← Back to all news
arXiv Machine Learning

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.

Related stories

arXiv Statistics ML
Oct 2

Convergence Analysis of the Wasserstein Proximal Algorithm beyond Geodesic Convexity

The paper presents a convergence analysis of the Wasserstein proximal algorithm without requiring geodesic convexity of the objective functional. By assuming a Wasserstein analog of the Euclidean Polyak–Łojasiewicz inequality, the authors prove that the algorithm achieves unbiased, linear convergence, improving on previous rates that required strong geodesic convexity. The analysis is extended to inexact proximal steps for geodesically semiconvex objectives, and numerical experiments show faster convergence than noisy gradient descent on mean‑field neural networks.

By Shuailong Zhu, Xiaohui Chen
More like this →
arXiv Machine Learning
Sep 14

Almost Sure Convergence Analysis of Stochastic Gradient Methods with Clipping and Additive Noise

The paper proves that stochastic gradient descent with gradient clipping and additive Gaussian noise (SGD‑CN) converges almost surely under smoothness and bounded noise assumptions, given standard decaying step sizes. The analysis extends to momentum variants such as the stochastic heavy ball and Nesterov's accelerated gradient, showing that careful energy constructions yield similar guarantees. These results provide stronger theoretical foundations for understanding the pathwise behaviour of clipped stochastic gradient methods in both convex and nonconvex regimes.

By Amartya Mukherjee, Jun Liu
safety
More like this →
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)
reinforcement-learning
More like this →
arXiv Statistics ML
Oct 26

Stochastic Inverse Problem: stability, regularization and Wasserstein gradient flow

The paper investigates stochastic inverse problems where the goal is to recover a probability distribution of an unknown parameter that explains observed data. It examines direct inversion, a variational formulation with regularization, and optimization via Wasserstein gradient flows, highlighting parallels and differences with deterministic inverse problems. A key finding is that the choice of metric—both in the loss function and during optimization—strongly influences stability and the optimizer’s properties.

By Qin Li, Maria Oprea, Li Wang, Yunan Yang
More like this →
Hugging Face Trending Papers
Aug 6

The Tamed Subgradient Unadjusted Langevin Algorithm beyond Convexity

We study the problem of sampling from target distributions whose potentials are simultaneously non-smooth, subject to superlinear gradient growth, and non-convex. We introduce the Subgradient Tamed Unadjusted Langevin Algorithm (SG-TULA), a discretisation of the Langevin diffusion that operates directly on subgradients, without relying on computationally demanding smoothing procedures.

llmsdiffusion
More like this →