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

arXiv: https://arxiv.org/abs/2606.28808 · HTML: https://mlsanity.com/paper/2606.28808/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- Builds on 26 of its 56 references that are in the map.
- First posted on arXiv 27 Jun 2026.

## Builds on (most cited first, 26 of 26)

- [Simple and Scalable Predictive Uncertainty Estimation using Deep Ensembles](https://mlsanity.com/paper/1612.01474/) (2016)
- [Non-convex learning via Stochastic Gradient Langevin Dynamics: a nonasymptotic analysis](https://mlsanity.com/paper/1702.03849/) (2017)
- [Theoretical guarantees for approximate sampling from smooth and log-concave densities](https://mlsanity.com/paper/1412.7392/) (2014)
- [Non-asymptotic convergence analysis for the Unadjusted Langevin Algorithm](https://mlsanity.com/paper/1507.05021/) (2015)
- [A Complete Recipe for Stochastic Gradient MCMC](https://mlsanity.com/paper/1506.04696/) (2015)
- [High-dimensional Bayesian inference via the Unadjusted Langevin Algorithm](https://mlsanity.com/paper/1605.01559/) (2016)
- [User-friendly guarantees for the Langevin Monte Carlo with inaccurate gradient](https://mlsanity.com/paper/1710.00095/) (2017)
- [Global Convergence of Langevin Dynamics Based Algorithms for Nonconvex Optimization](https://mlsanity.com/paper/1707.06618/) (2017)
- [Convergence of Langevin MCMC in KL-divergence](https://mlsanity.com/paper/1705.09048/) (2017)
- [Efficient Bayesian computation by proximal Markov chain Monte Carlo: when Langevin meets Moreau](https://mlsanity.com/paper/1612.07471/) (2016)
- [Consistency and fluctuations for stochastic gradient Langevin dynamics](https://mlsanity.com/paper/1409.0578/) (2014)
- [Convergence of Langevin Monte Carlo in Chi-Squared and Renyi Divergence](https://mlsanity.com/paper/2007.11612/) (2020)
- [Global Convergence of Stochastic Gradient Hamiltonian Monte Carlo for Non-Convex Stochastic Optimization: Non-Asymptotic Performance Bounds and Momentum-Based Acceleration](https://mlsanity.com/paper/1809.04618/) (2018)
- [Decentralized Stochastic Gradient Langevin Dynamics and Hamiltonian Monte Carlo](https://mlsanity.com/paper/2007.00590/) (2020)
- [Breaking Reversibility Accelerates Langevin Dynamics for Global Non-Convex Optimization](https://mlsanity.com/paper/1812.07725/) (2018)
- [Excess Risk of Two-Layer ReLU Neural Networks in Teacher-Student Settings and its Superiority to Kernel Methods](https://mlsanity.com/paper/2205.14818/) (2022)
- [Generalized EXTRA stochastic gradient Langevin dynamics](https://mlsanity.com/paper/2412.01993/) (2024)
- [Improving the convergence of reversible samplers](https://mlsanity.com/paper/1601.08118/) (2016)
- [Hessian-Free High-Resolution Nesterov Acceleration for Sampling](https://mlsanity.com/paper/2006.09230/) (2020)
- [Non-Reversible Langevin Algorithms for Constrained Sampling](https://mlsanity.com/paper/2501.11743/) (2025)
- [Non-Convex Optimization via Non-Reversible Stochastic Gradient Langevin Dynamics](https://mlsanity.com/paper/2004.02823/) (2020)
- [Gradient-adjusted underdamped Langevin dynamics for sampling](https://mlsanity.com/paper/2410.08987/) (2024)
- [DIGing--SGLD: Decentralized and Scalable Langevin Sampling over Time--Varying Networks](https://mlsanity.com/paper/2511.12836/) (2025)
- [Regime-Switching Langevin Monte Carlo Algorithms](https://mlsanity.com/paper/2509.00941/) (2025)
- [Sampling non-log-concave densities via Hessian-free high-resolution dynamics](https://mlsanity.com/paper/2601.02725/) (2026)
- [Accelerating Constrained Sampling: A Large Deviations Approach](https://mlsanity.com/paper/2506.07816/) (2025)

## Built on by (most cited first, 2 of 2)

- [Improved Analysis for Hessian-free High-resolution Monte Carlo Sampling](https://mlsanity.com/paper/2608.25052/) (2026)
- [Penalized Nonreversible Langevin for Constrained Sampling](https://mlsanity.com/paper/2609.25381/) (2026)

## Abstract (arXiv)

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.