Stochastic Gradient Descent over P2

Abstract:Stochastic gradient descent (SGD) admits diffusion approximations that replace the complicated randomness of stochastic gradients by Gaussian noise, providing a powerful tool for understanding its dynamics and long-time behavior. We investigate whether an analogous approximation principle holds for optimization over probability measures, where the objective is a functional defined on the Wasserstein space P2. The nonlinear geometry and infinite-dimensional nature of P2 prevent a direct extension of the classical Euclidean theory. Using Lions differentiability, we lift the problem to a linear Hilbert space, where higher-order differential calculus becomes available. We then construct a Gaussian random-field approximation whose velocity field matches the mean and covariance of the original stochastic gradient. By exploiting this moment matching through higher-order Taylor expansions, we show that the Gaussian approximation captures the SGD dynamics with second-order weak accuracy. Our result provides a rigorous foundation for replacing sample-driven randomness by analytically tractable Gaussian fluctuations in stochastic optimization over probability measures.

Machine Learning (stat.ML); Machine Learning (cs.LG); Probability (math.PR)

arXiv:2609.13343 [stat.ML]

(or arXiv:2609.13343v1 [stat.ML] for this version)

https://doi.org/10.48550/arXiv.2609.13343 Focus to learn more arXiv-issued DOI via DataCite (pending registration)

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