t=1}^n \mathbb{P}(U_{1,t} = u_1)$, $\forall u_1 \in \mathcal{U}_1$ and $\mathbb{P}(X_{1,T} = x_{1,T}, U_{2,T} = u_{2,T}) = \frac{1}{n} \sum_{t=1}^n \mathbb{P}(X_{1,t} = x_1, U_{2,t} = u_2)$, $\forall (x_1, u_2) \in \mathcal{X}_1 \times \mathcal{U}_2$ and using auxiliary random variables $U_1 = U_{1,T}, X_1 = X_{1,T}, U_2 = U_{2,T}$, we have

$$
\begin{align*}
\mathbb{E}[c_P(U_1^n)] &= \mathbb{E} \left[ \frac{1}{n} \sum_{t=1}^{n} U_{1,t}^2 \right] = \mathbb{E} [U_{1,T}^2] = \mathbb{E} [U_1^2], \\
\mathbb{E}[c_S(X_1^n, U_2^n)] &= \mathbb{E} \left[ \frac{1}{n} \sum_{t=1}^{n} (X_{1,t} - U_{2,t})^2 \right] \\
&= \mathbb{E} \left[ (X_{1,T} - U_{2,T})^2 \right] = \mathbb{E} \left[ (X_1 - U_2)^2 \right].
\end{align*}
$$

In conclusion, if a pair of costs $(P, S) \in \mathbb{R}^2$ is achievable, then for all $\varepsilon > 0$, there exists $\tilde{n} \in \mathbb{N}^*$ such that for all $n \ge \tilde{n}$, there exists a control design $c \in \mathcal{C}(n)$ such that its induced long-run costs

$$
\begin{align*}
& \mathbb{E}[|P - c_P(U_1^n)| + |S - c_S(X_1^n, U_2^n)|] \le \varepsilon \\
\Rightarrow & |P - \mathbb{E}[c_P(U_1^n)]| + |S - \mathbb{E}[c_S(X_1^n, U_2^n)]| \le \varepsilon \\
\Rightarrow & |P - \mathbb{E}[U_1^n]| + |S - \mathbb{E}[(X_1 - U_2)^2]| \le \varepsilon \tag{76}
\end{align*}
$$

The equation (76) is valid for all $\varepsilon > 0$. This shows that $(X_0, W_1, W_2, U_1, X_1, Y_1, U_2)$ satisfy (16), (17), (18).
Now, without loss of generality, we consider the joint Gaussian random variables $(X_0, W_1, W_2, U_1) \sim \mathcal{N}(0, K)$ optimal for problem (20), are centered with the covariance matrix

$$ K = \begin{pmatrix} Q & \rho_1 \sqrt{QV_1} & \rho_2 \sqrt{QV_2} & \rho_3 \sqrt{QP} \\ \rho_1 \sqrt{QV_1} & V_1 & \rho_4 \sqrt{V_1 V_2} & \rho_5 \sqrt{V_1 P} \\ \rho_2 \sqrt{QV_2} & \rho_4 \sqrt{V_1 V_2} & V_2 & \rho_6 \sqrt{V_2 P} \\ \rho_3 \sqrt{QP} & \rho_5 \sqrt{V_1 P} & \rho_6 \sqrt{V_2 P} & P \end{pmatrix}. \quad (79) $$

Since $X_0 \perp W_1$, $\rho_1 = 0$. Also, given the Markov chain $U_1 \leftarrow (X_0, W_1) \leftarrow W_2$, from a simple extension of Lemma C.1, we can obtain that $\rho_6$ is uniquely determined through the other four parameters by $\rho_6 = \rho_2\rho_3 + \rho_4\rho_5$. Moreover, other active correlation coefficients $(\rho_2, \rho_3, \rho_4, \rho_5) \in [-1, 1]^4$ are chosen such that

$$ \det(K) = QV_1V_2P(-1 + \rho_2^2 + \rho_4^2)(-1 + \rho_3^2 + \rho_5^2) \ge 0. $$

Given covariance matrix $K$ and (2), the covariance matrix $K_2$ of $(X_0, W_1, W_2, Y_1)$ could be easily computed, with a determinant given by

$$ \det(K_2) = QV_1V_2(-1 + \rho_2^2 + \rho_4^2)(P(-1 + \rho_3^2 + \rho_5^2) - N). $$

The positive semi-definite property of $K_2$ must also be satisfied with properly chosen $(\rho_2, \rho_3, \rho_4, \rho_5)$.

We have the following lemma determining the explicit expression of the information constraint (17) and the optimization object (20) under the joint Gaussian constraint.

**Lemma C.2.** Assume $(X_0, W_1, W_2, U_1) \sim \mathcal{N}(0, K)$, then

$$ I(W_1, W_2; Y_1) - I(W_2; X_0|W_1) = I(W_1; Y_1) - I(W_2; X_0|W_1, Y_1) \quad (80) $$

$$ = \frac{1}{2} \log \left( \frac{T_1}{T_1 - T_2} \right), \quad (81) $$

where the terms

$$ T_1 = (P + Q + N + 2\rho_3\sqrt{QP})(-1 + \rho_2^2 + \rho_4^2), $$

$$ T_2 = N\rho_2^2 + P\rho_2^2(1 - \rho_3^2) - P\rho_5^2(1 - \rho_4^2). $$

And the object to minimize writes

$$ \begin{aligned} & \mathbb{E} \left[ \left( X_1 - \mathbb{E}[X_1|W_1, W_2, Y_1] \right)^2 \right] \\ & = \frac{N \cdot f_1(\rho_2, \rho_3, \rho_4, \rho_5)}{(1 - \rho_4^2) \cdot N + f_1(\rho_2, \rho_3, \rho_4, \rho_5)}, \end{aligned} \quad (82) $$

where

$$ \begin{aligned} & f_1(\rho_2, \rho_3, \rho_4, \rho_5) \\ & = -P\rho_2^2\rho_3^2 - (Q + 2\rho_3\sqrt{PQ})(-1 + \rho_2^2 + \rho_4^2) \\ & \quad + P(1 - \rho_4^2)(1 - \rho_5^2). \end{aligned} $$

The proof of this lemma is given in Appendix C-B

Thus, from (81), the original information constraint is given explicitly by

$$ \frac{1}{2} \log \left( \frac{T_1}{T_1 - T_2} \right) \ge 0 \Leftrightarrow T_1 \ge T_2 \ge 0 \text{ or } T_1 \le T_2 \le 0. $$

Now, we examine the objective (82). First, if $1 - \rho_4^2 = 0$, from (82), we can get that

$$ \mathbb{E} \left[ \left( X_1 - \mathbb{E}[X_1 | W_1, W_2, Y_1] \right)^2 \right] = N. $$

Next, we focus on the case of $1 - \rho_4^2 \ne 0$. In this case, (82) is of the form

$$ \frac{N \cdot f(\rho_2, \rho_3, \rho_4, \rho_5)}{N + f(\rho_2, \rho_3, \rho_4, \rho_5)}, \quad (83) $$

where

$$ f(\rho_2, \rho_3, \rho_4, \rho_5) = f_1(\rho_2, \rho_3, \rho_4, \rho_5)/(1 - \rho_4^2). $$
Note that, the function $x \mapsto \frac{N \cdot x}{N+x}$ is nonnegative and strictly increasing over the region $(-\infty, -N] \cup [0, \infty)$. Therefore, our goal of minimizing (83) is now transformed to either minimizing the nonnegative object

$$f(\rho_2, \rho_3, \rho_4, \rho_5) \geq 0$$

or minimizing the negative object

$$f(\rho_2, \rho_3, \rho_4, \rho_5) \leq -N$$

subject to the following constraints:

1. $\det(K) \geq 0 \implies$ 
   $QV_1V_2P(-1 + \rho_2^2 + \rho_4^2)(-1 + \rho_3^2 + \rho_5^2)^{(A)} \geq 0,$

2. $\det(K_2) \geq 0 \implies$ 
   $QV_1V_2(-1 + \rho_2^2 + \rho_4^2)(P(-1 + \rho_3^2 + \rho_5^2) - N)^{(B)} \geq 0,$

3. $T_1 \geq T_2 \geq 0 \implies$ 
   $(Q + P + N + 2\rho_3\sqrt{QP})(-1 + \rho_2^2 + \rho_4^2)^{(C1)} \geq N\rho_2^2 + P\rho_2^2(1 - \rho_3^2) - P\rho_5^2(1 - \rho_4^2)^{(D1)} \geq 0,$ 
   or, $T_1 \leq T_2 \leq 0 \implies$ 
   $(Q + P + N + 2\rho_3\sqrt{QP})(-1 + \rho_2^2 + \rho_4^2)^{(C2)} \leq N\rho_2^2 + P\rho_2^2(1 - \rho_3^2) - P\rho_5^2(1 - \rho_4^2)^{(D2)} \leq 0.$

To simplify the above constraints, we consider the following two distinct cases:

Case 1, if $-1 + \rho_2^2 + \rho_4^2 \geq 0$, constraints (A) and (B) together yield $-1 + \rho_3^2 + \rho_5^2 \geq \frac{N}{P}$. Moreover, constraint (C1) gives us $f(\rho_2, \rho_3, \rho_4, \rho_5) \leq -N$. In this case, our optimization problem boils down to minimizing

$$f(\rho_2, \rho_3, \rho_4, \rho_5) \leq -N, \qquad (84)$$

subject to

$$ 
\begin{align} 
1. & -1 + \rho_2^2 + \rho_4^2 \geq 0, \notag \\ 
2. & -1 + \rho_3^2 + \rho_5^2 \geq \frac{N}{P}, \notag \\ 
3. & N\rho_2^2 + P\rho_2^2(1 - \rho_3^2) - P\rho_5^2(1 - \rho_4^2) \geq 0. \tag{85}
\end{align} 
$$

Notice that $f(\rho_2, \rho_3, \rho_4, \rho_5)$ is decreasing function of $\rho_5^2$. From (85), we get that $\rho_5^2 \leq \frac{N\rho_2^2 + P\rho_2^2(1-\rho_3^2)}{P(1-\rho_4^2)}$. Therefore, the optimizer is given by $(\rho_5^*)^2 = \frac{N\rho_2^2 + P\rho_2^2(1-\rho_3^2)}{P(1-\rho_4^2)}$. By plugging $(\rho_5^*)^2$ into (84), we obtain that

$$f(\rho_2, \rho_3, \rho_4, \rho_5^*) = \frac{(-1 + \rho_2^2 + \rho_4^2)(Q + P + 2\rho_3\sqrt{PQ}) - N\rho_2^2}{1 - \rho_4^2}.$$

Since

$$\frac{\partial f(\rho_2, \rho_3, \rho_4, \rho_5^*)}{\partial \rho_4^2} \leq 0,$$

we know that $f \to -\infty$ as $\rho_4^2 \to 1$. Moreover, since $\mathbb{E}[(X_1 - \mathbb{E}[X_1|W_1, W_2, Y_1])^2]$ is continuous and converges to $N$ when $\rho_4^2 \to 1$, in this case, the minimal value of $N$ is obtained at the boundary $\rho_4^2 = 1$.

Case 2, If $-1 + \rho_2^2 + \rho_4^2 \leq 0$, conditions (A) and (B) together give us $-1 + \rho_3^2 + \rho_5^2 \leq 0$, and (C2) gives us $f(\rho_2, \rho_3, \rho_4, \rho_5) \geq -N$ (but only $f \geq 0$ contributes to a nonnegative estimation cost). Therefore, in this case, our optimization problem boils down to minimizing

$$f(\rho_2, \rho_3, \rho_4, \rho_5) \geq 0, \qquad (86)$$

subject to

$$ 
\begin{align} 
1. & -1 + \rho_2^2 + \rho_4^2 \leq 0, \notag \\ 
2. & -1 + \rho_3^2 + \rho_5^2 \leq 0, \tag{87}
\end{align} 
$$
$$3. N\rho_2^2 + P\rho_2^2(1 - \rho_3^2) - P\rho_5^2(1 - \rho_4^2) \le 0. \qquad (88)$$

Since $f(\rho_2, \rho_3, \rho_4, \rho_5)$ is reduced especially when $\rho_5^2$ is increased, therefore, from (87), we get that $(\rho_5^*)^2 = 1 - \rho_3^2$. By replacing $(\rho_5^*)^2$ into (86), we get that

$$f(\rho_2, \rho_3, \rho_4, \rho_5^*) = \frac{(1 - \rho_2^2 - \rho_4^2)(\sqrt{P}\rho_3 + \sqrt{Q})^2}{(1 - \rho_4^2)}. \qquad (89)$$

Therefore, when $P \ge Q$, taking $\rho_3^* = -\sqrt{\frac{Q}{P}}$ and any $\rho_2, \rho_4$ satisfy the constraints results in the optimal value of $f(\rho_2, \rho_3^*, \rho_4, \rho_5^*) = 0$. In this case, $\mathbb{E}[(X_1 - \mathbb{E}[X_1|W_1, W_2, Y_1])^2] = 0$.

When $P < Q$, $f(\rho_2, \rho_3, \rho_4, \rho_5^*)$ is a decreasing function of $\rho_2^2$. The constraint (88) gives us the optimal value of $\rho_2^2$:

$$(\rho_2^*)^2 = \frac{P(1 - \rho_3^2)(1 - \rho_4^2)}{N + P(1 - \rho_3^2)}. \qquad (90)$$

Plugging the $(\rho_2^*)^2$ in (90) into (89), we have

$$f(\rho_2^*, \rho_3, \rho_4, \rho_5^*) = \frac{N(\sqrt{P}\rho_3 + \sqrt{Q})^2}{N + P(1 - \rho_3^2)}.$$

Then, taking $\frac{\partial f}{\partial \rho_3} = 0$ gives us the optimum

$$\rho_3^* = -\frac{P + N}{\sqrt{QP}},$$

which is valid only if the following condition holds

$$(\rho_3^*)^2 = \frac{(P + N)^2}{QP} \le 1 \implies \begin{cases} Q > 4N, \\ P \in [P_1, P_2], \end{cases} \qquad (91)$$

where

$$P_1 = \frac{1}{2} \left( Q - 2N - \sqrt{Q^2 - 4QN} \right), \\ P_2 = \frac{1}{2} \left( Q - 2N + \sqrt{Q^2 - 4QN} \right).$$

In this case, $(\rho_2^*)^2 = \frac{QP - (P+N)^2}{QP}$, and $f(\rho_2^*, \rho_3^*, \rho_4^*, \rho_5^*) = \frac{N(Q-N-P)}{N+P}$, which results in the estimation cost of

$$\mathbb{E}[(X_1 - \mathbb{E}[X_1|W_1, W_2, Y_1])^2] = \frac{N(Q - N - P)}{Q}.$$

In the case when the condition (91) is unmet, we always have $-\frac{P+N}{\sqrt{QP}} < -1$. Since $\frac{\partial f}{\partial \rho_3} > 0$, function $f$ increases when $\rho_3 \in [-1, 1]$ increases. Therefore, the minimal value of $f$ achieves at the left boundary $\rho_3^* = -1$, which gives us $\rho_2^* = \rho_5^* = 0$ and $f(\rho_2^*, \rho_3^*, \rho_4^*, \rho_5^*) = (\sqrt{Q} - \sqrt{P})^2$. Hence,

$$\mathbb{E}[(X_1 - \mathbb{E}[X_1|W_1, W_2, Y_1])^2] = \frac{N \cdot (\sqrt{Q} - \sqrt{P})^2}{N + (\sqrt{Q} - \sqrt{P})^2}.$$

Obviously, the minimal estimation cost value $N$ from case 1 is always larger than the cost derived in case 2. Therefore, summarizing our above analysis, the optimal Gaussian cost $S_G(P)$ is given by (22). $\square$

### A. Proof of Lemma C.1

From the Markov chain and joint Gaussian entropy, we have that

$$\begin{aligned} 0 &= I(X; Z|Y) \\ &= H(X, Y) + H(Y, Z) - H(Y) - H(X, Y, Z) \\ &= \frac{1}{2} \log \left( \frac{\det(\Sigma_{X,Y}) \cdot \det(\Sigma_{Y,Z})}{\sigma_Y^2 \cdot \det(\Sigma_{X,Y,Z})} \right), \end{aligned}$$

where all the information of the last step can be obtained from the covariance matrix (77). Therefore,

$$\begin{aligned} 0 &= \det(\Sigma_{X,Y}) \cdot \det(\Sigma_{Y,Z}) - \sigma_Y^2 \cdot \det(\Sigma_{X,Y,Z}) \\ &= PQ^2V(\rho_1\rho_3 - \rho_2)^2, \end{aligned}$$

which implies the result given in (78). $\square$
## B. Proof of Lemma C.2

The information constraint (17) can be rewritten in the following way:

$$
\begin{align*}
& I(W_1, W_2; Y_1) - I(W_2; X_0|W_1) \\
&= I(W_1; Y_1) + I(W_2; Y_1|W_1) - I(W_2; X_0|W_1) \\
&\stackrel{(a)}{=} I(W_1; Y_1) + I(W_2; Y_1|W_1) - I(W_2; X_0, Y_1|W_1) \\
&= I(W_1; Y_1) - I(W_2; X_0|W_1, Y_1) \\
&= \frac{1}{2} \log \left( \frac{\sigma_{W_1}^2 \cdot \sigma_{Y_1}^2 \cdot \det(K_2)}{\det(\Sigma_{W_1, W_2, Y_1}) \cdot \det(\Sigma_{X_0, W_1, Y_1})} \right) \\
&= \frac{1}{2} \log \left( \frac{T_1}{T_1 - T_2} \right),
\end{align*}
$$

where (a) comes from the Markov chain $Y_1 \leftarrow (X_0, W_1) \leftarrow W_2$, and thus $I(W_2; Y_1|X_0, W_1) = 0$. Additionally,

$$
\begin{align*}
& \mathbb{E} \left[ \left( X_1 - \mathbb{E}[X_1 | W_1, W_2, Y_1] \right)^2 \right] \\
&= \mathrm{Var} (X_1 | W_1, W_2, Y_1) \\
&\stackrel{(b)}{=} \sigma_{X_1}^2 - \Sigma_{X_1 W}^\top \Sigma_{W W}^{-1} \Sigma_{X_1 W} \\
&= \frac{N \cdot f_1(\rho_2, \rho_3, \rho_4, \rho_5)}{(1 - \rho_4^2) \cdot N + f_1(\rho_2, \rho_3, \rho_4, \rho_5)},
\end{align*}
$$

where step (b) is obtained using the Schur complement. Here, $W = (W_1, W_2, Y_1)^\top$, $\Sigma_{X_1 W} = (\sigma_{X_1, W_1}, \sigma_{X_1, W_2}, \sigma_{X_1, Y_1})^\top$, and $\Sigma_{W W}$ is the covariance matrix of $(W_1, W_2, Y_1)$. $\square$

# APPENDIX D

## PROOF OF COROLLARY III.4

From Theorem V.2, the information constraint for Witsenhausen counterexample with causal encoder and non-causal decoder with channel feedback $Y_1$ is

$$ I(W_1; Y_1) - I(U_2; X_0|W_1, Y_1) \geq 0. \qquad (92) $$

Under the joint Gaussian constraint, the MMSE estimator of $X_1$ is linear, i.e.,

$$ U_2 = \mathbb{E}[X_1 | W_1, W_2, Y_1] = a \cdot W_1 + b \cdot W_2 + c \cdot Y_1, $$

for some constants $a, b, c \in \mathbb{R}$. Substituting this into (92) yields

$$
\begin{align*}
& I(W_1; Y_1) - I(U_2; X_0|W_1, Y_1) \\
&= I(W_1; Y_1) - I(a \cdot W_1 + b \cdot W_2 + c \cdot Y_1; X_0|W_1, Y_1) \\
&= I(W_1; Y_1) - I(W_2; X_0|W_1, Y_1) \\
&= I(W_1, W_2; Y_1) - I(W_2; X_0|W_1),
\end{align*}
$$

where in the last equality we use the step (80) in Lemma C.2. Hence, this recovers the information constraint for the framework without channel feedback (17). Thus, the optimization domain for this optimization problem remains unchanged, and the optimal joint Gaussian cost is (22), establishing case 1) in the corollary.

As for 2), [57] showed that the optimal Gaussian cost for noncausal encoding with causal decoding, both with and without source feedforward, also coincides with (22).

Finally, for case 3), note that the Gaussian strategy (22) is simply a time-sharing mechanism of the best linear scheme. This applies even when both DMs are causal (achievability). Conversel, by cases 1) and 2), (22) also serves as a lowerbound of this causal Gaussian cost regardless of feedback or feedforward links (converse).

To conclude, we have established that all the causal decision-making configurations listed in the corollary, no matter with or without feedback or feedforward, share the same optimal joint Gaussian result. $\square$

# APPENDIX E

## PROOF FOR PROPOSITION IV.1

If the estimation cost is zero, then,

$$ \mathbb{E}[\mathrm{Var}(X_1|Y_1)] = \mathbb{E}[(X_1 - \mathbb{E}[X_1|Y_1])^2] = 0, $$

Hence, $X_1 = \mathbb{E}[X_1|Y_1]$ almost surely (a.s.).
Using Bayes' formula, given the independence of $X_1$ and the channel noise $Z_1$, the posterior distribution of $X_1$ given $Y_1 = y$ is

$$f_{X_1|Y_1}(x|y) = \frac{f_X(x)\phi_Z(y-x)}{f_Y(x)}, \forall x \in \text{supp}(X_1).$$

Now, suppose $\exists x_1 \neq x_2 \in \text{supp}(X_1)$, since $\phi_Z(y-x)$ is strictly positive for all $x \in \mathbb{R}$, then we have

$$f_{X_1|Y_1}(x_1|y) > 0, \quad f_{X_1|Y_1}(x_2|y) > 0,$$

which result in

$$\text{Var}(X_1|Y_1 = y) = \sum_{x_1,x_2} (x - \mathbb{E}[X_1|Y_1 = y])^2 f_{X_1|Y_1}(x|y) > 0.$$

for such a $y$. Hence,

$$\mathbb{E}[(X_1 - \mathbb{E}[X_1|Y_1])^2] = \mathbb{E}[\text{Var}(X_1|Y_1)] > 0.$$

Hence, by contradiction, we have $|\text{supp}(X_1)| = 1$, i.e., $X_1$ is a constant variable a.s.

Hence, we assume $X_1 = c$ for some constant $c \in \mathbb{R}$, $U_1 = X_1 - X_0 = c - X_0$, then we have

$$Q = \min_{c \in \mathbb{R}} \mathbb{E}[U_1^2], \text{ where } X_0 \sim \mathcal{N}(0, Q).$$

This is achieved if and only if $c = 0$, i.e., $U_1 = -X_0$.

<figure><img src="image_13.png" /></figure>

# APPENDIX F

## ACHIEVABILITY PROOF OF THEOREM V.2

We consider an arbitrary but fixed $\varepsilon > 0$ and assume the sequence $(X_0^n, W_1^n, U_1^n, X_1^n, Y_1^n, U_2^n)$ is generated i.i.d. according to a distribution that decomposes as $\mathcal{P}_{X_0,W_1}\mathcal{P}_{U_1|X_0,W_1}\mathcal{P}_{X_1,Y_1|X_0,U_1}\mathcal{P}_{U_2|X_0,W_1,Y_1}$, with $P = \mathbb{E}[U_1^2]$, $S = \mathbb{E}[(X_1 - U_2)^2]$. Let $\psi^{(n)}: \mathcal{X}_0^n \times \mathcal{W}_1^n \times \mathcal{U}_1^n \times \mathcal{X}_1^n \times \mathcal{Y}_1^n \times \mathcal{U}_2^n \to \{0,1\}$ denote an indicator function for sequences of length $n$ with

$$\psi^{(n)}(x_0^n, w_1^n, u_1^n, x_1^n, y_1^n, u_2^n) = \begin{cases} 1 & \text{if } |c_S(x_1^n, u_2^n) - S| \geq \frac{1}{12}\varepsilon \\ 0 & \text{otherwise.} \end{cases}$$

Using the weak law of large numbers (LLN) and the union bound we have

$$\mathbb{E}[\psi^{(n)}(X_0^n, W_1^n, U_1^n, X_1^n, Y_1^n, U_2^n)] \leq \delta_n \to 0,$$

as $n \to \infty$.

Now, we provide a coding scheme by considering a probability distribution $\mathcal{P}_{X_0,W_1,U_1,X_1,Y_1,U_2}$ that decomposes as (45) such that exists a $\delta > 0$ and a rate $R > 0$ such that

$$I(U_2; X_0, Y_1|W_1) + \delta \leq R \leq I(W_1, U_2; Y_1) - \delta. \quad (93)$$

where both the lower bound and the upper bound are also assumed to be finite. Let $c \in \mathcal{C}_f(Bn)$ a block-Markov random code over $B \in \mathbb{N}^*$ blocks each of length $n \in \mathbb{N}^*$. See the illustration in Figure 11.

*Random codebook:* We generate $|\mathcal{M}| = 2^{nR}$ sequences $W_1^n(m)$ i.i.d. $\sim \mathcal{P}_{W_1}$ with index $m \in \mathcal{M}$. For each index $m \in \mathcal{M}$, we generate the same number $|\mathcal{M}| = 2^{nR}$ sequences $U_2^n(m, \hat{m})$ with index $\hat{m} \in \mathcal{M}$ i.i.d. $\sim \mathcal{P}_{U_2|W_1}$ depending on sequence $W_1^n(m)$.

*Encoding scheme:* Let $m_b$ denote the message the encoder communicates at block $b \in [1: B]$. For the first block, WLOG, the encoder takes $m_1 = 1$ and returns $W_{1,1}^n = W_1^n(m_1)$. Then, at each transmission $b \in [2: B]$, it recalls the index $m_{b-1}$ and finds $m_b \in \mathcal{M}$ such that sequences

$$(X_{0,b-1}^n, Y_{1,b-1}^n, W_1^n(m_{b-1}), U_2^n(m_{b-1}, m_b)) \in \mathcal{A}_\varepsilon^{(n)}.$$

It deduces $W_{1,b}^n \triangleq W_1^n(m_b)$ for the current block $b$ and sequentially draws the symbol $U_{1,t,b}$ i.i.d. $\sim \mathcal{P}_{U_1|X_0,W_1}$ conditioning on $W_{1,t}(m_b)$ and $X_{0,b,t}$ received causally at each time instant $t \in [1:n]$.

*Decoding scheme:* The decoder first returns $\tilde{m}_1 = 1$ for block $b=1$. At block $b \in [2: B]$, it recalls $\tilde{m}_{b-1}$ from the last block and finds $\tilde{m}_b \in \mathcal{M}$ such that

$$(Y_{1,b}^n, W_1^n(\tilde{m}_b)) \in \mathcal{A}_\varepsilon^{(n)}(\mathcal{P}_{Y_1,W_1}), \\ (Y_{1,b-1}^n, W_1^n(\tilde{m}_{b-1}), U_2^n(\tilde{m}_{b-1}, \tilde{m}_b)) \in \mathcal{A}_\varepsilon^{(n)}(\mathcal{P}_{Y_1,W_1,U_2}).$$

We denote by $\tilde{W}_{1,b}^n = W_1^n(\tilde{m}_b)$ and $\tilde{U}_{2,b-1}^n = U_2^n(\tilde{m}_{b-1}, \tilde{m}_b)$ as our choice. For $b \in [1: B-1]$, the returned codeword $\tilde{U}_{2,b}^n$ serves as the reconstruction for the interim state $X_{1,b}^n$.
<figure><img src="image_14.png" /><figcaption>Fig. 11. Block-Markov codes for causal encoding noncausal decoding setup with channel feedback.</figcaption></figure>

**Termination block:** The decoder simply outputs $\tilde{U}_{2,B}^n = 0$. Usually, sequences are not jointly typical in the last block.

**Error analysis per block:** We first focus on the encoding error $\mathcal{E}^e$. For $b \in [2: B]$, let $\mathcal{E}_b^e(m_{b-1})$ denote the event of a failed encoding process during block $b$ given the knowledge of $m_{b-1}$, i.e., $\mathcal{E}_b^e(m_{b-1}) = \{\forall m_b \in \mathcal{M}, (X_{0,b-1}^n, Y_{1,b-1}^n, W_1^n(m_{b-1}), U_2^n(m_{b-1}, m_b)) \notin \mathcal{A}_\varepsilon^{(n)}(\mathcal{P}_{X_0,W_1,Y_1,U_2})\}$. Due to the independence between the codewords and Markov blocks, the probability of an encoding error in block $b$ given no encoding errors in the previous blocks is

$$ \mathbb{P}(\mathcal{E}_b^e(M_{b-1}) | \cap_{\beta=2}^{b-1} \bar{\mathcal{E}}_\beta^e(M_{\beta-1})) = \mathbb{P}(\mathcal{E}_b^e(M_{b-1})). $$

If $R \ge I(U_2; X_0, Y_1|W_1) + \delta$, following the Covering Lemma [78], we have $\mathbb{P}(\mathcal{E}_b^e(M_{b-1})) \to 0$ as $n \to \infty$, $\forall b \in [2: B]$. Thus, state sequences for the first $B-1$ blocks are successfully encoded with probability $\to 1$ as $n \to \infty$.

Next, we analyze the decoding error $\mathcal{E}^d$. For $b \in [2: B]$, let $\mathcal{E}_b^d(m_{b-1})$ denote the event of a failed decoding process in block $b$ given the past estimated index $m_{b-1}$. Furthermore, let $\mathcal{E}_{b-1}^{d,1}(m_{b-1})$ denote the event that the sequence $Y_{1,b-1}^n$ given $m_{b-1}$ is not jointly typical, i.e., $\mathcal{E}_{b-1}^{d,1}(m_{b-1}) = \{(Y_{1,b-1}^n, W_{1,b-1}^n(m_{b-1}), U_2^n(m_{b-1}, M_b)) \notin \mathcal{A}_\varepsilon^{(n)}(\mathcal{P}_{Y_1,W_1,U_2})\}$ and let $\mathcal{E}_b^{d,2}$ denote that the current sequence $Y_{1,b}^n$ is not jointly typical, i.e., $\mathcal{E}_b^{d,2} = \{(Y_{1,b}^n, W_1^n(m_b)) \notin \mathcal{A}_\varepsilon^{(n)}(\mathcal{P}_{Y_1,W_1})\}$. Note that, if a sequence with fewer terms $(Y_{1,b}^n, W_1^n(m_b))$ is atypical, it implies that a sequence with more terms $(Y_{1,b}^n, W_1^n(m_b), W_2^n(m_b, m_{b+1}))$ is also atypical. Therefore, we have $\mathcal{E}_b^{d,2} \subset \mathcal{E}_b^{d,1}(M_b)$. Then, the decoding error probability given no past decoding error or encoding error $\mathbb{P}(\mathcal{E}_b^d(m_{b-1}) | \cap_{\beta=2}^{b-1} \mathcal{E}_\beta^d(m_{\beta-1}) \cap \bar{\mathcal{E}}^e)$ can be upperbounded by

$$ \mathbb{P}(\mathcal{E}_b^d(m_{b-1}) | \cap_{\beta=2}^{b-1} \bar{\mathcal{E}}_\beta^d(m_{\beta-1}) \cap \bar{\mathcal{E}}^e \cap \bar{\mathcal{E}}_{b-1}^{d,1}(m_{b-1}) \cap \bar{\mathcal{E}}_b^{d,2}) \\ + \mathbb{P}(\mathcal{E}_{b-1}^{d,1}(m_{b-1}) \cup \mathcal{E}_b^{d,2} | \cap_{\beta=2}^{b-1} \bar{\mathcal{E}}_\beta^d(m_{\beta-1}) \cap \bar{\mathcal{E}}^e) \tag{94} $$

using the union bound. The first term of (94) can be upperbounded by

$$ \begin{aligned} & \mathbb{P}(\mathcal{E}_b^d(m_{b-1}) | \cap_{\beta=2}^{b-1} \bar{\mathcal{E}}_\beta^d(m_{\beta-1}) \cap \bar{\mathcal{E}}^e \cap \bar{\mathcal{E}}_{b-1}^{d,1}(m_{b-1}) \cap \bar{\mathcal{E}}_b^{d,2}) \\ & = \mathbb{P}(\exists m' \neq M_b, \text{ s.t. } \{W_1^n(m') \in \mathcal{A}_\varepsilon^{(n)}(W_1|Y_{1,b}^n)\} \cap \\ & \qquad \{U_2^n(M_{b-1}, m') \in \mathcal{A}_\varepsilon^{(n)}(U_2^n|Y_{1,b-1}^n, W_{1,b-1}^n)\}) \\ & \stackrel{(a)}{\le} 2^{-ne}, \end{aligned} $$

where (a) is due to the joint Packing Lemma [78] in order to satisfy both conditions at the same time given that $R \le I(W_1; Y_1) + I(U_2; Y_1|W_1) - 7\varepsilon$. Moreover, as for the second term, we have

$$ \begin{aligned} & \mathbb{P}(\mathcal{E}_{b-1}^{d,1}(m_{b-1}) \cup \mathcal{E}_b^{d,2} | \cap_{\beta=2}^{b-1} \bar{\mathcal{E}}_\beta^d(m_{\beta-1}) \cap \bar{\mathcal{E}}^e) \\ & \stackrel{(b)}{\le} \sum_{i=b-1,b} \mathbb{P}((Y_{1,i}^n, W_{1,i}^n, U_{2,i}^n) \notin \mathcal{A}_\varepsilon^{(n)} | \end{aligned} $$
$$ (X_{0,i}^n, Y_{1,i}^n, W_{1,i}^n, U_{2,i}^n) \in \mathcal{B}_{\varepsilon}^{(n)} $$

$$ \xrightarrow{n \to \infty} 0, $$

where (b) comes from the independence of each Markov block and the fact that $\{(Y_{1,b}^n, W_{1,b}^n) \notin \mathcal{A}_{\varepsilon}^{(n)}\} \subset \{(Y_{1,b}^n, W_{1,b}^n, U_{2,b}^n) \notin \mathcal{A}_{\varepsilon}^{(n)}\}$, and (c) comes from that $(Y_{1,b}^n, W_{1,b}^n, U_{2,b}^n) \subseteq (X_{1,b}^n, Y_{1,b}^n, W_{1,b}^n, U_{2,b}^n)$ where the latter sequence is already jointly typical given no encoding errors.

Therefore, by the above arguments, both the encoding and decoding errors vanish asymptotically. Since the generated sequences are jointly typical with high probability, the cost analysis and the existence of a desired control design $c \in \mathcal{C}_f(Bn)$ follow directly from the no-feedback achievability proof detailed in Appendix A.

# APPENDIX G

## CONVERSE PROOF OF THEOREM V.2

We consider a control design with feedback $c \in \mathcal{C}_f(n)$ of block length $n \in \mathbb{N}$ that achieves a pair of costs $(P, S)$. According to Csiszár sum identity, we have

$$
\begin{align*}
0 &= \sum_{t=1}^{n} I(X_0^{t-1}, Y_1^{t-1}; Y_{1,t} | Y_{1,t+1}^n) \\
  &\quad - \sum_{t=1}^{n} I(Y_{1,t+1}^n; X_{0,t}, Y_{1,t} | X_0^{t-1}, Y_1^{t-1}) \\
  &\le \sum_{t=1}^{n} I(X_0^{t-1}, Y_1^{t-1}, Y_{1,t+1}^n; Y_{1,t}) \\
  &\quad - \sum_{t=1}^{n} I(Y_{1,t+1}^n; X_{0,t}, Y_{1,t} | X_0^{t-1}, Y_1^{t-1}) \\
  &= \sum_{t=1}^{n} \left[ I(X_0^{t-1}, Y_1^{t-1}; Y_{1,t}) + I(Y_{1,t+1}^n; Y_{1,t} | X_0^{t-1}, Y_1^{t-1}) \right] \\
  &\quad - \sum_{t=1}^{n} I(Y_{1,t+1}^n; X_{0,t}, Y_{1,t} | X_0^{t-1}, Y_1^{t-1}) \\
  &= \sum_{t=1}^{n} I(X_0^{t-1}, Y_1^{t-1}; Y_{1,t}) - \sum_{t=1}^{n} I(Y_{1,t+1}^n; X_{0,t} | X_0^{t-1}, Y_1^{t-1}, Y_{1,t}) \\
  &\stackrel{(a)}{=} \sum_{t=1}^{n} I(X_0^{t-1}, Y_1^{t-1}; Y_{1,t}) - \sum_{t=1}^{n} \left[ I(Y_{1,t+1}^n; X_{0,t} | X_0^{t-1}, Y_1^{t-1}, Y_{1,t}) \right. \\
  &\qquad \left. + I(U_{2,t}; X_{0,t} | X_0^{t-1}, Y_1^{t-1}, Y_{1,t}, Y_{1,t+1}^n) \right] \\
  &= \sum_{t=1}^{n} I(X_0^{t-1}, Y_1^{t-1}; Y_{1,t}) \\
  &\quad - \sum_{t=1}^{n} I(U_{2,t}, Y_{1,t+1}^n; X_{0,t} | X_0^{t-1}, Y_1^{t-1}, Y_{1,t}) \\
  &\le \sum_{t=1}^{n} I(X_0^{t-1}, Y_1^{t-1}; Y_{1,t}) - \sum_{t=1}^{n} I(U_{2,t}; X_{0,t} | X_0^{t-1}, Y_1^{t-1}, Y_{1,t}) \\
  &\stackrel{(b)}{=} \sum_{t=1}^{n} I(W_{1,t}; Y_{1,t}) - \sum_{t=1}^{n} I(U_{2,t}; X_{0,t} | W_{1,t}, Y_{1,t}).
\end{align*}
$$

Here, (a) comes from the result of noncausal decoding, which induces the Markov chain $U_2^n \leftarrow Y_1^n \leftarrow X_0^n$, which is equivalent to $I(U_2^n; X_0^n | Y_1^n) = 0$. Therefore, according to the chain rule of mutual information, the term $I(U_{2,t}; X_{0,t} | X_0^{t-1}, Y_1^{t-1}, Y_{1,t}, Y_{1,t+1}^n) = 0$, for each $t \in \{1, ..., n\}$. (b) comes from the identification of $W_{1,t} = (X_0^{t-1}, Y_1^{t-1})$ for each $t \in [1: n]$. Therefore, the introduction of the auxiliary random variable $W_{1,t}$ gives us the following Markov chains:

* $X_{0,t}$ is independent of $W_{1,t} = (X_0^{t-1}, Y_1^{t-1})$, due to the i.i.d. source and the memoryless property of the channel.

* $(X_{1,t}, Y_{1,t}) \leftarrow (U_{1,t}, X_{0,t}) \leftarrow W_{1,t}$, because of the memoryless channel and the symbolwise generation of $X_1$ in (1) and the memoryless channel given in (2).
* $U_{2,t} \leftarrow (X_{0,t}, Y_{1,t}, W_{1,t}) \leftarrow (U_{1,t}, X_{1,t})$ comes from the property of causal encoding function with feedback and the deterministic symbolwise formulation (2). This Markov chain is proved in Lemma G.1 in Appendix G-A.

Given the above three Markov chains that holds for each time stage, we now define an independent time-mixing random variable $T \sim \text{Unif}\{1, ..., n\}$ such that

$$
\begin{align*}
& \sum_{t=1}^{n} I(W_{1,t}; Y_{1,t}) - \sum_{t=1}^{n} I(U_{2,t}; X_{0,t} | W_{1,t}, Y_{1,t}) \\
&= \sum_{t=1}^{n} I(W_{1,t}; Y_{1,t} | T = t) - \sum_{t=1}^{n} I(U_{2,t}; X_{0,t} | W_{1,t}, Y_{1,t}, T = t) \\
&\stackrel{(c)}{=} n \cdot (I(W_{1,T}; Y_{1,T} | T) - I(U_{2,T}; X_{0,T} | W_{1,T}, Y_{1,T}, T)) \\
&\le n \cdot (I(W_{1,T}; T | Y_{1,T}) - I(U_{2,T}; X_{0,T} | W_{1,T}, T, Y_{1,T})) \\
&\stackrel{(d)}{=} n \cdot (I(W_1; Y_1) - I(U_2; X_0 | W_1, Y_1))
\end{align*}
$$

where (c) comes from that we introduce the random variables of $W_{1,T}$, $Y_{1,T}$, $U_{2,T}$, $X_{0,T}$, and (d) comes from the identification of $W_1 \triangleq (W_{1,T}, T)$, $Y_1 \triangleq Y_{1,T}$, $U_2 \triangleq U_{2,T}$, $X_0 \triangleq X_{0,T}$. Therefore, these newly introduced random variables satisfy

* $X_0 \perp W_1$.

* $(X_1, Y_1) \leftarrow (X_0, U_1) \leftarrow W_1$.

* $U_2 \leftarrow (X_0, Y_1, W_1) \leftarrow (U_1, X_1)$.

Hence, the joint distribution over the random variables $(X_0, W_1, U_1, X_1, Y_1, U_2)$ decomposes of the form of (45) and satisfies the information constraint (46). Now we reformulate the n-stage long-run cost

$$ \gamma_p^n(c) = \mathbb{E} \left[ \frac{1}{n} \sum_{t=1}^{n} U_{1,t}^2 \right] = \mathbb{E}[U_1^2] \quad (95) $$

And also,

$$ \gamma_s^n(c) = \mathbb{E} \left[ \frac{1}{n} \sum_{t=1}^{n} (X_{1,t} - U_{2,t})^2 \right] = \mathbb{E}[(X_1 - U_2)^2] \quad (96) $$

Moreover, from (15), we also have

$$
\begin{align}
& \mathbb{E} \left[ |P - c_P(U_1^n)| + |S - c_S(X_1^n, U_2^n)| \right] \le \varepsilon \nonumber \\
& \implies |P - \mathbb{E}[c_P(U_1^n)]| + |S - \mathbb{E}[c_S(X_1^n, U_2^n)]| \le \varepsilon \nonumber \\
& \implies |P - \mathbb{E}[U_1^2]| + |S - \mathbb{E}[(X_1 - U_2)^2]| \le \varepsilon. \tag{97}
\end{align}
$$

This is valid for all $\varepsilon > 0$. This shows that the sequence $(X_0, W_1, U_1, X_1, Y_1, U_2)$ satisfy (45), (46), (47).

### A. Proof of the Markov chain

**Lemma G.1.** For every $t \in \{1, ..., n\}$, it holds that

$$ U_{2,t} \leftarrow (X_{0,t}, Y_{1,t}, W_{1,t}) \leftarrow (U_{1,t}, X_{1,t}). $$

**Proof of Lemma G.1.**

$$
\begin{align}
& \mathbb{P}(U_{2,t} | X_{0,t}, Y_{1,t}, W_{1,t}, U_{1,t}, X_{1,t}) \nonumber \\
&= \mathbb{P}(U_{2,t} | X_0^t, Y_1^t, U_{1,t}, X_{1,t}) \nonumber \\
&\stackrel{(e)}{=} \sum_{t=1}^{n} \mathbb{P}(X_{0,t+1}^n, U_{1,t+1}^n, Y_{1,t+1}^n, U_{2,t} | X_0^t, Y_1^t, U_{1,t}, X_{1,t}) \nonumber \\
&= \sum_{t=1}^{n} \mathbb{P}(X_{0,t+1}^n | X_0^t, Y_1^t, U_{1,t}, X_{1,t}) \tag{f1} \\
&\qquad \times \mathbb{P}(U_{1,t+1}^n, Y_{1,t+1}^n | X_0^n, Y_1^t, U_{1,t}, X_{1,t}) \tag{g1} \\
&\qquad \times \mathbb{P}(U_{2,t} | X_0^n, Y_1^t, U_{1,t}, X_{1,t}, U_{1,t+1}^n, Y_{1,t+1}^n), \tag{h1}
\end{align}
$$

where in equality (e) the sum is over the sequences $(X_{0,t+1}^n, U_{1,t+1}^n, Y_{1,t+1}^n)$. Now, each probability term above

* (f1) $\mathbb{P}(X_{0,t+1}^n | X_0^t, Y_1^t)$, because the source is generated i.i.d., and the encoder is causal.

* (g1) decomposes as follows

$$ \mathbb{P}(U_{1,t+1}^n, Y_{1,t+1}^n | X_0^n, Y_1^t, U_{1,t}, X_{1,t}) $$
$$
\begin{align*}
&= \prod_{i=t+1}^{n} \mathbb{P}(U_{1,i}, Y_{1,i} | U_{1,t+1}^{i-1}, Y_{1,t+1}^{i-1}, X_0^n, Y_1^t, U_{1,t}, X_{1,t}) \\
&= \prod_{i=t+1}^{n} \mathbb{P}(U_{1,i} | U_{1,t+1}^{i-1}, Y_{1,t+1}^{i-1}, X_0^n, Y_1^t, U_{1,t}, X_{1,t}) \tag{i1} \\
&\qquad \times \mathbb{P}(Y_{1,i} | U_{1,t+1}^{i-1}, Y_{1,t+1}^{i-1}, X_0^n, Y_1^t, U_{1,t}, X_{1,t}, U_{1,i}) \tag{j1} \\
&= \prod_{i=t+1}^{n} \mathbb{P}(U_{1,i} | U_{1,t+1}^{i-1}, Y_{1,t+1}^{i-1}, X_0^n, Y_1^t) \tag{i2} \\
&\qquad \times \mathbb{P}(Y_{1,i} | U_{1,t+1}^{i-1}, Y_{1,t+1}^{i-1}, X_0^n, Y_1^t, U_{1,i}) \tag{j2} \\
&= \mathbb{P}(U_{1,t+1}^n, Y_{1,t+1}^n | X_0^n, Y_1^t)
\end{align*}
$$

where (i1) → (i2) comes from the causal encoding function $f_{U_{1,i}|X_0^i, Y_1^{i-1}}^{(f,i)}$, and (j1) → (j2) is because of the memoryless channel $Y_{1,i} = X_{0,i} + U_{1,i} + Z_i$.

* (h1) = $\mathbb{P}(U_{2,t}|X_0^n, Y_1^t, U_{1,t+1}^n, Y_{1,t+1}^n)$ comes from the noncausal decoder $g_{U_{2,t}|Y_1^n}$. Hence $U_{2,t}$ is fully determined by $(Y_1^t, Y_{1,t+1}^n)$.

Therefore, we obtain that

$$
\begin{align*}
& \mathbb{P}(U_{2,t}|X_{0,t}, Y_{1,t}, W_{1,t}, U_{1,t}, X_{1,t}) \\
&= \sum_{X_{0,t+1}^n, U_{1,t+1}^n, Y_{1,t+1}^n} \mathbb{P}(X_{0,t+1}^n | X_0^t, Y_1^t) \\
&\quad \times \mathbb{P}(U_{1,t+1}^n, Y_{1,t+1}^n | X_0^n, Y_1^t) \\
&\quad \times \mathbb{P}(U_{2,t}|X_0^n, Y_1^t, U_{1,t+1}^n, Y_{1,t+1}^n) \\
&= \mathbb{P}(U_{2,t}|X_{0,t}, Y_{1,t}, W_{1,t}).
\end{align*}
$$

<figure><img src="image_15.png" /></figure>

## APPENDIX H

### PROOF OF THEOREM V.6

The power consumption induced by the scheme (51) is

$$
\begin{align*}
P &= \mathbb{E}[U_1^2] = \mathbb{E}[(W_1 + a \cdot \text{sign}(X_0) + bX_0)^2] \\
&= \mathbb{E}[(W_1)^2] + \mathbb{E}[(a \cdot \text{sign}(X_0) + bX_0)^2] \\
&= V_1 + a^2 + b^2Q + 2ab \cdot \mathbb{E}[|X_0|] \\
&= V_1 + a^2 + b^2Q + 2ab\sqrt{\frac{2Q}{\pi}}.
\end{align*}
$$

The information constraint (46) becomes

$$
\begin{align}
& I(W_1; Y_1) - I(U_2; X_0|W_1, Y_1) \nonumber \\
&= h(Y_1) - h(Y_1|W_1) - h(U_2|W_1, Y_1) + h(U_2|X_0, W_1, Y_1) \nonumber \\
&= h(Y_1) - h(Y_1|W_1) - h(X_1|W_1, Y_1) + \underbrace{h(X_1|X_0, W_1, Y_1)}_{=0} \nonumber \\
&= h(Y_1) - h(X_1, Y_1|W_1) \nonumber \\
&= h(Y_1) - h(X_1|W_1) - h(Y_1|X_1, W_1) \nonumber \\
&= h(Y_1) - h(X_1|W_1) - h(Z_1|X_1, W_1) \nonumber \\
&= h(Y_1) - h(X_1|W_1) - h(Z_1) \nonumber \\
&= h(Y_1) - h(X_1|W_1) - \frac{1}{2} \log 2\pi e N, \tag{98}
\end{align}
$$

where in the second equality, we replace $U_2$ by $X_1$ given the fact that (52). Now, we examine the (conditional) PDFs of $Y_1$ and $X_1|W_1$ respectively.

Given a fixed value of $X_0 = x$, the conditional distribution $Y_1|X_0 = x \sim \mathcal{N}(a \cdot \text{sign}(x) + \delta x, V_1 + N)$, where we denote $\delta \triangleq b + 1$ for simplicity. Hence,

$$f_{Y_1}(y) = \int_{-\infty}^{\infty} f_{Y_1|X_0}(y|x) f_{X_0}(x) dx$$
$$ 
\begin{aligned}
&= \int_{-\infty}^{0} f_{Y_1|X_0}(y|x)f_{X_0}(x)dx \\
&\quad + \int_{0}^{\infty} f_{Y_1|X_0}(y|x)f_{X_0}(x)dx \\
&= \frac{1}{2}\mathcal{S}\mathcal{N}(a, \sigma, \alpha) + \frac{1}{2}\mathcal{S}\mathcal{N}(-a, \sigma, -\alpha)
\end{aligned} 
\quad (99) $$

is a mixture of two skew-normal distributions, where $\sigma^2 = V_1 + \delta^2 Q + N$, $\alpha = \frac{\sqrt{Q\delta}}{\sqrt{V_1+N}}$. On the other hand, given that $X_0 \perp W_1$, we have

$$ 
\begin{aligned}
&h(X_1|W_1) \\
&= h(W_1 + a \cdot \text{sign}(X_0) + \delta X_0|W_1) \\
&= h(a \cdot \text{sign}(X_0) + \delta X_0|W_1) \\
&= h(a \cdot \text{sign}(X_0) + \delta X_0).
\end{aligned} 
$$

Therefore, by denoting $X \triangleq a \cdot \text{sign}(X_0) + \delta X_0$, we obtain that

$$ f_X(x) = \frac{1}{\sqrt{\delta^2 Q}} \phi\left(\frac{x-a}{\sqrt{\delta^2 Q}}\right) \mathbf{1}_{\text{sign}(\delta) \cdot (x-a) \ge 0} \\ 
+ \frac{1}{\sqrt{\delta^2 Q}} \phi\left(\frac{x+a}{\sqrt{\delta^2 Q}}\right) \mathbf{1}_{\text{sign}(\delta) \cdot (x+a) \le 0}, $$

which concludes our proof for Theorem V.6. $\square$

# APPENDIX I

## PROOF OF COROLLARY V.7

We look at the special case of $V_1 = 0, a = 0, b = 0$. In this way, the ZEC-f scheme (51) boils down to

$$ 
\begin{aligned}
&U_1 = 0, \\
&X_1 = X_0, \\
&Y_1 = X_0 + Z_1.
\end{aligned} 
$$

The power cost now becomes

$$ P = \mathbb{E}[U_1^2] = 0. $$

Moreover, the information constraint that we derived in (56) now becomes

$$ 
\begin{aligned}
&h(X_0 + Z_1) - h(X_0) - \frac{1}{2}\log 2\pi e N \\
&= \frac{1}{2}\log 2\pi e(Q + N) - \frac{1}{2}\log 2\pi eQ - \frac{1}{2}\log 2\pi eN \\
&= \frac{1}{2}\log \frac{Q + N}{2\pi eQN}.
\end{aligned} 
$$

In conclusion, if the information constraint (46) is nonnegative if and only if (59) is satisfied. $\square$

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