Composite Optimization in Open Multi-Agent Systems under Additive Errors

This paper studies distributed composite optimization in open multi-agent systems, where agents may join and leave the network over time. We propose a variant of Open ADMM in which the composite proximal step is approximated through a finite number of proximal-gradient iterations, enabling agents to handle composite objectives even when the exact proximal operator is not available in closed form. Since the inexact computation of the proximal step induces additive errors, we establish a convergence result for paracontractive open multi-agent systems subject to bounded additive perturbations that capture both inexact local updates and noisy communications. Under mild assumptions on the network topology, the proposed algorithm converges linearly to a neighborhood of the optimal consensus set, with an explicit bound on the steady-state tracking error. The theoretical guarantees are corroborated by numerical experiments on distributed learning applications, namely linear regression (supervised learning) and principal component analysis (unsupervised learning).