Adaptive Decentralized Composite Optimization via Three-Operator Splitting
This work addresses decentralized composite optimization for networked agents, offering incremental improvements in adaptive stepsize strategies.
The paper tackles decentralized optimization over networks with composite objectives, proposing an adaptive stepsize method via three-operator splitting and a new preconditioning metric, achieving sublinear convergence under convexity and linear convergence under strong convexity with numerical validation.
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.