Dual Averaging for Distributed Optimization: Convergence Analysis and Network Scaling

The goal of decentralized optimization over a network is to optimize a global objective formed by a sum of local (possibly nonsmooth) convex functions using only local computation and communication. It arises in various application domains, including distributed tracking and localization, multi-agent co-ordination, estimation in sensor networks, and large-scale optimization in machine learning. We develop and analyze distributed algorithms based on dual averaging of subgradients, and we provide sharp bounds on their convergence rates as a function of the network size and topology. Our method of analysis allows for a clear separation between the convergence of the optimization algorithm itself and the effects of communication constraints arising from the network structure. In particular, we show that the number of iterations required by our algorithm scales inversely in the spectral gap of the network. The sharpness of this prediction is confirmed both by theoretical lower bounds and simulations for various networks. Our approach includes both the cases of deterministic optimization and communication, as well as problems with stochastic optimization and/or communication.

💡 Research Summary

The paper addresses the fundamental problem of decentralized convex optimization in which a global objective is expressed as the sum of local convex (possibly nonsmooth) functions, each owned by a node in a communication network. The authors propose a distributed algorithm based on the dual‑averaging framework, originally introduced for centralized subgradient methods, and adapt it to a network setting by interleaving local subgradient accumulation with a consensus‑type averaging step.

Algorithmic structure. At each iteration every node i computes a subgradient g_i(t) of its local function f_i at the current estimate x_i(t). It then updates an accumulated dual variable z_i(t) = ∑_{s≤t} g_i(s). The communication step mixes these dual variables with neighbors using a symmetric, doubly‑stochastic weight matrix W that respects the graph topology: z_i(t+½) = ∑j W{ij} z_j(t). Finally, each node solves a proximal problem with a common strongly convex regularizer ψ (e.g., ½‖·‖²) and a stepsize α_t to obtain the next primal iterate x_i(t+1). The same ψ and stepsize schedule are used by all nodes, which greatly simplifies implementation.

Convergence analysis. The authors separate the error into two components: (1) the optimization error that would appear even in a fully centralized setting, and (2) the network error caused by imperfect consensus. By leveraging standard dual‑averaging bounds, they show that the optimization term decays as O(1/√T). The network term is governed by the spectral gap λ = 1 − σ₂(W) of the mixing matrix; specifically, the total error after T iterations satisfies

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