Stochastic Gradient Descent over P2

• URL: http://arxiv.org/abs/2609.13343v2
• Date: Tue, 15 Sep 2026 09:23:52 GMT
• Title: Stochastic Gradient Descent over P2
• Abstract summary: gradient descent (SGD) admits diffusion approximations that replace the complicated randomness of gradients by Gaussian noise.<n>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.
• Score: 4.6107344298094235
• License: http://creativecommons.org/publicdomain/zero/1.0/
• 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.

Related papers

• Probabilistic Gaussian Homotopy: A Probability-Space Continuation Framework for Nonconvex Optimization [5.977181405597602]
  We introduce Probabilistic Homotopy (PGH), a continuation probability for non Gaussian generalizations.<n>We show that PGH smooths regions at scale $0$ and recovers original objective as $ 0 original directions.
  arXiv  Detail & Related papers  (2026-03-13T19:32:29Z)
• Revisiting Zeroth-Order Optimization: Minimum-Variance Two-Point Estimators and Directionally Aligned Perturbations [57.179679246370114]
  We identify the distribution of random perturbations that minimizes the estimator's variance as the perturbation stepsize tends to zero.<n>Our findings reveal that such desired perturbations can align directionally with the true gradient, instead of maintaining a fixed length.
  arXiv  Detail & Related papers  (2025-10-22T19:06:39Z)
• Hessian-guided Perturbed Wasserstein Gradient Flows for Escaping Saddle Points [54.06226763868876]
  Wasserstein flow (WGF) is a common method to perform optimization over the space of measures.<n>We show that PWGF converges to a global optimum in terms of general non objectives.
  arXiv  Detail & Related papers  (2025-09-21T08:14:20Z)
• Gaussian Approximation and Multiplier Bootstrap for Stochastic Gradient Descent [14.19520637866741]
  We establish the non-asymptotic validity of the multiplier bootstrap procedure for constructing confidence sets.<n>We derive approximation rates in convex distance of order up to $1/sqrtn$.
  arXiv  Detail & Related papers  (2025-02-10T17:49:05Z)
• Gradient-Based Non-Linear Inverse Learning [2.6149030745627644]
  We study statistical inverse learning in the context of nonlinear inverse problems under random design.<n>We employ gradient descent (GD) and descent gradient (SGD) with mini-batching, both using constant step sizes.<n>Our analysis derives convergence rates for both algorithms under classical a priori assumptions on the smoothness of the target function.
  arXiv  Detail & Related papers  (2024-12-21T22:38:17Z)
• Limit Theorems for Stochastic Gradient Descent with Infinite Variance [51.4853131023238]
  We show that the gradient descent algorithm can be characterized as the stationary distribution of a suitably defined Ornstein-rnstein process driven by an appropriate L'evy process.<n>We also explore the applications of these results in linear regression and logistic regression models.
  arXiv  Detail & Related papers  (2024-10-21T09:39:10Z)
• Nonasymptotic Analysis of Stochastic Gradient Descent with the Richardson-Romberg Extrapolation [22.652143194356864]
  We address the problem of solving strongly convex and smooth problems using gradient descent (SGD) with a constant step size.<n>We provide an expansion of the mean-squared error of the resulting estimator with respect to the number of iterations $n$.<n>Our analysis relies on the properties of the SGDs viewed as a time-homogeneous Markov chain.
  arXiv  Detail & Related papers  (2024-10-07T15:02:48Z)
• Sampling from Gaussian Process Posteriors using Stochastic Gradient Descent [43.097493761380186]
  gradient algorithms are an efficient method of approximately solving linear systems. We show that gradient descent produces accurate predictions, even in cases where it does not converge quickly to the optimum. Experimentally, gradient descent achieves state-of-the-art performance on sufficiently large-scale or ill-conditioned regression tasks.
  arXiv  Detail & Related papers  (2023-06-20T15:07:37Z)
• Improved Convergence Rate of Stochastic Gradient Langevin Dynamics with Variance Reduction and its Application to Optimization [50.83356836818667]
  gradient Langevin Dynamics is one of the most fundamental algorithms to solve non-eps optimization problems. In this paper, we show two variants of this kind, namely the Variance Reduced Langevin Dynamics and the Recursive Gradient Langevin Dynamics.
  arXiv  Detail & Related papers  (2022-03-30T11:39:00Z)
• Nonconvex Stochastic Scaled-Gradient Descent and Generalized Eigenvector Problems [98.34292831923335]
  Motivated by the problem of online correlation analysis, we propose the emphStochastic Scaled-Gradient Descent (SSD) algorithm. We bring these ideas together in an application to online correlation analysis, deriving for the first time an optimal one-time-scale algorithm with an explicit rate of local convergence to normality.
  arXiv  Detail & Related papers  (2021-12-29T18:46:52Z)
• On the Double Descent of Random Features Models Trained with SGD [78.0918823643911]
  We study properties of random features (RF) regression in high dimensions optimized by gradient descent (SGD) We derive precise non-asymptotic error bounds of RF regression under both constant and adaptive step-size SGD setting. We observe the double descent phenomenon both theoretically and empirically.
  arXiv  Detail & Related papers  (2021-10-13T17:47:39Z)

This list is automatically generated from the titles and abstracts of the papers in this site.

This site does not guarantee the quality of this site (including all information) and is not responsible for any consequences.