Decentralized Non-Convex Learning with Linearly Coupled Constraints

TL;DR

This paper introduces a novel decentralized algorithm for solving nonconvex optimization problems with general linear constraints across multi-agent networks, achieving convergence to approximate solutions efficiently.

Contribution

It proposes the proximal dual consensus (PDC) algorithm, the first to handle nonconvex problems with general linear constraints in a decentralized setting.

Findings

Converges to an $ ext{epsilon}$-KKT solution within $ ext{O}(1/ ext{epsilon})$ iterations.

Allows cheap gradient descent per iteration without affecting convergence order.

Demonstrates good performance on regression and classification tasks with partial data observations.

Abstract

Motivated by the need for decentralized learning, this paper aims at designing a distributed algorithm for solving nonconvex problems with general linear constraints over a multi-agent network. In the considered problem, each agent owns some local information and a local variable for jointly minimizing a cost function, but local variables are coupled by linear constraints. Most of the existing methods for such problems are only applicable for convex problems or problems with specific linear constraints. There still lacks a distributed algorithm for such problems with general linear constraints and under nonconvex setting. In this paper, to tackle this problem, we propose a new algorithm, called "proximal dual consensus" (PDC) algorithm, which combines a proximal technique and a dual consensus method. We build the theoretical convergence conditions and show that the proposed PDC algorithm can converge to an $\epsilon$-Karush-Kuhn-Tucker solution within $\mathcal{O}(1/\epsilon)$ iterations. For computation reduction, the PDC algorithm can choose to perform cheap gradient descent per iteration while preserving the same order of $\mathcal{O}(1/\epsilon)$ iteration complexity. Numerical results are presented to demonstrate the good performance of the proposed algorithms for solving a regression problem and a classification problem over a network where agents have only partial observations of data features.