# Nonasymptotic analysis of Stochastic Gradient Hamiltonian Monte Carlo under local conditions for nonconvex optimization

arXiv: https://arxiv.org/abs/2002.05465 · HTML: https://mlsanity.com/paper/2002.05465/

## Key facts

- Cited by 13 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 13 Feb 2020.

## Built on by (most cited first, 13 of 13)

- [A proof of convergence for gradient descent in the training of artificial neural networks for constant target functions](https://mlsanity.com/paper/2102.09924/) (2021)
- [Kinetic Langevin MCMC Sampling Without Gradient Lipschitz Continuity -- the Strongly Convex Case](https://mlsanity.com/paper/2301.08039/) (2023)
- [A proof of convergence for the gradient descent optimization method with random initializations in the training of neural networks with ReLU activation for piecewise linear target functions](https://mlsanity.com/paper/2108.04620/) (2021)
- [A proof of convergence for stochastic gradient descent in the training of artificial neural networks with ReLU activation for constant target functions](https://mlsanity.com/paper/2104.00277/) (2021)
- [Convergence proof for stochastic gradient descent in the training of deep neural networks with ReLU activation for constant target functions](https://mlsanity.com/paper/2112.07369/) (2021)
- [Proximal Interacting Particle Langevin Algorithms](https://mlsanity.com/paper/2406.14292/) (2024)
- [Kinetic Interacting Particle Langevin Monte Carlo](https://mlsanity.com/paper/2407.05790/) (2024)
- [Decentralized Bayesian Learning with Metropolis-Adjusted Hamiltonian Monte Carlo](https://mlsanity.com/paper/2107.07211/) (2021)
- [Statistical Finite Elements via Langevin Dynamics](https://mlsanity.com/paper/2110.11131/) (2021)
- [Global convergence of optimized adaptive importance samplers](https://mlsanity.com/paper/2201.00409/) (2022)
- [Subsampling Error in Stochastic Gradient Langevin Diffusions](https://mlsanity.com/paper/2305.13882/) (2023)
- [Contractive kinetic Langevin samplers beyond global Lipschitz continuity](https://mlsanity.com/paper/2509.12031/) (2025)
- [Accurate Large-sample Uncertainty Quantification using Stochastic Gradient Markov Chain Monte Carlo](https://mlsanity.com/paper/2606.00293/) (2026)

## Abstract (arXiv)

We provide a nonasymptotic analysis of the convergence of the stochastic gradient Hamiltonian Monte Carlo (SGHMC) to a target measure in Wasserstein-2 distance without assuming log-concavity. Our analysis quantifies key theoretical properties of the SGHMC as a sampler under local conditions which significantly improves the findings of previous results. In particular, we prove that the Wasserstein-2 distance between the target and the law of the SGHMC is uniformly controlled by the step-size of the algorithm, therefore demonstrate that the SGHMC can provide high-precision results uniformly in the number of iterations. The analysis also allows us to obtain nonasymptotic bounds for nonconvex optimization problems under local conditions and implies that the SGHMC, when viewed as a nonconvex optimizer, converges to a global minimum with the best known rates. We apply our results to obtain nonasymptotic bounds for scalable Bayesian inference and nonasymptotic generalization bounds.