Consensus Error-Dependent Laplace Noise

• The paper [2509.11917] introduces consensus error-dependent Laplace noise, where the noise scale is directly tied to the local disagreement in finite-horizon LQR consensus.
• The mechanism dynamically adjusts Laplace noise based on real-time consensus error, ensuring differential privacy for local LQR weight matrices without fixed noise schedules.
• Design trade-offs balance privacy and performance, as noise energy scales with 1/epsilon², highlighting the impact of sensitivity and consensus mismatch on bounded consensus.

Searching arXiv for the exact paper and closely related consensus/privacy/noise papers to ground the article in current arXiv records. Consensus error-dependent Laplace noise is a privacy mechanism for networked consensus control in which the Laplace scale is tied to a local disagreement signal rather than fixed a priori. In the arXiv literature summarized here, its most direct formulation appears in the distributed finite-horizon LQR framework of "Distributed Finite-Horizon Optimal Control for Consensus with Differential Privacy Guarantees" (Ma et al., 15 Sep 2025). There, the mechanism perturbs communicated state information with Laplace noise whose magnitude depends on the current local consensus mismatch, while a time-dependent cost scaling is introduced so that the closed-loop system preserves differential privacy of the local LQR weighting matrices and still achieves bounded consensus. Closely related literatures study decaying Laplace noise, fixed-design Laplace initialization noise, or state-dependent Brownian diffusion, but those mechanisms are distinct in both noise law and adaptation rule (Nozari et al., 2015, Wang et al., 2023, Russo et al., 2016).

1. Conceptual scope and distinguishing features

In the formulation of (Ma et al., 15 Sep 2025), the multi-agent system has discrete-time single-integrator dynamics

$$x_i(t+1)=x_i(t)+u_i(t), \qquad i=1,2,\ldots,N,\; t\in\mathbb{N},$$

with $x_i(t)\in \mathbb{R}^n$ and $u_i(t)\in \mathbb{R}^n$. The agents communicate over a directed graph

$\mathcal{G}(\mathcal{V},\mathcal{E},\mathcal{A}),$

with adjacency matrix $\mathcal{A}=[a_{ij}]$, in-neighbor set

$$\mathcal{N}_i=\{j\in\mathcal{V}\mid (j,i)\in\mathcal{E}\},$$

and Laplacian $\mathcal{L}=[l_{ij}]$. The standing graph condition is that $\mathcal{G}$ contains a directed spanning tree (Ma et al., 15 Sep 2025).

The expression “consensus error-dependent Laplace noise” refers here to a mechanism in which the Laplace scale is proportional to a local consensus mismatch term,

$$\left\|\sum_{j\in\mathcal{N}_i} a_{ij}\bigl[\hat{x}_j(t)-x_i(t)\bigr]\right\|_\infty,$$

rather than to a fixed initialization variance or to an exogenous time-decay schedule (Ma et al., 15 Sep 2025). The noisy communicated state is

$\hat{x}_j(t)\triangleq x_j(t)+\eta_j(t),$

and the local mismatch is the quantity through which the current state of disagreement enters the privacy mechanism.

This construction differs sharply from several adjacent mechanisms. In "Differentially Private Average Consensus: Obstructions, Trade-Offs, and Optimal Algorithm Design" (Nozari et al., 2015), the Laplace scale is purely time-dependent,

$x_i(t)\in \mathbb{R}^n$0

with exponentially decaying Laplace noise and no dependence on the realized consensus mismatch. In "Differentially Private Consensus for Time-Delay Multi-agent Systems" (Wang et al., 13 Jun 2026), the Laplace scale is also time-varying only and may even increase with time, but it is designed from delayed-history sensitivity rather than from the current disagreement. In "Observer-based Differentially Private Consensus for Linear Multi-agent Systems" (Zong et al., 18 Dec 2025), the Laplace schedule is exogenous,

$x_i(t)\in \mathbb{R}^n$1

or

$x_i(t)\in \mathbb{R}^n$2

again without any formula $x_i(t)\in \mathbb{R}^n$3. By contrast, (Ma et al., 15 Sep 2025) makes the noise amplitude explicitly state-dependent through the local disagreement term.

A second distinguishing feature is the privacy target. In (Ma et al., 15 Sep 2025), the private data are not the agents’ states or initial conditions, but the local LQR weight matrices $x_i(t)\in \mathbb{R}^n$4, which encode each agent’s control preferences (Ma et al., 15 Sep 2025). This contrasts with average-consensus privacy papers that protect initial states via message perturbation or initialization perturbation (Nozari et al., 2015, Wang et al., 2023).

2. Finite-horizon LQR consensus formulation

The private quantities in (Ma et al., 15 Sep 2025) are

$x_i(t)\in \mathbb{R}^n$5

with

$x_i(t)\in \mathbb{R}^n$6

The database of sensitive information is

$x_i(t)\in \mathbb{R}^n$7

with admissible database space

$x_i(t)\in \mathbb{R}^n$8

The paper states that protecting these matrices protects each agent’s “control preferences” and local cost function (Ma et al., 15 Sep 2025).

The unperturbed distributed finite-horizon LQR cost for agent $x_i(t)\in \mathbb{R}^n$9 is introduced as

$u_i(t)\in \mathbb{R}^n$0

The privacy-preserving version replaces the neighbor state by a noisy state and introduces a time-dependent scaling factor $u_i(t)\in \mathbb{R}^n$1: $u_i(t)\in \mathbb{R}^n$2 Here

$u_i(t)\in \mathbb{R}^n$3

The resulting receding-horizon distributed control law is

$u_i(t)\in \mathbb{R}^n$4

where $u_i(t)\in \mathbb{R}^n$5 is the finite-horizon LQR gain determined by $u_i(t)\in \mathbb{R}^n$6, the horizon $u_i(t)\in \mathbb{R}^n$7, and the finite-horizon recursion (Ma et al., 15 Sep 2025). The closed-loop dynamics are then

$u_i(t)\in \mathbb{R}^n$8

This architecture is essential to the meaning of consensus error-dependent Laplace noise in (Ma et al., 15 Sep 2025). The gain sensitivity with respect to $u_i(t)\in \mathbb{R}^n$9 determines how a private change enters the dynamics, while the local mismatch term determines the magnitude of the noisy perturbation.

3. Error-dependent Laplace mechanism

The network-level disagreement operator is

$\mathcal{G}(\mathcal{V},\mathcal{E},\mathcal{A}),$0

and with

$\mathcal{G}(\mathcal{V},\mathcal{E},\mathcal{A}),$1

the consensus error is

$\mathcal{G}(\mathcal{V},\mathcal{E},\mathcal{A}),$2

Its squared norm

$\mathcal{G}(\mathcal{V},\mathcal{E},\mathcal{A}),$3

is the disagreement energy,

$\mathcal{G}(\mathcal{V},\mathcal{E},\mathcal{A}),$4

The paper’s noise law is local rather than global, but this consensus-error formalism clarifies what “error-dependent” means (Ma et al., 15 Sep 2025).

The sensitivity of the finite-horizon LQR gain is defined by

$\mathcal{G}(\mathcal{V},\mathcal{E},\mathcal{A}),$5

This quantity is bounded when the feasible sets $\mathcal{G}(\mathcal{V},\mathcal{E},\mathcal{A}),$6 are bounded (Ma et al., 15 Sep 2025).

The Laplace mechanism is then defined by

$\mathcal{G}(\mathcal{V},\mathcal{E},\mathcal{A}),$7

with

$\mathcal{G}(\mathcal{V},\mathcal{E},\mathcal{A}),$8

Equivalently, each component of $\mathcal{G}(\mathcal{V},\mathcal{E},\mathcal{A}),$9 is sampled i.i.d. from a zero-mean Laplace law with scale

$\mathcal{A}=[a_{ij}]$0

This formula exhibits four coupled dependencies already stated in (Ma et al., 15 Sep 2025): the time-varying state/cost scaling $\mathcal{A}=[a_{ij}]$1, the auxiliary design sequence $\mathcal{A}=[a_{ij}]$2, the gain sensitivity $\mathcal{A}=[a_{ij}]$3, and the current local consensus mismatch. The mechanism is therefore simultaneously time-varying, agent-specific, privacy-budget-dependent, and disagreement-dependent.

A common misconception is to equate any decaying Laplace mechanism with an error-dependent one. The literature summarized here does not support that identification. In (Nozari et al., 2015), the scale is fixed by $\mathcal{A}=[a_{ij}]$4, not by realized disagreement. In (Zong et al., 18 Dec 2025), the scale is $\mathcal{A}=[a_{ij}]$5 or $\mathcal{A}=[a_{ij}]$6, again not a function of $\mathcal{A}=[a_{ij}]$7. In (Wang et al., 13 Jun 2026), the scale may increase with time,

$\mathcal{A}=[a_{ij}]$8

but still remains time-index-dependent rather than disagreement-dependent. The precise hallmark of the mechanism in (Ma et al., 15 Sep 2025) is the appearance of the local consensus mismatch directly inside the Laplace scale.

4. Differential privacy of local weight matrices

The adjacency relation in (Ma et al., 15 Sep 2025) is defined on the database

$\mathcal{A}=[a_{ij}]$9

after canonical normalization of each pair. The text states that privacy is defined for the pair $$\mathcal{N}_i=\{j\in\mathcal{V}\mid (j,i)\in\mathcal{E}\},$$0 up to a normalization because the LQR gain is scale-invariant with respect to simultaneous scaling of $$\mathcal{N}_i=\{j\in\mathcal{V}\mid (j,i)\in\mathcal{E}\},$$1 and $$\mathcal{N}_i=\{j\in\mathcal{V}\mid (j,i)\in\mathcal{E}\},$$2 (Ma et al., 15 Sep 2025).

The mechanism satisfies the standard definition of $$\mathcal{N}_i=\{j\in\mathcal{V}\mid (j,i)\in\mathcal{E}\},$$3-differential privacy: $$\mathcal{N}_i=\{j\in\mathcal{V}\mid (j,i)\in\mathcal{E}\},$$4 for adjacent $$\mathcal{N}_i=\{j\in\mathcal{V}\mid (j,i)\in\mathcal{E}\},$$5 and measurable $$\mathcal{N}_i=\{j\in\mathcal{V}\mid (j,i)\in\mathcal{E}\},$$6. At time $$\mathcal{N}_i=\{j\in\mathcal{V}\mid (j,i)\in\mathcal{E}\},$$7, the relevant mechanism output is

$$\mathcal{N}_i=\{j\in\mathcal{V}\mid (j,i)\in\mathcal{E}\},$$8

If $$\mathcal{N}_i=\{j\in\mathcal{V}\mid (j,i)\in\mathcal{E}\},$$9 and $\mathcal{L}=[l_{ij}]$0 differ only in agent $\mathcal{L}=[l_{ij}]$1’s weight pair, the sensitivity of $\mathcal{L}=[l_{ij}]$2 is bounded by

$\mathcal{L}=[l_{ij}]$3

The scale in the Laplace law is calibrated exactly to this quantity, with the factor $\mathcal{L}=[l_{ij}]$4, and the paper invokes the standard Laplace mechanism lemma to obtain per-step privacy (Ma et al., 15 Sep 2025).

Under the summability condition

$\mathcal{L}=[l_{ij}]$5

Theorem 1 of (Ma et al., 15 Sep 2025) states that $\mathcal{L}=[l_{ij}]$6 preserves

$\mathcal{L}=[l_{ij}]$7

-differential privacy at time $\mathcal{L}=[l_{ij}]$8, and the full infinite-horizon mechanism

$\mathcal{L}=[l_{ij}]$9

preserves

$\mathcal{G}$0

-differential privacy. The proof uses the Adaptive Sequential Composition Theorem because later states depend on earlier noisy states (Ma et al., 15 Sep 2025).

This privacy formulation is structurally different from earlier DP consensus papers. In (Nozari et al., 2015), privacy concerns agents’ initial states and the global eavesdropper’s observation of the message trajectory. In (Wang et al., 2023), privacy also targets initial data and uses a correlated-zero-sum initialization mechanism, with a dedicated Laplace-noise design at the initialization stage rather than continual message perturbation. By contrast, (Ma et al., 15 Sep 2025) uses continual state perturbation to hide local LQR preference matrices rather than initial conditions.

5. Consensus dynamics, second moments, and privacy–performance trade-off

With stacked variables

$\mathcal{G}$1

block-diagonal gain matrix

$\mathcal{G}$2

and shorthand

$\mathcal{G}$3

the closed-loop network dynamics are

$\mathcal{G}$4

and the disagreement dynamics are

$\mathcal{G}$5

The Lyapunov function is

$\mathcal{G}$6

A key technical constant is

$\mathcal{G}$7

together with

$\mathcal{G}$8

The privacy-dependent coefficient is

$\mathcal{G}$9

This formula makes explicit that the privacy penalty scales like $$\left\|\sum_{j\in\mathcal{N}_i} a_{ij}\bigl[\hat{x}_j(t)-x_i(t)\bigr]\right\|_\infty,$$0 (Ma et al., 15 Sep 2025).

Because a Laplace variable with scale $$\left\|\sum_{j\in\mathcal{N}_i} a_{ij}\bigl[\hat{x}_j(t)-x_i(t)\bigr]\right\|_\infty,$$1 has second moment $$\left\|\sum_{j\in\mathcal{N}_i} a_{ij}\bigl[\hat{x}_j(t)-x_i(t)\bigr]\right\|_\infty,$$2, the paper derives the per-agent noise-energy identity

$$\left\|\sum_{j\in\mathcal{N}_i} a_{ij}\bigl[\hat{x}_j(t)-x_i(t)\bigr]\right\|_\infty,$$3

The local mismatch term is then bounded by the previous global disagreement and previous noise energy, leading to

$$\left\|\sum_{j\in\mathcal{N}_i} a_{ij}\bigl[\hat{x}_j(t)-x_i(t)\bigr]\right\|_\infty,$$4

The sufficient design conditions used for bounded consensus are

$$\left\|\sum_{j\in\mathcal{N}_i} a_{ij}\bigl[\hat{x}_j(t)-x_i(t)\bigr]\right\|_\infty,$$5

and

$$\left\|\sum_{j\in\mathcal{N}_i} a_{ij}\bigl[\hat{x}_j(t)-x_i(t)\bigr]\right\|_\infty,$$6

Under these assumptions, Theorem 2 of (Ma et al., 15 Sep 2025) proves that

$$\left\|\sum_{j\in\mathcal{N}_i} a_{ij}\bigl[\hat{x}_j(t)-x_i(t)\bigr]\right\|_\infty,$$7

that is, bounded consensus in mean square.

The asymptotic error analysis introduces

$$\left\|\sum_{j\in\mathcal{N}_i} a_{ij}\bigl[\hat{x}_j(t)-x_i(t)\bigr]\right\|_\infty,$$8

and assumes

$$\left\|\sum_{j\in\mathcal{N}_i} a_{ij}\bigl[\hat{x}_j(t)-x_i(t)\bigr]\right\|_\infty,$$9

where

$\hat{x}_j(t)\triangleq x_j(t)+\eta_j(t),$0

The resulting asymptotic bound is

$\hat{x}_j(t)\triangleq x_j(t)+\eta_j(t),$1

with $\hat{x}_j(t)\triangleq x_j(t)+\eta_j(t),$2 satisfying

$\hat{x}_j(t)\triangleq x_j(t)+\eta_j(t),$3

The formula isolates the privacy-dependent contribution through $\hat{x}_j(t)\triangleq x_j(t)+\eta_j(t),$4, so stronger privacy enlarges the bound through its $\hat{x}_j(t)\triangleq x_j(t)+\eta_j(t),$5 dependence (Ma et al., 15 Sep 2025).

This trade-off is conceptually different from those in DP average-consensus papers protecting initial states. In (Wang et al., 2023), the asymptotic mean-square error of the Laplace design is

$\hat{x}_j(t)\triangleq x_j(t)+\eta_j(t),$6

with privacy achieved by fixed-design initialization noise rather than state-adaptive message perturbation. In (Nozari et al., 2015), the asymptotic variance of the consensus value is

$\hat{x}_j(t)\triangleq x_j(t)+\eta_j(t),$7

and the optimal design collapses to one-shot perturbation of initial states. These are distinct privacy–accuracy regimes from the error-dependent LQR mechanism of (Ma et al., 15 Sep 2025).

6. Relation to neighboring literatures and recurrent misconceptions

The most important clarification is that consensus error-dependent Laplace noise is not synonymous with privacy-preserving consensus noise in general. Several nearby literatures must be separated carefully.

First, error-dependent noise and Laplace noise do not generally coincide. "On noise-induced synchronization and consensus" (Russo et al., 2016) studies state-dependent multiplicative noise in Itô SDEs, including diffusion terms that vanish on the synchronization manifold and scale with disagreement, but the increments are Brownian rather than Laplace. The paper therefore informs the “consensus-error-dependent” part of the topic, but not the “Laplace” part.

Second, Laplace privacy mechanisms in consensus are often not error-dependent. In (Nozari et al., 2015), the Laplace mechanism is exponentially decaying in time and not a function of the current residual. In (Wang et al., 2023), Laplace noise is used only at initialization, through a correlated-zero-sum DiShuf stage and a single additional Laplace variable at a secure agent. In (Zong et al., 18 Dec 2025), observer-based MASs use exogenously scheduled decaying Laplace noise, and the paper explicitly does not study $\hat{x}_j(t)\triangleq x_j(t)+\eta_j(t),$8. In (Wang et al., 13 Jun 2026), the delayed-consensus mechanism uses time-varying Laplace scales tied to explicit sensitivity bounds for delayed initial histories, and the scale may even increase with time, but it is not an explicit function of the current disagreement.

Third, privacy-preserving consensus papers often use non-Laplace noise families altogether. "A Privacy Preserving Randomized Gossip Algorithm via Controlled Noise Insertion" (Hanzely et al., 2019) uses per-node Gaussian noise with deterministic geometric decay and inter-iteration correlation created by subtracting previously injected noise and adding a smaller fresh perturbation. Its noise depends on a node-local update counter rather than on consensus error, disagreement, or residual, and it does not formalize differential privacy. "On the Influence of Noise in Randomized Consensus Algorithms" (Vizuete et al., 2020) gives a variance-based asymptotic disagreement analysis for additive noise in randomized consensus; fixed-scale Laplace noise fits the second-moment framework through $\hat{x}_j(t)\triangleq x_j(t)+\eta_j(t),$9, but state-dependent Laplace noise falls outside the paper’s exact formulas without new arguments.

Within this broader landscape, (Ma et al., 15 Sep 2025) occupies a specific niche. It combines a finite-horizon LQR consensus controller, a privacy objective on local weight matrices rather than states, and a Laplace scale that is explicitly proportional to a local disagreement term. This suggests a design pattern in which privacy calibration follows the actual pathway by which private parameters influence the control law: gain sensitivity enters through $x_i(t)\in \mathbb{R}^n$00, and the magnitude of the perturbation is modulated by the consensus mismatch that multiplies the gain in the control update. A plausible implication is that the mechanism suppresses unnecessary noise when local disagreement is already small, but the bounded-consensus guarantee in the paper is stated in terms of the explicit summability and contraction conditions on $x_i(t)\in \mathbb{R}^n$01, $x_i(t)\in \mathbb{R}^n$02, and the gain-dependent constants, not through a separate optimality theorem (Ma et al., 15 Sep 2025).

The simulation instance in (Ma et al., 15 Sep 2025) illustrates this mechanism on a 4-agent directed graph with

$x_i(t)\in \mathbb{R}^n$03

and sequences

$x_i(t)\in \mathbb{R}^n$04

The reported cumulative privacy leakage over $x_i(t)\in \mathbb{R}^n$05 to $x_i(t)\in \mathbb{R}^n$06 is

$x_i(t)\in \mathbb{R}^n$07

and the state trajectories approach a bounded consensus region under the proposed mechanism (Ma et al., 15 Sep 2025). This example does not establish superiority over fixed-scale Laplace or Gaussian alternatives, but it does instantiate the theory’s central claim: consensus error-dependent Laplace noise can be embedded into a distributed finite-horizon optimal-control protocol so as to protect local weight matrices while preserving bounded consensus.