An Inexact Semi-Proximal ALM with Adaptive Communication-Free Local Solves for Decentralized Composite Optimization

Decentralized optimization enables agents to collaboratively solve learning and decision-making problems using only local computation and neighbor communication. In many practical networks, communication is substantially more expensive than local computation, motivating algorithms that perform multiple local updates between successive exchanges. Augmented Lagrangian methods provide an effective framework for composite optimization, but their decentralized subproblems are typically coupled through the network and may require repeated communication during the inner iterations. To alleviate this issue, we propose CFI-spALM, a Communication-Free Inexact Semi-Proximal Augmented Lagrangian Method that uses a structured semi-proximal term to decompose the coupled subproblem into independent nodewise problems. Each agent adaptively performs local proximal-gradient updates under an absolute or relative stopping rule, requiring only one model-vector exchange per outer iteration. For convex composite objectives, we establish global convergence, sublinear communication complexity, and linear convergence under metric subregularity. Experiments on logistic regression and rank-deficient LASSO demonstrate that CFI-spALM can achieve the prescribed optimization target with fewer model-vector exchanges than the comparison methods.