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Distributed Learning over Noisy Communication Networks

Emrah Akyol, Marcos Vasconcelos

TL;DR

This work analyzes distributed coordination over graphs under explicit noisy communication, modeling neighbor observations via BSC and BEC channels. It uncovers a fast regime where channel-averaged updates induce a Gibbs sampler on a scaled potential with attenuation κ, and a snapshot regime where single noisy realizations yield a generally nonreversible Markov chain whose drift matches the fast regime to first order at high temperature; a finite-$K$ interpolation connects these extremes. The key contributions include an exact Gibbs structure in the fast regime, a high-temperature expansion for snapshot dynamics, a finite-$K$ framework, and extensions to heterogeneous link reliabilities with a clear communication-theoretic interpretation. The results provide practical guidance for designing communication-aware distributed learning systems, showing how reliability and budget tradeoffs translate into effective temperature and coordination quality, with broad implications for networked sensing, robotics, and IoT applications.

Abstract

We study binary coordination games over graphs under log-linear learning when neighbor actions are conveyed through explicit noisy communication links. Each edge is modeled as either a binary symmetric channel (BSC) or a binary erasure channel (BEC). We analyze two operational regimes. For binary symmetric and binary erasure channels, we provide a structural characterization of the induced learning dynamics. In a fast-communication regime, agents update using channel-averaged payoffs; the resulting learning dynamics coincide with a Gibbs sampler for a scaled coordination potential, where channel reliability enters only through a scalar attenuation coefficient. In a snapshot regime, agents update from a single noisy realization and ignore channel statistics; the induced Markov chain is generally nonreversible, but admits a high-temperature expansion whose drift matches that of the fast Gibbs sampler with the same attenuation. We further formalize a finite-$K$ communication budget, which interpolates between snapshot and fast behavior as the number of channel uses per update grows. This viewpoint yields a communication-theoretic interpretation in terms of retransmissions and repetition coding, and extends naturally to heterogeneous link reliabilities via effective edge weights. Numerical experiments illustrate the theory and quantify the tradeoff between communication resources and steady-state coordination quality.

Distributed Learning over Noisy Communication Networks

TL;DR

This work analyzes distributed coordination over graphs under explicit noisy communication, modeling neighbor observations via BSC and BEC channels. It uncovers a fast regime where channel-averaged updates induce a Gibbs sampler on a scaled potential with attenuation κ, and a snapshot regime where single noisy realizations yield a generally nonreversible Markov chain whose drift matches the fast regime to first order at high temperature; a finite- interpolation connects these extremes. The key contributions include an exact Gibbs structure in the fast regime, a high-temperature expansion for snapshot dynamics, a finite- framework, and extensions to heterogeneous link reliabilities with a clear communication-theoretic interpretation. The results provide practical guidance for designing communication-aware distributed learning systems, showing how reliability and budget tradeoffs translate into effective temperature and coordination quality, with broad implications for networked sensing, robotics, and IoT applications.

Abstract

We study binary coordination games over graphs under log-linear learning when neighbor actions are conveyed through explicit noisy communication links. Each edge is modeled as either a binary symmetric channel (BSC) or a binary erasure channel (BEC). We analyze two operational regimes. For binary symmetric and binary erasure channels, we provide a structural characterization of the induced learning dynamics. In a fast-communication regime, agents update using channel-averaged payoffs; the resulting learning dynamics coincide with a Gibbs sampler for a scaled coordination potential, where channel reliability enters only through a scalar attenuation coefficient. In a snapshot regime, agents update from a single noisy realization and ignore channel statistics; the induced Markov chain is generally nonreversible, but admits a high-temperature expansion whose drift matches that of the fast Gibbs sampler with the same attenuation. We further formalize a finite- communication budget, which interpolates between snapshot and fast behavior as the number of channel uses per update grows. This viewpoint yields a communication-theoretic interpretation in terms of retransmissions and repetition coding, and extends naturally to heterogeneous link reliabilities via effective edge weights. Numerical experiments illustrate the theory and quantify the tradeoff between communication resources and steady-state coordination quality.
Paper Structure (37 sections, 14 theorems, 91 equations, 4 figures)

This paper contains 37 sections, 14 theorems, 91 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Contributions
  3. Related Work
  4. Model and Preliminaries
  5. Coordination game and potential
  6. Communication model
  7. Binary symmetric channel
  8. Binary erasure channel
  9. Asynchronous log-linear learning
  10. Snapshot and Fast Communication Regimes
  11. Snapshot regime
  12. High-temperature expansion
  13. Fast regime
  14. Fast logit update
  15. Exact Gibbs structure
  16. ...and 22 more sections

Key Result

Theorem 1

The Markov chain with transition matrix $P_\beta^{\mathrm{S}}$ is irreducible and aperiodic. Hence it admits a unique stationary distribution $\pi_\beta^{\mathrm{S}}$.

Figures (4)

  • Figure 1: Means and 95% confidence intervals of the steady-state coordination potential for snapshot and fast communication over BSC links across different network topologies. In the high-temperature regime ( $\beta=0.5$) the two dynamics exhibit comparable performance, as predicted by the first-order approximation. At lower temperatures ($\beta=2$) fast communication significantly outperforms snapshot communication and exhibits reduced variability, owing to channel-averaged updates, whereas snapshot communication displays higher variance due to its dependence on single-shot channel observations.
  • Figure 2: Means and 95% confidence intervals of the steady-state coordination potential for snapshot and fast communication over BEC links across different network topologies. In the high-temperature regime ( $\beta=0.5$) the two dynamics exhibit comparable performance, as predicted by the first-order approximation. At lower temperatures ($\beta=2$) fast communication significantly outperforms snapshot communication and exhibits reduced variability, owing to channel-averaged updates, whereas snapshot communication displays higher variance due to its dependence on single-shot channel observations.
  • Figure 3: Steady-state coordination performance under finite-$K$ communication over BSC links across different network topologies, with the fast (channel-averaged) regime shown as a reference line. The parameter $K$ denotes the number of independent channel uses per neighbor and update. As $K$ increases, finite-$K$ communication progressively averages out channel noise and the resulting performance converges toward the fast regime. The improvement is monotone but exhibits diminishing returns, indicating that moderate values of $K$ capture most of the benefit of expectation-based updates.
  • Figure 4: Steady-state coordination performance under heterogeneous BSC links, where edge reliabilities are drawn from a specified range. As link heterogeneity increases, the performance gap between fast and snapshot communication widens. Fast communication mitigates the effect of unreliable edges through expectation-based updates, while snapshot communication exhibits greater sensitivity to link variability.

Theorems & Definitions (28)

  • Theorem 1
  • proof
  • Lemma 1
  • proof
  • Theorem 2
  • proof
  • Corollary 1
  • proof
  • Remark 1
  • Lemma 2
  • ...and 18 more