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

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.

💡 Research Summary

The paper addresses the problem of accelerating decentralized optimization over directed communication graphs by dynamically adapting the edge weights that govern inter‑agent message exchanges. Traditional directed decentralized gradient descent (Di‑DGD) relies on a fixed weighted adjacency matrix A; its convergence speed is heavily influenced by the spectral gap of A and the heterogeneity of local data, often leading to slow progress when the graph is poorly conditioned or data are highly non‑i.i.d.

To overcome these limitations, the authors propose a novel framework called Dynamic Directed Decentralized Gradient (D³GD). The core idea is to treat the edge‑weight matrix as a time‑varying variable that is jointly optimized with the primary decision variables θ. Starting from the standard Di‑DGD recursion, they introduce a Lyapunov‑type “cost‑to‑go” function L_k that captures both the global objective value at the weighted average of the local parameters and a consensus error term. By analyzing the one‑step decrease of L_k, they isolate a term that depends solely on the adjacency matrix A. This term is formalized as a design function J_k(A;Θ_k), where Θ_k stacks all local parameters at iteration k. J_k consists of two parts: (i) a quadratic term measuring the deviation of the current parameters from the Perron‑weighted average, and (ii) a cross term that couples the current parameters with the local gradients. Minimizing J_k would maximize the decrease of L_k, thereby accelerating convergence.

Directly solving the minimization problem at each iteration is infeasible for three reasons: (1) the resulting A may violate the irreducibility and stochasticity assumptions required for Di‑DGD; (2) the Perron eigenvector π_A would change abruptly, breaking the analysis; (3) J_k depends on global information (all Θ_k, all gradients). The authors therefore design a practical algorithm that (a) updates A on a slower time‑scale using a convex combination with an initial feasible matrix A_0 (parameter δ controls the mixing), ensuring that the stochasticity and connectivity properties are preserved; and (b) performs a coordinate‑wise projected gradient descent on each row of A, which corresponds to the communication strategy of a single agent. The gradient of J_k with respect to a row a_i involves terms that require global averages (π_A^TΘ_k and π_A^T∇F_k). To compute these in a fully decentralized way, the authors introduce two dynamic consensus trackers, z_k and q_k, that respectively estimate the Perron‑weighted average of the parameters and of the gradients. These trackers follow standard push‑sum‑like updates (equations (15)–(16)) and converge to the desired averages as A evolves slowly.

Algorithm 1 (D³GD) thus alternates between (i) a Di‑DGD step using the current A_k, (ii) a mixing step A_k←(1−δ)A_k+δA_0, and (iii) a row‑wise projected gradient update a_i←Π_{A_G,i}