Optimization and Analysis of Distributed Averaging with Short Node Memory

In this paper, we demonstrate, both theoretically and by numerical examples, that adding a local prediction component to the update rule can significantly improve the convergence rate of distributed averaging algorithms. We focus on the case where the local predictor is a linear combination of the node’s two previous values (i.e., two memory taps), and our update rule computes a combination of the predictor and the usual weighted linear combination of values received from neighbouring nodes. We derive the optimal mixing parameter for combining the predictor with the neighbors’ values, and carry out a theoretical analysis of the improvement in convergence rate that can be obtained using this acceleration methodology. For a chain topology on n nodes, this leads to a factor of n improvement over the one-step algorithm, and for a two-dimensional grid, our approach achieves a factor of n^1/2 improvement, in terms of the number of iterations required to reach a prescribed level of accuracy.

💡 Research Summary

The paper introduces a simple yet powerful acceleration technique for the classic distributed averaging (consensus) problem. In the standard one‑step algorithm each node updates its state by taking a weighted average of its own current value and the values received from its neighbors. The convergence speed of this method is limited by the spectral properties of the graph Laplacian: the second‑smallest eigenvalue (algebraic connectivity) determines how fast the error contracts.

To overcome this limitation the authors equip every node with a very short memory: the two most recent states are stored and linearly combined into a predictor
(p_i(t)=\alpha x_i(t-1)+(1-\alpha)x_i(t-2)).
The new update rule mixes this predictor with the usual neighbor‑average:
(x_i(t+1)=\beta p_i(t)+(1-\beta)\sum_{j\in\mathcal N_i} w_{ij}x_j(t)).
Here (w_{ij}) are stochastic weights derived from the Laplacian (e.g., Metropolis–Hastings), (\alpha\in