Cross-round convergence guarantee for Monte Carlo GTV-KL

Establish a cross-round convergence guarantee for the GTV-KL algorithm when the Monte Carlo surrogate is recomputed with fresh samples and neighbor parameters evolve across communication rounds.

Background

The GTV-KL coupling uses a Monte Carlo estimate of the Kullback–Leibler divergence between neighboring Gaussian mixture models. With neighbors held fixed during a round, the resulting surrogate is unbiased and smooth, so descent can be shown for each individual round’s surrogate. The unresolved issue is obtaining a convergence result across rounds, where neighbor models change and fresh Monte Carlo samples are drawn.

The paper provides a stationarity guarantee only for the smooth, closed-form GTV-MMD objective under one local step per communication round. Extending an analogous guarantee to the stochastic, evolving-surrogate setting of GTV-KL remains unresolved.

References

For GTV-KL, the Monte-Carlo surrogate eq:kl_mc with frozen neighbors is unbiased and smooth on $\mathcal{W}$, yielding descent of each round's surrogate; a cross-round guarantee, with fresh samples, is left open.

eq:kl_mc:

$D{p^{(i')}}{p^{(i)}} \frac{1}{#1{\mathrm{nbr}}}\!\sum_{r=1}^{#1{\mathrm{nbr}}} \log\frac{p^{(i')}(x^{(r)})}{p^{(i)}(x^{(r)})}, \;\; x^{(r)}\!\sim\! p^{(i')}, $

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