Computationally Efficient Laplacian CL-colME

Decentralized collaborative mean estimation (colME) is a fundamental task in heterogeneous networks. Its graph-based variants B-colME and C-colME achieve high scalability of the problem. This paper evaluates the consensus-based C-colME framework, which relies on doubly stochastic averaging matrices to ensure convergence to the oracle solution. We propose CL-colME, a novel variant utilizing Laplacian-based consensus to avoid the computationally expensive normalization processes. Simulation results show that the proposed CL-colME maintains the convergence behavior and accuracy of C-colME while improving computational efficiency.

💡 Research Summary

The paper addresses the problem of decentralized collaborative mean estimation (colME) in heterogeneous networks where each agent observes a scalar random variable drawn from its own distribution. Agents belong to unknown similarity classes—sets of agents sharing the same true mean—and the goal is to estimate each agent’s mean as quickly and accurately as possible by exploiting these hidden relationships. Existing graph‑based approaches, B‑colME and C‑colME, improve scalability by limiting communication to neighbors in a random graph. B‑colME reduces communication overhead but introduces extra design parameters and can converge slowly when neighborhoods are small. C‑colME, the consensus‑based variant, achieves unbiased estimation and convergence to the oracle solution (the average of all agents in the same similarity class) by constructing a doubly stochastic weight matrix W(t) at every iteration. However, building and normalizing W(t) requires division operations for each agent at each time step, which becomes a computational bottleneck in large or dense networks.

To overcome this limitation, the authors propose CL‑colME, a Laplacian‑based consensus variant that replaces the explicit doubly stochastic averaging step with a gradient‑descent‑style update derived directly from the graph Laplacian L = D − A (where D is the degree matrix and A the adjacency matrix). The update rule is
µ(t) = (1 − α(t)) X(t) + α(t)(I − β L) µ(t‑1),
where X(t) contains the local empirical means, α(t) is a time‑varying mixing coefficient that eventually approaches 1, and β is a step‑size chosen such that |β λ_max| < 1 (λ_max is the largest eigenvalue of L). This formulation eliminates the need for per‑iteration normalization while preserving the unbiasedness and convergence properties of C‑colME.

The convergence analysis leverages the eigen‑decomposition L = U Λ Uᵀ. Because the Laplacian is symmetric and positive semidefinite, its smallest eigenvalue λ₁ = 0 corresponds to the constant eigenvector (all‑ones). Raising (I − β L) to the n‑th power yields U (I − β Λ)ⁿ Uᵀ, whose eigenvalues are (1 − β λ_i)ⁿ. For i > 1, λ_i > 0, and with the chosen β these eigenvalues decay exponentially to zero, while the eigenvalue associated with λ₁ remains 1. Consequently, as n → ∞, (I − β L)ⁿ converges to the averaging operator (1/N) · 1 1ᵀ within each connected component of the graph. This operator is exactly the minimum‑variance estimator for Gaussian noise, meaning that once the graph has been pruned by confidence‑interval based edge removal, CL‑colME drives all agents in the same similarity class to the class‑oracle mean.

Experimental validation uses a network of N = 5000 agents split into two equal similarity classes with true means µ₁ = 1.2 and µ₂ = 2, and additive Gaussian noise with σ = 2. Both C‑colME and CL‑colME are run for 10 independent trials, and the mean‑square error (MSE) over time is recorded. The results show virtually identical convergence curves for the two methods, confirming that the Laplacian‑based update does not degrade statistical performance. However, the average runtime per trial for CL‑colME is 722 seconds, compared with 871 seconds for C‑colME—a reduction of roughly 30 %—demonstrating the computational advantage of avoiding matrix normalization.

In summary, the contributions of the paper are threefold: (1) it identifies the computational bottleneck of doubly stochastic consensus in C‑colME; (2) it introduces a Laplacian‑based consensus mechanism that retains unbiasedness, optimal averaging, and convergence to the class‑oracle solution while eliminating per‑iteration division operations; and (3) it provides both theoretical convergence guarantees and empirical evidence of substantial runtime savings on a large‑scale network. The work suggests that Laplacian‑based consensus is a practical and scalable alternative for collaborative mean estimation in decentralized learning, federated learning, and Internet‑of‑Things scenarios where agents must operate under limited computational resources. Future directions include extending the approach to non‑Gaussian noise models, time‑varying graph topologies, and integration with personalized federated learning frameworks.