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”

Extending them would require replacing di(wt) by the direction that client i actually contributes to the aggregate, namely the average of its τ local LP directionsut i := τ −1 Pτ −1k=0 di(wt i,k), taking ¯ut = Pi ρiut i as the shared direction with effective step αητ (αthe server learning rate of Algorithm 1), and imposing the descent-margin condition on ut i evaluated at the shared point wt. We do not carry this out here, because the LP direction di(·) is a solution map of a linear program whose active set changes with w and is in general not continuous in w; the drift ∥ut i − di(wt)∥ therefore cannot be bounded by O(L0τ ηB) or folded into ∆t i without an additional stability argument for the LP solution, and we make no such claim.

— Collaborative Personalized Preference Alignment for LLMs under Data Deficiency  (2610.05898 - Yang et al., 5 Oct 2026) in Appendix H.4, paragraph following Equation (55), p. 29