Exact Average Consensus under Noisy Communication Links: A Decentralized Gradient Perspective

I. INTRODUCTION

• Exact average consensus is a coordination reference for distributed sensing and cooperative control, where deviations may propagate into later decisions.Agents seek agreement on the average of their initial state values.Source: §I · p. 1
• Without modification, link-level disturbances can make the network average follow a random walk, preventing convergence to a fixed consensus value.Prior work models disturbances including channel fading, quantization errors, and additive link noise.Source: §I · p. 1
• Stochastic-approximation methods with decaying stepsizes typically converge to an unbiased random variable with nonzero variance, rather than the exact initial average.Monte Carlo averaging has been used to compensate for random deviation at additional energy and runtime cost.Source: §I · pp. 1–2
• The decentralized-gradient formulation motivates an update combining current state, decayed neighbor contributions, and a decayed initial-state anchor; suitable stepsizes yield almost-sure exact consensus under MDS link disturbances.The perspective also interprets related methods, including noise-resilient push-sum, dynamic average consensus, and over-the-air distributed consensus.Source: §I · p. 2

II. PRELIMINARIES ON AVERAGE CONSENSUS WITH NOISY COMMUNICATION LINKS

• The network is modeled as a time-invariant undirected graph, and the mixing matrix is symmetric and doubly stochastic with contraction away from consensus.The spectral radius of W − 11⊤/N is less than one.Source: §II · p. 2
• The mixing matrix assigns zero weight to non-neighboring agents, while its row and column sums equal one.These properties support the stated average-consensus behavior.Source: §II · p. 2
• Under the stated mixing assumption, the noise-free linear iteration achieves asymptotic average consensus at the agents’ initial-state average.The mixing matrix has a simple eigenvalue at one, while its other eigenvalues lie strictly between −1 and 1.Source: §II · p. 2

A. Assumptions on Link-Level Disturbances

• The disturbance vector collects agent-level perturbations that may arise from channel noise, packet losses, or quantization errors.The text defines e_i[t] as the disturbance observed by agent i at iteration t.Source: A. Assumptions on Link-Level… · p. 2
• The aggregate disturbance is assumed to be an MDS with uniformly bounded conditional second moments, with e[0] equal to zero.The MDS is defined relative to the filtration generated by information available over time.Source: A. Assumptions on Link-Level… · p. 2
• The model allows temporal dependence, time-varying variance, and correlation among agent-wise disturbances, making it more general than i.i.d. noise.AWGN is included as a special case satisfying the stated conditions.Source: A. Assumptions on Link-Level… · p. 2

B. Average Consensus under Link-Level Disturbances

• Almost-sure average consensus requires convergence to a random variable whose expectation is the initial average and whose variance is finite.Exact almost-sure average consensus is the special case where that variable has zero variance.Source: B. Average Consensus under Link-Level… · p. 3
• The analysis represents the iteration using a weighted Laplacian and a scaled mixing matrix whose disagreement-subspace eigenvalues lie strictly within (−1, 1).The scaled eigenvalues are formed from the original eigenvalues and the stepsize.Source: B. Average Consensus under Link-Level… · p. 3
• Under the stepsize and disturbance assumptions, the standard iteration achieves almost-sure average consensus but generally not exact almost-sure average consensus.Exact consensus requires a constant limiting value equal to the initial average, whereas the general definition allows an unbiased finite-variance random limit.Source: B. Average Consensus under Link-Level… · p. 3

• The distributed optimization formulation has the initial network average as its unique optimizer under the consensus constraint.Source: §III · p. 3
• AAC augments the noisy consensus iteration with a decaying anchor toward each agent’s initial state and achieves exact average consensus under the stated conditions.The anchor is a local gradient correction in a two-stepsize noisy DGD-type recursion.Source: §III · pp. 3–4
• The anchor’s effect must become asymptotically negligible relative to disagreement-subspace contraction; the stated spectral-radius condition may hold only after a finite time.Source: §III · p. 3

• The squared distance to the initial-average consensus vector separates orthogonally into average bias and disagreement components.Source: §IV · p. 4
• AAC achieves exact consensus, whereas PAC achieves consensus to a random limit, distinguishing E-ASAC from ordinary ASAC.Source: §IV · p. 4
• Under Assumptions 1–3, the corresponding condition, and the stated stepsize conditions, AAC converges almost surely to the exact average.Source: §IV · p. 4

• The disturbance-driven average evolves as a martingale, and square-summable stepsizes make its accumulated-noise martingale L2-bounded and almost-surely convergent.Source: A. Random Drift without Anchoring · p. 5
• A non-degenerate disturbance projection onto the consensus subspace gives the limiting average positive variance, ruling out exact consensus even when disagreement vanishes.Source: A. Random Drift without Anchoring · pp. 4–5
• PAC’s disturbances drive the network average to a random limit while disagreement vanishes, leaving consensus at that random value rather than the initial average.The decomposition attributes the residual consensus error to accumulated disturbance-induced drift in the average.Source: A. Random Drift without Anchoring · p. 5

B. The Effects of Anchoring: How Does It Help?

• The anchoring term contracts the average bias toward its initial value, counteracting the random drift caused by accumulated disturbances.The paper identifies the average disturbance as the source of drift in the network average.Source: B. The Effects of Anchoring: How Does… · p. 5
• The proof decomposes the error into average bias and disagreement, showing that both vanish almost surely for exact average consensus.The average-bias result is established separately from the disagreement analysis.Source: B. The Effects of Anchoring: How Does… · p. 6
• AAC drives the network average back toward its initial value while the consensus update contracts disagreement.Together, these mechanisms prevent drift from the initial network average.Source: B. The Effects of Anchoring: How Does… · pp. 5–6

C. Unified Interpretation of Representative Methods

• Anchoring provides a common interpretation of representative noise-robust consensus methods, rather than treating them as isolated constructions.The paper identifies the principle as introducing an anchor toward a desired target to suppress noise accumulation.Source: C. Unified Interpretation of… · pp. 6–7
• NR-PushSum injects weighted initial values as anchors, and its subgradient-push interpretation assigns an anchoring role to the local subgradient term.The method is described as a push-sum counterpart of anchoring over directed graphs.Source: C. Unified Interpretation of… · p. 6
• Dynamic average consensus replaces the fixed initial-state anchor with a time-varying reference, supporting tracking of an evolving target while remaining robust to noise.The paper presents this as a dynamic counterpart of the anchoring mechanism.Source: C. Unified Interpretation of… · pp. 6–7
• Non-coherent OTA consensus has the same anchoring structure as AAC, providing an engineering realization in a noisy wireless setting.The paper notes that earlier OTA work implicitly used this anchoring idea.Source: C. Unified Interpretation of… · p. 7

V. NUMERICAL SIMULATIONS

• The simulation uses a connected undirected Erdős–Rényi graph with 36 agents and 3-bit unbiased stochastic quantization over the stated input range.The initial states are independently drawn uniformly from [−100, 100], and the quantization disturbance is described as MDS-type.Source: §V · p. 7
• AAC’s average consensus error continues to decay toward zero, while PAC’s error stabilizes at a nonzero level under quantization noise.The reported ACE is averaged over 100 realizations; PAC reaches consensus rapidly but its limiting value deviates from the initial average.Source: §V · pp. 7–8, Figure 2 · p. 8
• AAC’s slower convergence is attributed to the anchoring term’s continued pull toward each agent’s initial state.The state trajectories are shown for 2 × 10^4 iterations.Source: Figure 1 · p. 7, §V · p. 8

VI. CONCLUSIONS

• AAC achieves almost-sure convergence to the exact average under persistent link-level disturbances modeled as an MDS with uniformly bounded conditional second moments.The paper contrasts this with standard SA-based methods, which generally converge to a random variable whose expectation is the initial average.Source: §VI · p. 8
• The contribution is a principled interpretation and rigorous convergence analysis of an anchoring-based mechanism, rather than a completely new consensus algorithm.The perspective also unifies representative methods under a common viewpoint.Source: §VI · p. 8
• Numerical results highlight a trade-off: anchoring improves robustness and steady-state accuracy at the expense of convergence speed.The paper presents anchoring as a useful, interpretable design for exact average consensus under persistent disturbances.Source: §VI · p. 8