Cost of Actuation Error (CAE)

• Cost of Actuation Error (CAE) is a semantic performance metric that assigns tailored penalties to state reconstruction mismatches in remote monitoring and control systems.
• CAE differentiates the impact of errors using an ordered pair cost matrix, enabling asymmetric, application-specific penalties and guiding policy design under wireless unreliability.
• CAE formulations support policy optimization and extensions to multi-source scheduling, age-aware metrics, and goal-oriented communication frameworks.

Searching arXiv for papers on Cost of Actuation Error (CAE) and closely related goal-oriented/semantic remote estimation. Cost of Actuation Error (CAE) is a semantic, actuation-oriented performance metric for remote monitoring, estimation, and control systems in which a receiver acts on a reconstructed state rather than merely storing or displaying it. In the CAE formulation introduced for remote reconstruction of Markov sources over wireless channels, an estimation mismatch is not treated as a uniform binary event; instead, each mismatch pair between the true source state and the reconstructed state is assigned an application-specific penalty, so that the long-run metric reflects the expected consequence of acting on the wrong estimate rather than only the frequency of estimation error (Salimnejad et al., 2023). Subsequent work places CAE within a broader goal-oriented communication literature, extends it to multi-source scheduling, and relates it to age-aware semantic metrics and constrained optimization under wireless unreliability (Fountoulakis et al., 2023, Luo et al., 2023, Luo et al., 2024, Pomaje et al., 11 Aug 2025, Elessawy et al., 15 May 2026).

1. Formal definition and conceptual scope

CAE is defined through a state-pair cost matrix. In the foundational Markov-source remote actuation setting, if the true state at time slot $t$ is $X_t=i$ and the reconstructed state is $\hat X_t=j\neq i$, then the actuation mismatch incurs cost $C_{i,j}$, assumed fixed over time. The average cost of actuation error for an $N$-state DTMC source is

$$\bar{C}_{A} = \sum_{i=0}^{N-1}\sum_{\substack{j=0 \ j\neq i}}^{N-1} C_{i,j}\pi_{i,j},$$

where $\pi_{i,j}$ is the stationary probability of the joint erroneous state $(X_t,\hat X_t)=(i,j)$ (Salimnejad et al., 2023). This makes CAE a stationary expected per-slot penalty over mismatch states rather than a raw instantaneous distortion variable.

The defining feature of CAE is that it depends on the ordered pair $(X_t,\hat X_t)$, not merely on whether $X_t\neq \hat X_t$. The metric therefore captures “the significance or the non-commutative effects of an error at the receiver side since different errors may have different impact on the system,” as stated in the Markov-source reconstruction work (Salimnejad et al., 2023). This permits asymmetric penalties $X_t=i$0, state-importance asymmetry, and application-dependent consequence modeling.

A closely related finite-state formulation appears in goal-oriented wireless autonomous systems, where the source state is $X_t=i$1, the receiver estimate is $X_t=i$2, and the long-term average CAE is

$X_t=i$3

with instantaneous pairwise cost $X_t=i$4 and an average transmission-cost constraint (Fountoulakis et al., 2023). In that setting, CAE is the explicit optimization objective rather than an auxiliary evaluation metric.

A multi-source generalization defines the per-slot CAE as

$X_t=i$5

where $X_t=i$6 encodes source significance and $X_t=i$7 is a source-specific actuation-error cost map, generally non-commutative (Luo et al., 2023). This formulation introduces a two-level significance structure: source significance via $X_t=i$8, and state-pair significance via $X_t=i$9.

In a two-state goal-oriented AoI framework, CAE is written as an instantaneous mismatch cost

$\hat X_t=j\neq i$0

with $\hat X_t=j\neq i$1 for $\hat X_t=j\neq i$2 and $\hat X_t=j\neq i$3 otherwise, and long-term average

$\hat X_t=j\neq i$4

subject to a semantic constraint $\hat X_t=j\neq i$5 (Pomaje et al., 11 Aug 2025). This emphasizes that CAE can also appear as a feasibility constraint rather than an optimization objective.

2. System models in which CAE is used

CAE arose in remote monitoring and actuation systems where a sampler observes a Markov source, transmits over an unreliable wireless channel, and the receiver acts on the reconstructed state (Salimnejad et al., 2023). In that model, sampling action $\hat X_t=j\neq i$6, transmission action $\hat X_t=j\neq i$7, channel success indicator $\hat X_t=j\neq i$8, true state $\hat X_t=j\neq i$9, and reconstructed state $C_{i,j}$0 define the closed loop. The wireless success probability is

$C_{i,j}$1

with instantaneous and error-free ACK/NACK feedback (Salimnejad et al., 2023).

Two source classes are considered there. The first is a symmetric $C_{i,j}$2-state DTMC with

$C_{i,j}$3

and $C_{i,j}$4. The second is an $C_{i,j}$5-state Birth-Death Markov Process (BDMP) with nearest-neighbor transitions (Salimnejad et al., 2023). Since CAE depends on stationary mismatch probabilities, the source transition structure directly affects the metric.

Later work extends CAE to multiple sources in resource-constrained systems. There, $C_{i,j}$6 sources evolve as finite-state DTMCs $C_{i,j}$7, an agent selects at most one source per slot using $C_{i,j}$8, transmission is unreliable with success probability $C_{i,j}$9, and the receiver updates according to

$N$0

The actuator applies $N$1, so actuation consequences enter implicitly through $N$2 (Luo et al., 2023).

Another branch studies two-state remote estimation with normal and alarm states, in which the channel has i.i.d. packet drops, ACK/NACK feedback is instantaneous and error-free, and the receiver uses a sample-and-hold estimate

$N$3

That framework treats CAE as the mismatch-only special case of a broader age-aware semantic formulation (Luo et al., 2024).

The MPR extension considers two independent binary Markov sources $N$4 sharing a wireless multi-packet reception channel. Successful-update indicators $N$5 define synchronize-or-hold estimation

$N$6

and effective update probabilities $N$7 couple the PHY/MAC layer to CAE (Elessawy et al., 15 May 2026).

3. Relation to reconstruction error, distortion, and semantic metrics

CAE differs from ordinary reconstruction error because the latter only records whether the estimate is wrong. In the Markov-source reconstruction work, the binary time-averaged reconstruction error is

$N$8

while CAE weights mismatch states by their actuation consequences (Salimnejad et al., 2023). Reconstruction-error variance,

$N$9

and duration-based metrics such as consecutive error and cost of memory error likewise ignore which wrong state pair occurred (Salimnejad et al., 2023).

The distinction is central in wireless autonomous systems. There, the baseline policy that transmits whenever $$\bar{C}_{A} = \sum_{i=0}^{N-1}\sum_{\substack{j=0 \ j\neq i}}^{N-1} C_{i,j}\pi_{i,j},$$0 can achieve lower reconstruction error than CAE-aware policies for rapidly varying sources, yet still yield higher CAE because it expends resources on semantically unimportant mismatches and can miss harmful ones (Fountoulakis et al., 2023). This establishes a recurrent point in the literature: lower estimation error does not imply lower actuation cost.

The age-aware remote-estimation framework makes the hierarchy explicit. It states that distortion and CAE “only consider $$\bar{C}_{A} = \sum_{i=0}^{N-1}\sum_{\substack{j=0 \ j\neq i}}^{N-1} C_{i,j}\pi_{i,j},$$1,” whereas age metrics depend on $$\bar{C}_{A} = \sum_{i=0}^{N-1}\sum_{\substack{j=0 \ j\neq i}}^{N-1} C_{i,j}\pi_{i,j},$$2, and its own state is the augmented triple

$$\bar{C}_{A} = \sum_{i=0}^{N-1}\sum_{\substack{j=0 \ j\neq i}}^{N-1} C_{i,j}\pi_{i,j},$$3

with separate Age of Missed Alarm (AoMA) and Age of False Alarm (AoFA) processes (Luo et al., 2024). In that paper, CAE is recovered as the special case “where the lasting impact vanishes,” meaning mismatch-type asymmetry remains but age accumulation is removed (Luo et al., 2024).

The AoI-with-CAE-constraint formulation sharpens a related misconception. It argues that optimizing AoI alone is insufficient because CAE also depends on source dynamics and semantic penalties; faster source dynamics can produce higher CAE under the same average AoI, and different AoI trajectories can induce different CAE under identical average AoI (Pomaje et al., 11 Aug 2025). In that model, the expected instantaneous CAE is

$$\bar{C}_{A} = \sum_{i=0}^{N-1}\sum_{\substack{j=0 \ j\neq i}}^{N-1} C_{i,j}\pi_{i,j},$$4

where

$$\bar{C}_{A} = \sum_{i=0}^{N-1}\sum_{\substack{j=0 \ j\neq i}}^{N-1} C_{i,j}\pi_{i,j},$$5

and $$\bar{C}_{A} = \sum_{i=0}^{N-1}\sum_{\substack{j=0 \ j\neq i}}^{N-1} C_{i,j}\pi_{i,j},$$6 is the successful-update probability (Pomaje et al., 11 Aug 2025). This closed form directly exhibits semantic dependence beyond freshness alone.

4. Closed-form analysis for Markov sources

The original reconstruction-and-actuation paper does not provide one universal closed form for arbitrary $$\bar{C}_{A} = \sum_{i=0}^{N-1}\sum_{\substack{j=0 \ j\neq i}}^{N-1} C_{i,j}\pi_{i,j},$$7 and arbitrary $$\bar{C}_{A} = \sum_{i=0}^{N-1}\sum_{\substack{j=0 \ j\neq i}}^{N-1} C_{i,j}\pi_{i,j},$$8; instead, CAE is obtained by deriving the stationary distribution of the joint Markov chain $$\bar{C}_{A} = \sum_{i=0}^{N-1}\sum_{\substack{j=0 \ j\neq i}}^{N-1} C_{i,j}\pi_{i,j},$$9 and substituting into $\pi_{i,j}$0 (Salimnejad et al., 2023). The procedure is: construct the 2D DTMC, compute stationary probabilities $\pi_{i,j}$1, weight erroneous states by $\pi_{i,j}$2, and sum.

For the two-state DTMC under a randomized stationary policy, Lemma 4 gives stationary probabilities

$\pi_{i,j}$3

and hence

$\pi_{i,j}$4

Under symmetric costs $\pi_{i,j}$5, this reduces to a constant multiple of the off-diagonal stationary mass (Salimnejad et al., 2023).

For the two-state BDMP under randomized stationary policy, Lemma 5 gives

$\pi_{i,j}$6

again yielding

$\pi_{i,j}$7

(Salimnejad et al., 2023).

For the three-state DTMC under randomized stationary policy, the paper gives

$\pi_{i,j}$8

so CAE becomes

$\pi_{i,j}$9

(Salimnejad et al., 2023). The appendices provide corresponding $(X_t,\hat X_t)=(i,j)$0 expressions for three-state DTMC and BDMP under randomized stationary, change-aware, and semantics-aware policies (Salimnejad et al., 2023).

A distinct simplification emerges in the MPR binary-source model. For source $(X_t,\hat X_t)=(i,j)$1, the paper defines

$(X_t,\hat X_t)=(i,j)$2

and proves that for $(X_t,\hat X_t)=(i,j)$3,

$(X_t,\hat X_t)=(i,j)$4

Consequently,

$(X_t,\hat X_t)=(i,j)$5

and, since $(X_t,\hat X_t)=(i,j)$6,

$(X_t,\hat X_t)=(i,j)$7

The closed-form expression is

$(X_t,\hat X_t)=(i,j)$8

(Elessawy et al., 15 May 2026). In this binary stationary-randomized setting, CAE is therefore proportional to real-time reconstruction error, and semantic penalties reweight sources rather than altering the structure of the optimization problem.

5. Policy design and optimization with CAE

The role of CAE in optimization varies across papers. In the original Markov-source reconstruction paper, CAE is an evaluation metric, not an objective. The two optimization problems minimize time-averaged reconstruction error and average consecutive error under sampling-cost constraints; neither directly minimizes $(X_t,\hat X_t)=(i,j)$9 (Salimnejad et al., 2023). The paper explicitly does not provide CAE plots or a CAE-specific optimized policy (Salimnejad et al., 2023).

By contrast, "Goal-oriented Policies for Cost of Actuation Error Minimization in Wireless Autonomous Systems" makes CAE the primary constrained objective: $(X_t,\hat X_t)$0 Using Lagrangian relaxation, it solves an infinite-horizon average-cost CMDP, with the optimal constrained policy represented as a mixture of two deterministic policies associated with neighboring Lagrange multipliers (Fountoulakis et al., 2023). The paper also proposes a low-complexity drift-plus-penalty scheme based on the one-step expected CAE proxy

$(X_t,\hat X_t)$1

and online decision rule

$(X_t,\hat X_t)$2

(Fountoulakis et al., 2023).

The multi-source scheduling paper formulates long-term average CAE minimization under an average communication-resource constraint: $(X_t,\hat X_t)$3 with per-slot communication cost

$(X_t,\hat X_t)$4

It develops a drift-plus-penalty policy for known statistics and a Lyapunov-optimization-based deep reinforcement learning policy for unknown environments (Luo et al., 2023). The per-slot expected CAE contribution of source $(X_t,\hat X_t)$5, for subsystem state $(X_t,\hat X_t)$6, is

$(X_t,\hat X_t)$7

This makes the CAE reduction from a candidate transmission explicit (Luo et al., 2023).

In the AoI-constrained formulation, CAE enters as a feasibility condition rather than a minimization target. The main problem is

$(X_t,\hat X_t)$8

subject to $(X_t,\hat X_t)$9 and $X_t\neq \hat X_t$0 (Pomaje et al., 11 Aug 2025). Under stationary randomized policies, the CAE constraint becomes linear in the successful-update probability: $X_t\neq \hat X_t$1 which underpins the tractability and approximation analysis (Pomaje et al., 11 Aug 2025).

6. Generalizations, reinterpretations, and neighboring frameworks

A broad semantics-aware generalization is given by the AoMA/AoFA framework for two-state remote estimation. There, the weighted mismatch-age cost is

$X_t\neq \hat X_t$2

or equivalently

$X_t\neq \hat X_t$3

with $X_t\neq \hat X_t$4 controlling the relative importance of missed alarms and false alarms (Luo et al., 2024). The paper explicitly states that distortion and CAE are special cases “where the lasting impact vanishes,” so CAE is positioned there as an instantaneous mismatch-only semantic loss (Luo et al., 2024).

The MPR analysis generalizes CAE across jointly scheduled sources sharing a wireless channel. Its core structural contribution is the mapping

$X_t\neq \hat X_t$5

showing how actuation-centric mismatch costs can be analyzed directly from PHY/MAC update probabilities (Elessawy et al., 15 May 2026).

Some papers do not define CAE explicitly but offer closely related actuation-risk quantities. In the runtime actuarial control framework for autonomous AI agents, each side-effect-bearing action is priced against a safe default through a conservative reserve quote

$X_t\neq \hat X_t$6

and the conceptual counterfactual loss increment is

$X_t\neq \hat X_t$7

This suggests a runtime, counterfactual, safe-default-relative interpretation of actuation cost, though the paper does not use the CAE terminology (Chen, 25 May 2026). A plausible implication is that CAE can be extended beyond wireless estimation to action-gating and reserve-allocation problems where the relevant baseline is a contractual safe default rather than perfect state reconstruction.

Similarly, the causal decision-making framework MiCCD optimizes intervention cost subject to a counterfactual restoration constraint

$X_t\neq \hat X_t$8

with cost $X_t\neq \hat X_t$9, but does not itself define a CAE metric (Cai et al., 13 May 2025). This suggests a broader family of actuation-error notions built from intervention expenditure plus counterfactual failure risk, though such a metric is not explicitly introduced there.

7. Limitations, interpretations, and points of caution

Several limitations recur across the literature. The foundational CAE papers represent actuation consequences through a pairwise penalty matrix but do not explicitly model the plant, the control law, or the executed action variable. In that sense, CAE is usually a surrogate task-loss metric indexed by $X_t=i$00, not a full closed-loop control cost (Salimnejad et al., 2023, Fountoulakis et al., 2023). This abstraction is deliberate and useful, but it should not be conflated with a plant-level performance functional.

Another recurring limitation is that some papers define CAE analytically without numerically instantiating a cost matrix. The Markov-source reconstruction papers provide the stationary formulas needed to compute $X_t=i$01 under several policies, but they do not specify a concrete $X_t=i$02, provide CAE plots, or optimize CAE directly (Salimnejad et al., 2023). As a result, the metric’s semantic value is clear, but empirical CAE comparisons depend on a user-supplied application matrix.

A further caution concerns binary special cases. In the MPR binary-source model, the equality $X_t=i$03 implies that CAE is proportional to reconstruction error after a source-specific weight transformation (Elessawy et al., 15 May 2026). This should not be mistaken for a universal fact. The paper itself notes that the simplification does not generally hold for multi-state sources or more general dynamic policies, where directional mismatch probabilities can differ and CAE becomes genuinely distinct from mismatch rate (Elessawy et al., 15 May 2026).

Finally, age-aware work identifies a blind spot of pure CAE formulations: CAE captures instantaneous mismatch significance but not the lasting impact of unresolved errors (Luo et al., 2024). The missed-alarm and false-alarm age processes were proposed precisely because a CAE-optimal policy may transmit only on one error class and ignore the other, incurring large persistence costs over time (Luo et al., 2024). This does not refute CAE; it places it in a hierarchy of semantic metrics.

CAE is therefore best understood as the canonical mismatch-consequence metric in goal-oriented communication: more expressive than distortion or binary error, but narrower than age-augmented or full counterfactual intervention frameworks. Its central contribution is to shift evaluation and design from whether information is wrong to how costly it is to act on that wrong information (Salimnejad et al., 2023, Luo et al., 2023, Luo et al., 2024).