Recent work \cite{ arifgroup} introduced Federated Proximal Gradient \textbf{ (\texttt{ FedProxGrad})} for solving non-convex composite optimization problems in group fair federated learning. However, the original analysis established convergence only to a \textit{ noise-dominated neighborhood of stationarity}, with explicit dependence on a variance-induced noise floor. In this work, we provide an improved asymptotic convergence analysis for a generalized \texttt{ FedProxGrad}-type analytical framework with inexact local proximal solutions and explicit fairness regularization. We call this extended analytical framework \textbf{ DS \texttt{ FedProxGrad}} (Decay Step Size \texttt{ FedProxGrad}). Under a Robbins-Monro step-size schedule \cite{ robbins1951stochastic} and a mild decay condition on local inexactness, we prove that $\liminf_{ r\to\infty} \mathbb{ E}[\|\nabla F(\mathbf{ x}^r)\|^2] = 0$, i.e., the algorithm is asymptotically stationary and the convergence rate does not depend on a variance-induced noise floor.
Categories: cs.LG stat.ML