Adaptive Decentralized Composite Optimization via Three-Operator Splitting

TL;DR

This paper introduces a decentralized optimization method that adaptively adjusts stepsizes using local backtracking and a three-operator splitting approach, achieving robust convergence guarantees.

Contribution

It presents a novel decentralized optimization algorithm with adaptive stepsize adjustment based on a three-operator splitting and a new preconditioning metric.

Findings

Converges sublinearly under convexity.

Achieves linear convergence under strong convexity and partial smoothness.

Numerical results validate the effectiveness of the adaptive stepsize strategy.

Abstract

The paper studies decentralized optimization over networks, where agents minimize a sum of {\it locally} smooth (strongly) convex losses and plus a nonsmooth convex extended value term. We propose decentralized methods wherein agents {\it adaptively} adjust their stepsize via local backtracking procedures coupled with lightweight min-consensus protocols. Our design stems from a three-operator splitting factorization applied to an equivalent reformulation of the problem. The reformulation is endowed with a new BCV preconditioning metric (Bertsekas-O'Connor-Vandenberghe), which enables efficient decentralized implementation and local stepsize adjustments. We establish robust convergence guarantees. Under mere convexity, the proposed methods converge with a sublinear rate. Under strong convexity of the sum-function, and assuming the nonsmooth component is partly smooth, we further prove linear convergence. Numerical experiments corroborate the theory and highlight the effectiveness of the proposed adaptive stepsize strategy.