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Computationally Efficient Laplacian CL-colME

Nikola Stankovic

TL;DR

This work tackles decentralized collaborative mean estimation in heterogeneous networks where online identification of similar data sources is crucial. It introduces CL-colME, a Laplacian-based consensus variant that omits the construction and normalization of a doubly stochastic averaging matrix, replacing it with a smoothing step on the graph Laplacian $\mathbf{L}=\mathbf{D}-\mathbf{A}$ and a gradient-like update $\boldsymbol{\mu}(t)\leftarrow(\mathbf{I}-\beta\mathbf{L})\boldsymbol{\mu}(t-1)$ (in the regime $\alpha(t)\rightarrow 1$). The approach preserves class-wise unbiased convergence to oracle means within connected components while reducing per-iteration cost, as supported by convergence arguments and spectral analysis. Numerical results on a 5,000-agent example show CL-colME achieves comparable accuracy to C-colME with notable speedups, illustrating scalability advantages for large-scale decentralized learning in heterogeneous settings.

Abstract

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.

Computationally Efficient Laplacian CL-colME

TL;DR

This work tackles decentralized collaborative mean estimation in heterogeneous networks where online identification of similar data sources is crucial. It introduces CL-colME, a Laplacian-based consensus variant that omits the construction and normalization of a doubly stochastic averaging matrix, replacing it with a smoothing step on the graph Laplacian and a gradient-like update (in the regime ). The approach preserves class-wise unbiased convergence to oracle means within connected components while reducing per-iteration cost, as supported by convergence arguments and spectral analysis. Numerical results on a 5,000-agent example show CL-colME achieves comparable accuracy to C-colME with notable speedups, illustrating scalability advantages for large-scale decentralized learning in heterogeneous settings.

Abstract

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.
Paper Structure (10 sections, 22 equations, 1 figure)

This paper contains 10 sections, 22 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Problem Formulation
  3. Local Means
  4. Confidence Intervals and Similarity Detection
  5. Graph-Based Collaboration and Pruning
  6. Expected Separation Time
  7. Consensus-based Approach C-colME
  8. Consensus Laplacian-Based Variant CL-colME
  9. Numerical Example
  10. Conclusion

Figures (1)

  • Figure 1: C-colME and CL-colME ($N=5000$): Total MSE of the estimation averaged over agents at time instants $t$ (blue line). The results are averaged over 10 realizations with $\beta=0.1$. Calculation time for CL-colME is 722s per realization, on average, while for C-colME it was 871s. Local solution (green dashed line) and oracle solution (red dotted line) are also given.