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# Zero Estimation Cost Strategy for Witsenhausen Counterexample with Causal Encoder

Mengyuan Zhao, Tobias Oechtering, Maël Le Treust

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## ► To cite this version:

Mengyuan Zhao, Tobias Oechtering, Maël Le Treust. Zero Estimation Cost Strategy for Witsenhausen Counterexample with Causal Encoder. ISIT 2025 - IEEE International Symposium on Information Theory, Jun 2025, Ann Arbor, MI, United States. pp.1-6. hal-05314537

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https://hal.science/hal-05314537v1

Submitted on 14 Oct 2025

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Distributed under a Creative Commons Attribution 4.0 International License
# Zero Estimation Cost Strategy for Witsenhausen Counterexample with Causal Encoder

Mengyuan Zhao and Tobias J. Oechtering

Division of Information Science and Engineering

KTH Royal Institute of Technology

10044 Stockholm, Sweden

Maël Le Treust

Univ. Rennes, CNRS, Inria, IRISA UMR 6074

35000 Rennes, France

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

## I. INTRODUCTION

In 1968, Witsenhausen proposed his renowned counterexample, highlighting the suboptimality of affine strategies in the Linear Quadratic Gaussian (LQG) settings with non-classical information pattern [1]. This counterexample has since become a prominent toy example in the study of distributed decision-making [2]–[5] and information-theoretic control [6]–[13].

The vector-valued extension of Witsenhausen counterexample [14] facilitates the application of many information-theoretic approaches [15]–[18] to analyze this open problem. Among these, the coordination coding method [19]–[21] has proven powerful in building cooperative behavior among different decision-makers, yielding novel bounds and insights into distributed decision-making problems.

Recent advances in this direction for Witsenhausen counterexample are the single-letter characterizations for the optimal power-estimation cost trade-off region in various causal decision-making frameworks, including causal decoding [22], causal coding with feedback [23], and, in particular, causal encoding [24]. The auxiliary random variables (aux. RVs) involved in the single-letter expressions, not only capture the asymptotic behavior of the costs, but also explicitly carry the

<figure><img src="image_4.png" /><figcaption>Fig. 1. Causal-encoding noncausal-decoding vector-valued Witsenhausen counterexample.</figcaption></figure>

dual role of control in Witsenhausen counterexample: joint state control and information communication.

Building upon this idea, the design of aux. RVs that can be efficiently communicated becomes a key question. In [25], we explored control designs within the class of jointly Gaussian aux. RVs and determined the optimal estimation cost as a function of power cost. This optimal Gaussian scheme is shown to operate as a time-sharing mechanism between two affine strategies. However, it is outperformed by Witsenhausen's two-point strategy, which communicates the controlled system state more efficiently by designing it to be a binary sign symbol.

Inspired by the two-point strategy, we propose a zero estimation cost (ZEC) approach for the causal-encoding setup based on the single-letter coordination coding result, where one aux. RV is designed to be Gaussian, and the other one is discrete representing the sign of the source state. The ZEC scheme requests the two aux. RVs to describe the controlled system state $X_1$ deterministically. Due to block coding, these two aux. RVs are subsequently revealed to the noncausal decoder, which facilitates the decoder to perfectly reconstruct the system state, thereby achieving zero estimation cost. Simulation results show that the ZEC strategy significantly reduces the power required for zero-cost system state reconstruction at the decoder, offering a substantial improvement over existing single-shot methods. Next, we introduce a more general non-zero estimation cost (Non-ZEC) scheme by incorporating a test channel on top of the discrete aux. RV. This extended scheme reduces the necessary power cost by allowing a trade-off with estimation accuracy, which is shown numerically, to be a time-sharing operation between the original two-point scheme and our proposed ZEC strategy. By exploiting block-coding gain, our proposed approach strictly outperforms the two-point strategy and greatly enhances the power-estimation

<footnote>This work is supported by Swedish Research Council (VR) under grant 2020-03884. The work of Maël Le Treust is supported in part by PEPR NF FOUNDS ANR-22-PEFT-0010.</footnote>
trade-off for Witsenhausen counterexample.

This paper is organized as follows: Section II formulates the problem and recapitulates some foundational results. Section III introduces the ZEC scheme and its performance analysis. Section IV extends this to the Non-ZEC scheme with numerical simulations. Lastly, a conclusion follows in Section V.

## II. SYSTEM MODEL

Let us consider the vector-valued Witsenhausen counterexample setup with causal source states and channel noises that 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 $Q, N \in \mathbb{R}^+$, where $\mathbb{I}$ is the identity matrix, see Figure 1. We denote by $X_1$ the memoryless interim system state and $Y_1$ the output of the memoryless additive channel, generated by

$$X_1 = X_0 + U_1 \quad \text{with } X_0 \sim \mathcal{N}(0, Q), \quad (1)$$

$$Y_1 = X_1 + Z_1 = X_0 + U_1 + Z_1 \quad \text{with } Z_1 \sim \mathcal{N}(0, N). \quad (2)$$

We denote by $\mathcal{P}_{X_0} = \mathcal{N}(0, Q)$ the generative Gaussian probability distribution of the source, and by $\mathcal{P}_{X_1, Y_1|X_0, U_1}$ the channel probability distribution according to (1) and (2).

We define the control design for this setup, its induced cost functions, and the achievable cost pairs as follows:

**Definition 1.** For $n \in \mathbb{N}$, a “control design” with causal encoder and noncausal decoder is a tuple of stochastic functions $c = (\{f_{U_1,t|X_0^t}\}_{t=1}^n, g_{U_2^n|Y_1^n})$ defined by

$$f_{U_1,t|X_0^t}: X_0^t \to \mathcal{U}_1, \quad g_{U_2^n|Y_1^n}: \mathcal{Y}_1^n \to \mathcal{U}_2^n, \quad (3)$$

which induces a distribution over sequences of symbols:

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

**Definition 2.** We define the two long-run cost functions $c_P(u_1^n) = \frac{1}{n} \sum_{t=1}^n (u_1,t)^2$ and $c_S(x_1^n, x_2^n) = \frac{1}{n} \sum_{t=1}^n (x_1,t - u_2,t)^2$. 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 $c$ as in (3) such that

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

The following theorem is the single-letter characterization for the optimal cost region formed by all achievable cost pairs.

**Theorem 3** ([24, Theorem II.3]). The pair of Witsenhausen costs $(P, S)$ is achievable if and only if there exists a joint distribution over the random variables $(X_0, W_1, W_2, U_1, X_1, Y_1, U_2)$ that decomposes according to

$$\mathcal{P}_{X_0} \mathcal{P}_{W_1} \mathcal{P}_{W_2} |_{X_0, W_1} \mathcal{P}_{U_1} |_{X_0, W_1} \mathcal{P}_{X_1, Y_1} |_{X_0, U_1} \mathcal{P}_{U_2} |_{W_1, W_2, Y_1}, \quad (4)$$

such that

$$\begin{align} I(W_1, W_2; Y_1) - I(W_2; X_0|W_1) \ge 0, \tag{5} \\ P = \mathbb{E}[U_1^2], \quad S = \mathbb{E}[(X_1 - U_2)^2], \nonumber \end{align}$$

where $\mathcal{P}_{X_0}$ and $\mathcal{P}_{X_1, Y_1|X_0, U_1}$ are two given Gaussian distributions, and $W_1, W_2$ are two aux. RVs.

The two aux. 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 [26]. $W_2$ 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.

**Remark 1.** The following Markov chains follow from the joint probability distribution (4):

$$\left\{ \begin{array}{l} X_0 \text{ is independent of } W_1, \\ U_1 \leftarrow (X_0, W_1) \leftarrow W_2, \\ (X_1, Y_1) \leftarrow (X_0, U_1) \leftarrow (W_1, W_2), \\ U_2 \leftarrow (W_1, W_2, Y_1) \leftarrow (X_0, U_1, X_1). \end{array} \right. \quad (6)$$

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 non-causal decoding and symbol-wise reconstruction.

In order to investigate the optimal achievable cost pairs, we focus on the lower boundary of the two-dimensional power-estimation cost region characterized in Theorem 3: For a given power cost $P \ge 0$, we aim to determine the minimum estimation cost $S$ achievable at the decoder. Since the minimum mean square error (MMSE) estimation provides the optimal decoding policy and is given by the conditional expectation, we have the following lemma:

**Lemma 4.** Given a power cost parameter $P \ge 0$, the optimal estimation cost $S^*(P)$ is given by

$$\begin{align} S^*(P) &= \inf_{P \in \mathcal{P}(P)} \mathbb{E} \left[ \left( X_1 - \mathbb{E}[X_1|W_1, W_2, Y_1] \right)^2 \right], \tag{7} \\ \mathbb{P}(P) &= \left\{ \begin{array}{@{}l@{}} (\mathcal{P}_{W_1}, \mathcal{P}_{W_2|X_0, W_1}, \mathcal{P}_{U_1|X_0, W_1}) \quad \text{s.t. } P = \mathbb{E}[U_1^2], \\ I(W_1, W_2; Y_1) - I(W_2; X_0|W_1) \ge 0 \end{array} \right\}. \nonumber \end{align}$$

Next, we revisit Witsenhausen’s two-point strategy. Our findings in [25] show it outperforms both the best affine [25, Lemma III.3] and optimal joint Gaussian strategies [25, Theorem III.4] for some values of $Q, N$.

**Theorem 5** ([1, Theorem 2]). For parameter $a \ge 0$, Witsenhausen’s two-point strategy is given by

$$U_1 = a \cdot \mathrm{sign}(X_0) - X_0. \quad (8)$$

The power and estimation costs are given by

$$P_2(a) = Q + a \left( a - 2\sqrt{\frac{2Q}{\pi}} \right), \quad (9)$$

$$S_2(a) = a^2 \sqrt{\frac{2\pi}{N}} \phi \left( \frac{a}{\sqrt{N}} \right) \int \frac{\phi \left( \frac{y_1}{\sqrt{N}} \right)}{\cosh \left( \frac{ay_1}{N} \right)} dy_1, \quad (10)$$

where $\phi(x) = \frac{1}{\sqrt{2\pi}} e^{-\frac{x^2}{2}}$ and the optimal receiver’s strategy is given by $\mathbb{E}[X_1|Y_1 = y_1] = a \tanh(\frac{ay_1}{N})$.
The core idea of the two-point strategy is to cancel the continuous source state $X_0$, such that the encoder designates the system state $X_1 = U_1 + X_0 = a \cdot \text{sign}(X_0)$ to be binary, making it easier for the decoder to estimate. Inspired by this approach, in the next section, we introduce the ZEC scheme based on our coordination coding result in Theorem 3. The ZEC method leverages the block-coding gain to reconstruct $X_1$ and achieves a zero estimation cost even at a significantly low power budget.

### III. THE ZERO ESTIMATION COST SCHEME

We propose the following design of aux. RVs involved in Theorem 3, where $W_1$ is continuous Gaussian and $W_2$ is discrete binary<fnref n="1" />. For given parameters $V_1 \ge 0, a \ge 0$,

$$ 
\begin{aligned}
& X_0 \sim \mathcal{N}(0, Q), \\
& W_1 \sim \mathcal{N}(0, V_1), \\
& W_2 = a \cdot S, \text{ where } S = \text{sign}(X_0), \\
& U_1 = W_1 + a \cdot S - X_0, \\
& X_1 = U_1 + X_0 = W_1 + a \cdot S = W_1 + W_2, \\
& Y_1 = X_1 + Z_1 = W_1 + W_2 + Z_1, \quad Z_1 \sim \mathcal{N}(0, N).
\end{aligned} 
\quad (11) $$

In this joint control-communication scheme, the control action $U_1$ subtracts the Gaussian source state $X_0$ as in (8) and embeds the two aux. RVs ($W_1, W_2$). This makes the system state $X_1$ offset the uncertainty of the source state and becomes a deterministic function of ($W_1, W_2$). These two aux. RVs, due to block coding, can be communicated to the receiver efficiently. Hence, the noncausal decoder has access to sufficient information and outputs the MMSE estimation

$$ \mathbb{E}[X_1|W_1, W_2, Y_1] = \mathbb{E}[W_1 + W_2|W_1, W_2, Y_1] = W_1 + W_2 $$

which perfectly reconstructs $X_1$, resulting in a zero estimation cost. Recall that, according to Definition 2, the estimation cost $S = 0$ is achievable means that as the number of transmission $n \to \infty$, the averaged long-fun cost function $c_S(X_1^n, U_2^n)$ could be made as close to zero as possible.

Moreover, the power cost needed for this system is

$$ 
\begin{aligned}
P &= \mathbb{E}[U_1^2] = \mathbb{E}[(W_1 + a \cdot S - X_0)^2] \\
&= V_1 + \left(Q + a^2 - 2a\sqrt{\frac{2Q}{\pi}}\right).
\end{aligned} 
$$

Therefore, for a fixed power cost $P \ge 0$, $V_1$ is uniquely determined by the parameter $a$ which needs to satisfy the following condition

$$ V_1 = P - \left( Q + a^2 - 2a\sqrt{\frac{2Q}{\pi}} \right) \ge 0. \quad (12) $$

The information constraint (5) in bits becomes

$$ 
\begin{aligned}
I(W_1, W_2; Y_1) - I(W_2; X_0|W_1) \\
&= h(Y_1) - h(Y_1|W_1, W_2) - h(W_2|W_1) + h(W_2|X_0, W_1) \\
&= h(Y_1) - \frac{1}{2}\log_2(2\pi e N) - 1 \ge 0,
\end{aligned} 
\quad (13) $$

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

$$ f_{Y_1}(y) = \frac{1}{2\sqrt{(V_1 + N)}} \left[ \phi\left(\frac{y - a}{\sqrt{V_1 + N}}\right) + \phi\left(\frac{y + a}{\sqrt{V_1 + N}}\right) \right], \quad (14) $$

where $\phi(x) = \frac{1}{\sqrt{2\pi}} \exp\left(\frac{x^2}{2}\right)$ is the standard Gaussian p.d.f. Since there is no closed form for the entropy of Gaussian mixture distributions, methods discussed in [28, 29] can be employed for numerical simulation.

We denote the set of parameters $a \ge 0$ that satisfy the power cost constraint (12) and the information constraint (13) by

$$ \mathcal{A}^0(P) = \{a \ge 0 : h(Y_1) - \frac{1}{2}\log_2(2\pi e N) \ge 1\}, \quad (15) $$

and $V_1 = P - \left( Q + a^2 - 2a\sqrt{\frac{2Q}{\pi}} \right) \ge 0$.

Based on the above derivation, the optimal cost function for the ZEC system is summarized in the following theorem:

**Theorem 6.** Given the power cost $P \ge 0$, the MMSE estimation cost for the ZEC coding scheme (11) is given by

$$ S_{\text{ZEC}}(P) = 0, \text{ for } P \ge P^*, \quad (16) $$

where the value

$$ P^* = \min\{P : \mathcal{A}^0(P) \neq \emptyset\}. \quad (17) $$

To satisfy the second nonnegative constraint of the admissible condition in (15), the minimum required power cost $P^*$ for the ZEC scheme must satisfy

$$ P^* \ge \min_a \left( Q + a^2 - 2a\sqrt{\frac{2Q}{\pi}} \right) = Q \left( 1 - \frac{2}{\pi} \right) = P_2^{\min}, $$

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

To illustrate the performance of the ZEC scheme, we compare its cost function $S_{\text{ZEC}}(P)$, with that of the original two-point strategy $S_2(P)$ given in (9)-(10), the best affine strategy $S_\ell(P)$ [25, Lemma III.3], and the optimal joint Gaussian strategy $S_G(P)$ [25, Theorem III.4] at $Q = 1, N = 0.15$ in Figure 2. As we can see, using only a slightly higher power cost than $P_2^{\min} = 0.363$, the ZEC scheme can already achieve a zero-estimation-cost system state reconstruction when $P \ge P^* = 0.383$. In contrast, all the other strategies, $S_2(P)$, $S_\ell(P)$, and $S_G(P)$, only achieve zero-estimation-cost reconstruction at a significantly bigger power cost of $P = Q = 1$.

Figure 3 illustrates how the minimum required power $P^*$ given in (17) varies with different values of $N$ at $Q = 1$. Notably, for small noise levels (e.g., $N \le 0.07$), we observe that $P^* = P_2^{\min} = 0.363$, indicating that zero-cost estimation can be achieved without any additional power expenditure compared to the original two-point strategy. Furthermore,

<footnote n="1">¹This approach is similar to the hybrid coding scheme [27], which also combines digital and analog coding.</footnote>
<figure><img src="image_5.png" /><figcaption>Fig. 2. Comparison of the four cost functions $S_{\text{ZEC}}(P)$, $S_2(P)$, $S_{\ell}(P)$, and $S_G(P)$ at $Q = 1, N = 0.15$. Our proposed scheme strictly outperforms the other strategies and achieves a zero-estimation-cost state reconstruction when $P \ge P^* = 0.383$.</figcaption></figure>

when $N \ge 0.3$, that is, scenarios when the two-point strategy no longer surpasses the optimal affine approach, our proposed scheme continues to achieve zero-cost estimation requiring only a power budget of $P \ge P^* = 0.501$. Note that this is only about half of the power budget needed by other strategies $S_{\ell}(P)$, $S_G(P)$, $S_2(P)$ for achieving zero-cost system state estimation. However, when $N \ge 0.65$, $P^* \ge Q = 1$, meaning that the ZEC scheme can no longer provide a zero-estimation-cost block-coding gain in high-noise regimes.

<figure><img src="image_6.png" /><figcaption>Fig. 3. Variation of $P^*$ as a function of the noise level $N$ when $Q = 1$. $P^* = P_{\min}^2 = 0.363$ when $N$ is small, and increases to $P^* = Q = 1$ for $N \ge 0.65$.</figcaption></figure>

Next, we extend the ZEC strategy to a more general Non-ZEC scheme. This extension introduces a trade-off between estimation accuracy and power cost, enabling a power cost reduction.

## IV. THE NON-ZERO ESTIMATION COST SCHEME

In this section, we apply a test channel between the aux. RV $W_2$ and the source state $X_0$ with a cross-over probability $\gamma$. Given nonnegative parameters $V_1, a, \gamma$, we consider

$$ 
\begin{array}{l}
\begin{array}{l}
X_0 \sim \mathcal{N}(0, Q), \\
W_1 \sim \mathcal{N}(0, V_1), \\
W_2 = \begin{cases} a \cdot S & \text{with probability } 1-\gamma, \\ -a \cdot S & \text{with probability } \gamma, \end{cases} \\
\text{where } S = \text{sign}(X_0), \\
U_1 = W_1 + a \cdot S - X_0, \\
X_1 = U_1 + X_0 = W_1 + a \cdot S, \\
Y_1 = X_1 + Z_1 = W_1 + a \cdot S + Z_1, \quad Z_1 \sim \mathcal{N}(0, N).
\end{array}
\end{array} 
\tag{18} $$

The design of $W_2$ as a binary RV dependent on $X_0$ rather than a deterministic variable introduces additional randomness, which affects the estimation precision of $X_1$ from $(W_1, W_2, Y_1)$. Note that, if $\gamma = 0$ or $1$, the above system degrades to a deterministic relation, where $X_1 = W_1 + W_2$ or $X_1 = W_1 - W_2$, respectively. In either case, the estimation cost is zero, hence we recover the ZEC scheme. Furthermore, it can be easily shown that the MMSE remains the same for $\gamma$ and $1-\gamma$. Therefore, we can restrict our analysis to $\gamma \in [0, 0.5]$, without loss of generality.

To simplify the presentation of the estimation cost result presented in Theorem 7, we define the following quantities:

$$ 
\begin{align*}
G_0(w_1) &= \frac{1}{\sqrt{V_1}} \phi \left( \frac{w_1}{\sqrt{V_1}} \right), \\
G_1(w_1, y_1) &= \frac{1}{\sqrt{N}} \phi \left( \frac{y_1 - (w_1 + a)}{\sqrt{N}} \right), \\
G_2(w_1, y_1) &= \frac{1}{\sqrt{N}} \phi \left( \frac{y_1 - (w_1 - a)}{\sqrt{N}} \right), \\
I(w_1, y_1) &= G_0 \left[ \frac{[(1-\gamma)G_1 - \gamma G_2]^2}{(1-\gamma)G_1 + \gamma G_2} + \frac{[\gamma G_1 - (1-\gamma)G_2]^2}{\gamma G_1 + (1-\gamma)G_2} \right].
\end{align*} 
$$

Similarly, we define the set of admissible parameters

$$ 
\begin{array}{l}
\mathcal{A}^{\gamma}(P) = \{a \ge 0 : h(Y_1) + H_2(\gamma) - h(Y_1|W_1, W_2) \ge 1, \\
\text{and } V_1 = P - \left( Q + a^2 - 2a\sqrt{\frac{2Q}{\pi}} \right) \ge 0 \}, \quad (19)
\end{array} 
$$

where $h(Y_1)$ is the entropy calculated from the distribution defined in (14), $H_2(\gamma) = -\gamma \log_2(\gamma) - (1-\gamma) \log_2(1-\gamma)$ is a binary entropy, and $h(Y_1|W_1, W_2)$ is the following Gaussian mixture differential entropy

$$ h \left( (1 - \gamma) \frac{1}{\sqrt{N}} \phi \left( \frac{y - a}{\sqrt{N}} \right) + \gamma \frac{1}{\sqrt{N}} \phi \left( \frac{y + a}{\sqrt{N}} \right) \right). $$

By setting $\gamma = 0$ (and also $\gamma = 1$), (19) recovers the admissible parameter set $\mathcal{A}^0(P)$ given in (15) of the ZEC scheme. Therefore,

$$ \mathcal{A}^0(P) \subseteq \mathcal{A}^\gamma(P), \tag{20} $$

indicating that the Non-ZEC scheme permits lower power costs compared to the ZEC scheme.
The full proof using the above quantities is provided in the extended version of this paper [30]. Additionally, the derivation of the ZEC cost function sketched in equations (12) to (15) can also be seen as a special case of the derivation of the Non-ZEC scheme by setting $\gamma = 0$.

**Theorem 7.** Given $P \ge P_2^{\min}$, the optimal estimation cost induced by the Non-ZEC scheme (18) is given by

$$S_{\text{Non-ZEC}}(P) = \min_{a \in \mathcal{A}^{\gamma}(P), \gamma \in [0, 0.5]} \{F(a, \gamma, P)\}, \quad (21)$$

where,

$$F(a, \gamma, P) = a^2 - \frac{a^2}{2} \iint I(w_1, y_1) dw_1 dy_1. \quad (22)$$

In this theorem, $F(a, \gamma, P)$ represents the achievable estimation cost for a given pair $(a, \gamma)$ at the power budget $P \ge P_2^{\min}$, and we can minimize $F$ over all admissible parameters $a \in \mathcal{A}^{\gamma}(P)$, $\gamma \in [0, 0.5]$ to get the optimized result (21). Moreover, by plugging in $\gamma = 0$, $S_{\text{Non-ZEC}}(P)$ boils down to $S_{\text{ZEC}}(P) = 0$ when $P \ge P^*$ given in (17).

<figure><img src="image_7.png" /><figcaption>Fig. 4. Evolution of $F(a, \gamma, P)$ with different values of $\gamma = 0, 0.05, 0.1, 0.5$. When $\gamma = 0$, $F(a, 0, P) = S_{\text{ZEC}}(P)$ in (16), and when $\gamma = 0.5$, the lower boundary of $F(a, 0.5, P)$ recovers that of the two-point strategy (yellow dashed curve).</figcaption></figure>

To analyze the performance of the Non-ZEC scheme, we examine how different values of $\gamma \in [0, 0.5]$ affect its achievable cost region. Figure 4 illustrates the function $F(a, \gamma, P)$ in (22) by plotting all the admissible points $a \in \mathcal{A}^{\gamma}(P)$ at each $P \ge P_2^{\min}$, resulting in a 2-dimensional region, for fixed parameters of $\gamma \in \{0, 0.01, 0.1, 0.5\}$ compared to the original two-point strategy at $Q = 1, N = 0.15$.

As shown in Figure 4, $F(a, 0, P)$ aligns with the ZEC cost function $S_{\text{ZEC}}(P)$ in (16), as expected. Gradually increasing $\gamma$, which introduces more randomness in $W_2$, allows for a reduction in power cost at the expense of sacrificing estimation accuracy. Ultimately, when $\gamma = 0.5$, which means $W_2$ provides no information about the source $X_0$ at all, the required

<figure><img src="image_8.png" /><figcaption>Fig. 5. Comparison of the four closed-form cost functions $S_{\text{Non-ZEC}}(P)$, $S_2(P)$, $S_{\ell}(P)$, $S_G(P)$, and the induced time-sharing cost $S_{t-s}(P)$ at $Q = 1, N = 0.15$.</figcaption></figure>

power budget reaches its minimum, and the lower boundary of the achievable cost $F(a, 0.5, P)$ aligns with that of the original two-point strategy.

In Figure 5, We plot the optimized achievable estimation cost function $S_{\text{Non-ZEC}}(P)$ in (21) and compare it with the other three cost functions $S_2(P)$, $S_{\ell}(P)$ and $S_G(P)$. The numerical results indicate that the Non-ZEC scheme functions effectively as a time-sharing mechanism between the ZEC scheme and the original two-point strategy. Moreover, $S_{t-s}(P)$ represents the estimation cost resulting from the time-sharing operation between the optimal Gaussian strategy $S_G(P)$ and the ZEC strategy $S_{\text{ZEC}}(P)$, which in this scenario, is shown to be superior to the Non-ZEC scheme.

## V. CONCLUSION

Our proposed joint control-communication schemes improve the overall power-estimation performance of Witsenhausen counterexample. In particular, the ZEC scheme significantly reduces the minimum power budget required for zero-estimation-cost system state reconstruction at the decoder from $P = Q$ to $P^*$. However, it remains unknown whether $P^*$ is the universal minimum power to achieve zero-cost estimation in the causal-encoding noncausal-decoding scenario. Furthermore, the current numerical results show that the Non-ZEC scheme is outperformed by the time-sharing mechanism between the optimal joint Gaussian strategy and the ZEC scheme. Whether there exist scenarios such that the Non-ZEC scheme can outperform this time-sharing mechanism remains unknown.

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