Variance Reduction for Stochastic Gradient Generalized Non-reversible Langevin Monte Carlo Algorithms

• URL: http://arxiv.org/abs/2606.28808v1
• Date: Sat, 27 Jun 2026 08:25:48 GMT
• Title: Variance Reduction for Stochastic Gradient Generalized Non-reversible Langevin Monte Carlo Algorithms
• Abstract summary: We study the leading-order fluctuation of gradient-Maruyama estimators for non-reversible Langevin dynamics.<n>We prove that the empirical average over a horizon of order the inverse squared stepsize satisfies a central limit theorem in the vanishing-stepsize regime.
• Score: 7.336637417508747
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: We study the leading-order fluctuation of stochastic gradient Euler-Maruyama estimators for generalized non-reversible Langevin dynamics. Under structural assumptions tailored to the small-stepsize central limit theorem and under an unbiased stochastic gradient oracle, we prove that the empirical average over a horizon of order the inverse squared stepsize satisfies a central limit theorem in the vanishing-stepsize regime. The limiting variance is characterized through the Poisson equation of the limiting full-gradient diffusion. We then rewrite this constant in an operator form that links it to the continuous-time asymptotic variance and, under standard operator-theoretic assumptions, derive a sufficient condition under which an anti-symmetric perturbation strictly reduces the leading-order fluctuation constant relative to the reversible baseline. We also identify bounded smooth predictive observables that re directly covered by the main theorem. As a separate Gaussian calculation beyond the bounded-test-function regime, we obtain closed-form formulas for quadratic Hamiltonians and linear observables. The framework covers non-reversible Langevin dynamics and augmented-state examples including Hessian-free high-resolution dynamics and a positive-definite subclass of gradient-adjusted underdamped Langevin dynamics that allow stochastic gradients. Numerical experiments on basic examples and Bayesian linear regression using synthetic data, and Bayesian logistic regression using real data support the predicted Gaussian fluctuations and show that the non-reversible schemes consistently reduce the root mean squared error (RMSE) relative to their reversible baselines.

Related papers

• Residual-Based Time Discretization on Nonlinear Approximation Manifolds: Analysis and Gaussian Applications [0.0]
  We study time-discrete parametric approximations of evolution equations in Hilbert spaces based on residual minimization.<n>Two natural residual formulations are considered: discretization followed by parametrization of the evolution equation, and discretization of the Dirac--Frenkel variational principle governing the parameter dynamics.
  arXiv  Detail & Related papers  (2026-07-16T14:57:44Z)
• A Mean-Field Analysis of Multi-Head Self-Attention under Cross-Entropy Training [0.0]
  This paper develops a mean-field theory for a single-layer causal multi-head self-attention model trained by cross-entropy minimization.<n>In the infinite-head limit, the averaged attention logits define a risk functional on probability measures, whose first variation generates a nonlinear Wasserstein gradient-flow equation.<n>We study the long-time behavior of the PDE: energy dissipation, convergence to the stationary set under compactness, convergence to a single stationary measure under topological or Kurdyka--ojasiewicz assumptions, and explicit convergence rates under gradient-domination conditions.
  arXiv  Detail & Related papers  (2026-06-09T06:38:27Z)
• On Higher-Order Geometric Refinements of Classical Covariance Asymptotics: An Approach via Intrinsic and Extrinsic Information Geometry [0.0]
  In curved models, including mixtures, curved exponential families, latent-variable models, and manifold-root parameter spaces, finite-sample behavior can deviate systematically from predictions.<n>We develop a coordinate-invariant, curvature-aware refinement by viewing a regular parametric family as a ((,g)) with Fisher-constrained-Rao metric.<n>We describe the resolved metric, the role of the real log canonical threshold in learning rates and posterior mean-squared error, and a curvature-based covariance expansion on the resolved space that recovers the regular theory as a special case.
  arXiv  Detail & Related papers  (2026-04-14T13:40:13Z)
• On the Convergence of Stochastic Gradient Descent with Perturbed Forward-Backward Passes [15.63629978994481]
  We present the first comprehensive theoretical analysis of this gradient cascade setting.<n>We identify conditions under which perturbations do not deteriorate the gradient convergence order.
  arXiv  Detail & Related papers  (2026-02-24T07:47:15Z)
• 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)
• Graph-based Clustering Revisited: A Relaxation of Kernel $k$-Means Perspective [73.18641268511318]
  We propose a graph-based clustering algorithm that only relaxes the orthonormal constraint to derive clustering results.<n>To ensure a doubly constraint into a gradient, we transform the non-negative constraint into a class probability parameter.
  arXiv  Detail & Related papers  (2025-09-23T09:14:39Z)
• 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)
• Nonparametric regression on random geometric graphs sampled from submanifolds [2.3326951882644553]
  We analyze the frequentist behaviour of the posterior distribution arising from Bayesian priors designed through random basis expansion. We prove that the posterior contraction rates of such methods are minimax optimal (up to logarithmic factors) for any positive smoothness index.
  arXiv  Detail & Related papers  (2024-05-31T15:18:44Z)
• Last-Iterate Convergence of Adaptive Riemannian Gradient Descent for Equilibrium Computation [52.73824786627612]
  This paper establishes new convergence results for textitgeodesic strongly monotone games.<n>Our key result shows that RGD attains last-iterate linear convergence in a textitgeometry-agnostic fashion.<n>Overall, this paper presents the first geometry-agnostic last-iterate convergence analysis for games beyond the Euclidean settings.
  arXiv  Detail & Related papers  (2023-06-29T01:20:44Z)
• Implicit Bias of Gradient Descent for Logistic Regression at the Edge of Stability [69.01076284478151]
  In machine learning optimization, gradient descent (GD) often operates at the edge of stability (EoS) This paper studies the convergence and implicit bias of constant-stepsize GD for logistic regression on linearly separable data in the EoS regime.
  arXiv  Detail & Related papers  (2023-05-19T16:24:47Z)
• 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.