Stochastic Gradient Descent over P2
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 Gau...
| Authors | Maria Oprea, Qin Li, Yunan Yang |
|---|---|
| Primary category | stat.ML |
| Published | 11 September 2026 |
| Last updated at source | 15 September 2026 |
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