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

2026-06-27

AI Summary

This work addresses the high variance of estimators in stochastic gradient generalized irreversible Langevin Monte Carlo by proposing a novel sampling method that integrates antisymmetric perturbations with stochastic gradients. By analyzing the leading-order fluctuations of the stochastic gradient Euler–Maruyama estimator under irreversible Langevin dynamics, the study provides the first rigorous proof that such perturbations significantly reduce asymptotic variance in the small-step-size limit and establishes a corresponding central limit theorem. The theoretical framework combines Poisson equations, operator theory, and closed-form Gaussian computations, and extends naturally to Hessian-free and other augmented-state systems. Numerical experiments on Bayesian linear and logistic regression tasks demonstrate that the proposed method consistently achieves lower root-mean-square error and substantially improves sampling efficiency compared to reversible baselines.

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.

Problem

• variance reduction
• stochastic gradient
• non-reversible Langevin dynamics
• asymptotic variance
• Monte Carlo sampling

Innovation

• variance reduction
• non-reversible Langevin dynamics
• stochastic gradient Monte Carlo
• central limit theorem
• Poisson equation