Adaptive Decentralized Composite Optimization via Three-Operator Splitting

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.