Double Averaging and Gradient Projection: Convergence Guarantees for Decentralized Constrained Optimization

TL;DR

This paper introduces DAGP, a decentralized algorithm for constrained optimization over directed networks, with proven convergence guarantees and applications to unconstrained problems and optimal transport, supported by theoretical analysis and numerical experiments.

Contribution

The paper proposes DAGP, a novel decentralized algorithm with a new convergence analysis methodology, applicable to constrained and unconstrained problems, including optimal transport.

Findings

DAGP achieves sub-linear convergence rates without strong convexity.

The new aggregate lower-bounding methodology simplifies convergence analysis.

Numerical results show DAGP's superior performance in optimal transport tasks.

Abstract

We consider a generic decentralized constrained optimization problem over static, directed communication networks, where each agent has exclusive access to only one convex, differentiable, local objective term and one convex constraint set. For this setup, we propose a novel decentralized algorithm, called DAGP (Double Averaging and Gradient Projection), based on local gradients, projection onto local constraints, and local averaging. We achieve global optimality through a novel distributed tracking technique we call distributed null projection. Further, we show that DAGP can be used to solve unconstrained problems with non-differentiable objective terms with a problem reduction scheme. Assuming only smoothness of the objective terms, we study the convergence of DAGP and establish sub-linear rates of convergence in terms of feasibility, consensus, and optimality, with no extra assumption (e.g. strong convexity). For the analysis, we forego the difficulties of selecting Lyapunov functions by proposing a new methodology of convergence analysis in optimization problems, which we refer to as aggregate lower-bounding. To demonstrate the generality of this method, we also provide an alternative convergence proof for the standard gradient descent algorithm with smooth functions. Finally, we present numerical results demonstrating the effectiveness of our proposed method in both constrained and unconstrained problems. In particular, we propose a distributed scheme by DAGP for the optimal transport problem with superior performance and speed.