# On Stationary-Point Hitting Time and Ergodicity of Stochastic Gradient Langevin Dynamics

arXiv: https://arxiv.org/abs/1904.13016 · HTML: https://mlsanity.com/paper/1904.13016/

## Key facts

- Cited by 14 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 30 Apr 2019.

## Built on by (most cited first, 14 of 14)

- [Stochastic Gradient Langevin Dynamics with Variance Reduction](https://mlsanity.com/paper/2102.06759/) (2021)
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- [Taming Score-Based Denoisers in ADMM: A Convergent Plug-and-Play Framework](https://mlsanity.com/paper/2603.10281/) (2026)
- [Improved Langevin Monte Carlo for stochastic optimization via landscape modification](https://mlsanity.com/paper/2302.03973/) (2023)
- [The Expected Loss of Preconditioned Langevin Dynamics Reveals the Hessian Rank](https://mlsanity.com/paper/2402.13810/) (2024)
- [Can We Do Better Than Random Start? The Power of Data Outsourcing](https://mlsanity.com/paper/2205.08098/) (2022)
- [Jump-Diffusion Langevin Dynamics for Multimodal Posterior Sampling](https://mlsanity.com/paper/2211.01774/) (2022)
- [Approximation to Deep Q-Network by Stochastic Delay Differential Equations](https://mlsanity.com/paper/2505.00382/) (2025)
- [Distribution of hitting times for dissipative random dynamical systems on ℝd, with application to stochastic gradient descent](https://mlsanity.com/paper/2609.30274/) (2026)

## Abstract (arXiv)

Stochastic gradient Langevin dynamics (SGLD) is a fundamental algorithm in stochastic optimization. Recent work by Zhang et al. [2017] presents an analysis for the hitting time of SGLD for the first and second order stationary points. The proof in Zhang et al. [2017] is a two-stage procedure through bounding the Cheeger's constant, which is rather complicated and leads to loose bounds. In this paper, using intuitions from stochastic differential equations, we provide a direct analysis for the hitting times of SGLD to the first and second order stationary points. Our analysis is straightforward. It only relies on basic linear algebra and probability theory tools. Our direct analysis also leads to tighter bounds comparing to Zhang et al. [2017] and shows the explicit dependence of the hitting time on different factors, including dimensionality, smoothness, noise strength, and step size effects. Under suitable conditions, we show that the hitting time of SGLD to first-order stationary points can be dimension-independent. Moreover, we apply our analysis to study several important online estimation problems in machine learning, including linear regression, matrix factorization, and online PCA.