arXiv Machine Learning
Jul 22

Linear convergence of proximal descent schemes on the Wasserstein space

arXiv:2411. 15067v2 Announce Type: replace-cross Abstract: We investigate proximal descent methods, inspired by the minimizing movement scheme introduced by Jordan, Kinderlehrer and Otto, for optimizing entropy-regularized functionals on the Wasserstein space.

By Razvan-Andrei Lascu, Mateusz B. Majka, David \v{S}i\v{s}ka, {\L}ukasz Szpruch
arXiv Machine Learning
Sep 17

Wasserstein Formulation of Reinforcement Learning. An Optimal Transport Perspective on Policy Optimization

The paper introduces a geometric framework for reinforcement learning that treats policies as mappings into the Wasserstein space of action probabilities. It establishes a Riemannian structure induced by stationary distributions, defines the tangent space of policies, and characterizes geodesics while addressing measurability concerns. The authors formulate a general RL optimization problem, construct a gradient flow via Otto's calculus, compute the gradient and Hessian of the energy, and demonstrate the approach with numerical examples for low‑dimensional problems and neural‑network‑parameterized policies for high‑dimensional settings.

By Mathias Dus (IRMA)
arXiv Machine Learning
Sep 15

Stochastic Gradient Descent over P2

The paper develops a diffusion approximation for stochastic gradient descent (SGD) when the optimization target is a functional on the Wasserstein space ℝ2. By lifting the problem to a Hilbert space via Lions differentiability, the authors construct a Gaussian random-field approximation whose velocity field matches the mean and covariance of the original stochastic gradient. They prove that this Gaussian approximation achieves second‑order weak accuracy, providing a rigorous basis for replacing sample‑driven randomness with analytically tractable Gaussian fluctuations in stochastic optimization over probability measures.

By Maria Oprea, Qin Li, Yunan Yang