Convergence analysis for multiple stale-neighbor local steps

Establish a convergence guarantee for the projected-gradient GTVMin algorithm when each communication round performs more than one local update using stale neighbor parameters.

Background

The algorithm freezes neighboring parameters during a communication round while each node performs multiple local projected-gradient steps. For more than one local step, later updates therefore use increasingly stale neighbor information. The experiments use five local steps as a communication-saving heuristic, but the paper does not establish convergence for this setting.

The formal convergence proposition is restricted to the GTV-MMD instance with exactly one local step per round and a suitable step size. An analysis covering the experimentally used multi-step regime is explicitly identified as open.

References

For $T_{\mathrm{loc}}>1$, later local steps use stale neighbor parameters; the $T_{\mathrm{loc}}=5$ used in the experiments is a communication-saving heuristic, also open.

— Federated Soft Clustering via Generalized Total Variation Minimization  (2609.19202 - Abdurakhmanova et al., 16 Sep 2026) in Section 3, “Algorithm and Convergence”