# Improved Convergence Rate of Stochastic Gradient Langevin Dynamics with Variance Reduction and its Application to Optimization

arXiv: https://arxiv.org/abs/2203.16217 · HTML: https://mlsanity.com/paper/2203.16217/

## Key facts

- Cited by 16 papers in the map (data through 14 Oct 2026).
- Builds on 15 of its 50 references that are in the map.
- First posted on arXiv 30 Mar 2022.

## Builds on (most cited first, 15 of 15)

- [Non-convex learning via Stochastic Gradient Langevin Dynamics: a nonasymptotic analysis](https://mlsanity.com/paper/1702.03849/) (2017)
- [Rapid Convergence of the Unadjusted Langevin Algorithm: Isoperimetry Suffices](https://mlsanity.com/paper/1903.08568/) (2019)
- [SARAH: A Novel Method for Machine Learning Problems Using Stochastic Recursive Gradient](https://mlsanity.com/paper/1703.00102/) (2017)
- [Theoretical guarantees for approximate sampling from smooth and log-concave densities](https://mlsanity.com/paper/1412.7392/) (2014)
- [Sampling as optimization in the space of measures: The Langevin dynamics as a composite optimization problem](https://mlsanity.com/paper/1802.08089/) (2018)
- [Global Convergence of Langevin Dynamics Based Algorithms for Nonconvex Optimization](https://mlsanity.com/paper/1707.06618/) (2017)
- [Towards a Theory of Non-Log-Concave Sampling: First-Order Stationarity Guarantees for Langevin Monte Carlo](https://mlsanity.com/paper/2202.05214/) (2022)
- [ProxSARAH: An Efficient Algorithmic Framework for Stochastic Composite Nonconvex Optimization](https://mlsanity.com/paper/1902.05679/) (2019)
- [Faster Convergence of Stochastic Gradient Langevin Dynamics for Non-Log-Concave Sampling](https://mlsanity.com/paper/2010.09597/) (2020)
- [SSRGD: Simple Stochastic Recursive Gradient Descent for Escaping Saddle Points](https://mlsanity.com/paper/1904.09265/) (2019)
- [Riemannian Langevin Algorithm for Solving Semidefinite Programs](https://mlsanity.com/paper/2010.11176/) (2020)
- [Stochastic Recursive Gradient Algorithm for Nonconvex Optimization](https://mlsanity.com/paper/1705.07261/) (2017)
- [On the Theory of Variance Reduction for Stochastic Gradient Monte Carlo](https://mlsanity.com/paper/1802.05431/) (2018)
- [Stochastic Gradient Langevin Dynamics with Variance Reduction](https://mlsanity.com/paper/2102.06759/) (2021)
- [Approximation to stochastic variance reduced gradient Langevin dynamics by stochastic delay differential equations](https://mlsanity.com/paper/2106.04357/) (2021)

## Built on by (most cited first, 16 of 16)

- [Convergence of mean-field Langevin dynamics: Time and space discretization, stochastic gradient, and variance reduction](https://mlsanity.com/paper/2306.07221/) (2023)
- [Algorithmic Stability of Heavy-Tailed SGD with General Loss Functions](https://mlsanity.com/paper/2301.11885/) (2023)
- [Time-Independent Information-Theoretic Generalization Bounds for SGLD](https://mlsanity.com/paper/2311.01046/) (2023)
- [Utilising the CLT Structure in Stochastic Gradient based Sampling : Improved Analysis and Faster Algorithms](https://mlsanity.com/paper/2206.03792/) (2022)
- [Provably Fast Finite Particle Variants of SVGD via Virtual Particle Stochastic Approximation](https://mlsanity.com/paper/2305.17558/) (2023)
- [On Divergence Measures for Training GFlowNets](https://mlsanity.com/paper/2410.09355/) (2024)
- [Quantum Speedups for Sampling and Non-convex Optimization with Stochastic Oracles](https://mlsanity.com/paper/2504.03626/) (2025)
- [Improved Langevin Monte Carlo for stochastic optimization via landscape modification](https://mlsanity.com/paper/2302.03973/) (2023)
- [Subsampling Error in Stochastic Gradient Langevin Diffusions](https://mlsanity.com/paper/2305.13882/) (2023)
- [Isoperimetry is All We Need: Langevin Posterior Sampling for RL with Sublinear Regret](https://mlsanity.com/paper/2412.20824/) (2024)
- [Beyond Propagation of Chaos: A Stochastic Algorithm for Mean Field Optimization](https://mlsanity.com/paper/2503.13115/) (2025)
- [Accelerating Convergence in Global Non-Convex Optimization with Reversible Diffusion](https://mlsanity.com/paper/2305.11493/) (2023)
- [Continuous-time Riemannian SGD and SVRG Flows on Wasserstein Probabilistic Space](https://mlsanity.com/paper/2401.13530/) (2024)
- [Langevin Multiplicative Weights Update with Applications in Polynomial Portfolio Management](https://mlsanity.com/paper/2502.19210/) (2025)
- [Flatness-Aware Stochastic Gradient Langevin Dynamics](https://mlsanity.com/paper/2510.02174/) (2025)
- [Generative Bayesian Filtering for State Estimation](https://mlsanity.com/paper/2607.20521/) (2026)

## Abstract (arXiv)

The stochastic gradient Langevin Dynamics is one of the most fundamental algorithms to solve sampling problems and non-convex optimization appearing in several machine learning applications. Especially, its variance reduced versions have nowadays gained particular attention. In this paper, we study two variants of this kind, namely, the Stochastic Variance Reduced Gradient Langevin Dynamics and the Stochastic Recursive Gradient Langevin Dynamics. We prove their convergence to the objective distribution in terms of KL-divergence under the sole assumptions of smoothness and Log-Sobolev inequality which are weaker conditions than those used in prior works for these algorithms. With the batch size and the inner loop length set to √n, the gradient complexity to achieve an ε-precision is tildeO((n+dn1/2ε-1)γ2 L2α-2), which is an improvement from any previous analyses. We also show some essential applications of our result to non-convex optimization.