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Decentralized Learning with Dynamically Refined Edge Weights: A Data-Dependent Framework

Rongxing Du, Hoi-To Wai

TL;DR

The paper addresses decentralized optimization over directed graphs by proposing D$^3$GD, a data-dependent dynamic edge-weight refinement method. It builds a cost-to-go–based Lyapunov function $L_k$ and a design function $J_k$ to guide weight updates, enabling faster, topology-aware convergence with a fully decentralized implementation. A new finite-time convergence analysis for Di-DGD on smooth (possibly non-convex) objectives is established, and numerical experiments show 30-40% speedups with edge weights adapting to data similarity. The work demonstrates practical acceleration for multi-agent learning on directed networks and reveals a link between data heterogeneity and learned topology.

Abstract

This paper aims to accelerate decentralized optimization by strategically designing the edge weights used in the agent-to-agent message exchanges. We propose a Dynamic Directed Decentralized Gradient (D3GD) framework and show that the proposed data-dependent framework is a practical alternative to the classical directed DGD (Di-DGD) algorithm for learning on directed graphs. To obtain a strategy for edge weights refinement, we derive a design function inspired by the cost-to-go function in a new convergence analysis for Di-DGD. This results in a data-dependent dynamical design for the edge weights. A fully decentralized version of D3GD is developed such that each agent refines its communication strategy using only neighbor's information. Numerical experiments show that D3GD accelerates convergence towards stationary solution by 30-40\% over Di-DGD, and learns edge weights that adapt to data similarity.

Decentralized Learning with Dynamically Refined Edge Weights: A Data-Dependent Framework

TL;DR

The paper addresses decentralized optimization over directed graphs by proposing DGD, a data-dependent dynamic edge-weight refinement method. It builds a cost-to-go–based Lyapunov function and a design function to guide weight updates, enabling faster, topology-aware convergence with a fully decentralized implementation. A new finite-time convergence analysis for Di-DGD on smooth (possibly non-convex) objectives is established, and numerical experiments show 30-40% speedups with edge weights adapting to data similarity. The work demonstrates practical acceleration for multi-agent learning on directed networks and reveals a link between data heterogeneity and learned topology.

Abstract

This paper aims to accelerate decentralized optimization by strategically designing the edge weights used in the agent-to-agent message exchanges. We propose a Dynamic Directed Decentralized Gradient (D3GD) framework and show that the proposed data-dependent framework is a practical alternative to the classical directed DGD (Di-DGD) algorithm for learning on directed graphs. To obtain a strategy for edge weights refinement, we derive a design function inspired by the cost-to-go function in a new convergence analysis for Di-DGD. This results in a data-dependent dynamical design for the edge weights. A fully decentralized version of D3GD is developed such that each agent refines its communication strategy using only neighbor's information. Numerical experiments show that D3GD accelerates convergence towards stationary solution by 30-40\% over Di-DGD, and learns edge weights that adapt to data similarity.
Paper Structure (10 sections, 3 theorems, 47 equations, 2 figures, 1 algorithm)

This paper contains 10 sections, 3 theorems, 47 equations, 2 figures, 1 algorithm.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Dynamic Di-DGD Algorithm
  4. Centralized Dynamic Algorithm
  5. Practical D$^3$GD Algorithms
  6. Numerical Experiments
  7. Conclusion
  8. Preliminary Lemmas
  9. Proof of Theorem 2.5
  10. Proof of Eq. (8)

Key Result

Theorem 2.5

Under Assumptions assu:strong_connect-assu:data_hetero. Define $C_{\pi 1} = \sum_{i=1}^n (1-\pi_i)^2 + (n-1) \pi_i^2$, $C_{\pi 2} = \sum_{i=1}^n \frac{1}{\pi^{2}_i}$. Let $\gamma_k \leq \frac{n\rho_{\bm A}}{2L\sqrt{C_{\pi1}(C_{\pi2}+C_0)}}$ for some constant $C_0$. Then, for any $T \geq 1$, the iter where $\widehat{\bm{\theta}}^k = \sum_{i=1}^n \pi_{i, {\bm A}} \bm{\theta}_i^k$ is the weighted ave

Figures (2)

  • Figure 1: Convergence of D$^3$GD and Di-DGD on sigmoid loss minimization. (Left) Stationarity measure $\frac{1}{n} \sum_{i=1}^n \| {\nabla} F( \bm{\theta}_i^k ) \|^2$. (Right) Total disagreement $\frac{1}{n^2}\sum_{i,j} \| \bm{\theta}_i^t - \bm{\theta}_j^t \|^2$.
  • Figure 2: Visualizations of the edge weights.

Theorems & Definitions (4)

  • Theorem 2.5
  • Lemma A.1
  • Lemma A.2
  • proof