Stochastic optimization on matrices and a graphon McKean–Vlasov limit

We study stochastic gradient descent for functions of large symmetric matrices that are unchanged by simultaneous permutations of rows and columns. As matrix dimensions grow with bounded entries, the optimization trajectories converge to deterministic curves of graphons. In the small-noise regime, the limit is a graphon gradient flow. With suitably scaled reflected Brownian noise, the limit is described by reflected stochastic differential equations, extending the classical McKean–Vlasov framework to graphons. The analysis develops exchangeable arrays of reflected diffusions and a corresponding notion of propagation of chaos for large matrices.

Published in The Annals of Applied Probability, 36(1), 355–392, February 2026. First available online on 18 February 2026.

Resources:

1. Journal article.
2. arXiv preprint.
3. BibTeX citation.