# Improved Analysis for Hessian-free High-resolution Monte Carlo Sampling

arXiv: https://arxiv.org/abs/2608.25052 · HTML: https://mlsanity.com/paper/2608.25052/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 29 of its 59 references that are in the map.
- First posted on arXiv 25 Aug 2026.

## Builds on (most cited first, 29 of 29)

- [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)
- [Stochastic model-based minimization of weakly convex functions](https://mlsanity.com/paper/1803.06523/) (2018)
- [High-dimensional Bayesian inference via the Unadjusted Langevin Algorithm](https://mlsanity.com/paper/1605.01559/) (2016)
- [Underdamped Langevin MCMC: A non-asymptotic analysis](https://mlsanity.com/paper/1707.03663/) (2017)
- [User-friendly guarantees for the Langevin Monte Carlo with inaccurate gradient](https://mlsanity.com/paper/1710.00095/) (2017)
- [Convergence of Langevin MCMC in KL-divergence](https://mlsanity.com/paper/1705.09048/) (2017)
- [Understanding the Acceleration Phenomenon via High-Resolution Differential Equations](https://mlsanity.com/paper/1810.08907/) (2018)
- [Sharp convergence rates for Langevin dynamics in the nonconvex setting](https://mlsanity.com/paper/1805.01648/) (2018)
- [Consistency and fluctuations for stochastic gradient Langevin dynamics](https://mlsanity.com/paper/1409.0578/) (2014)
- [On sampling from a log-concave density using kinetic Langevin diffusions](https://mlsanity.com/paper/1807.09382/) (2018)
- [Posterior Concentration for Sparse Deep Learning](https://mlsanity.com/paper/1803.09138/) (2018)
- [Convergence of Langevin Monte Carlo in Chi-Squared and Renyi Divergence](https://mlsanity.com/paper/2007.11612/) (2020)
- [Nonasymptotic estimates for Stochastic Gradient Langevin Dynamics under local conditions in nonconvex optimization](https://mlsanity.com/paper/1910.02008/) (2019)
- [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)
- [Improved Discretization Analysis for Underdamped Langevin Monte Carlo](https://mlsanity.com/paper/2302.08049/) (2023)
- [On stochastic gradient Langevin dynamics with dependent data streams in the logconcave case](https://mlsanity.com/paper/1812.02709/) (2018)
- [Decentralized Stochastic Gradient Langevin Dynamics and Hamiltonian Monte Carlo](https://mlsanity.com/paper/2007.00590/) (2020)
- [On stochastic gradient Langevin dynamics with dependent data streams: the fully non-convex case](https://mlsanity.com/paper/1905.13142/) (2019)
- [Breaking Reversibility Accelerates Langevin Dynamics for Global Non-Convex Optimization](https://mlsanity.com/paper/1812.07725/) (2018)
- [Generalized EXTRA stochastic gradient Langevin dynamics](https://mlsanity.com/paper/2412.01993/) (2024)
- [Hessian-Free High-Resolution Nesterov Acceleration for Sampling](https://mlsanity.com/paper/2006.09230/) (2020)
- [Posterior concentrations of fully-connected Bayesian neural networks with general priors on the weights](https://mlsanity.com/paper/2403.14225/) (2024)
- [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)
- [Sampling non-log-concave densities via Hessian-free high-resolution dynamics](https://mlsanity.com/paper/2601.02725/) (2026)
- [Variance Reduction for Stochastic Gradient Generalized Non-reversible Langevin Monte Carlo Algorithms](https://mlsanity.com/paper/2606.28808/) (2026)
- [Global Optimization Algorithm through High-Resolution Sampling](https://mlsanity.com/paper/2410.13737/) (2024)

## Built on by (most cited first, 1 of 1)

- [Penalized Nonreversible Langevin for Constrained Sampling](https://mlsanity.com/paper/2609.25381/) (2026)

## Abstract (arXiv)

Hessian-free high-resolution (HFHR) dynamics augments underdamped Langevin dynamics (ULD) with reversible position diffusion for sampling problems that arise in machine learning. We establish an explicit quantitative contraction rate for HFHR dynamics under a position Poincar'e inequality, weighted Hessian and Laplacian bounds, and a compact Sobolev embedding, where the potential function is not necessarily convex. An adapted time-augmented Poincar'e inequality yields an explicit rate that improves upon the contraction rate of the underdamped Langevin dynamics. We also give a weak-solution construction and a self-contained spectral proof of the divergence lemma underlying the argument. For HFHR Monte Carlo (HFHRMC) algorithm, which is based on a discretization scheme of HFHR dynamics, we use a path-space Girsanov argument to obtain a non-asymptotic convergence bound and an explicit iteration complexity in total variation distance. The bounds hold for every α≥0 and γ>0 and remain regular at the ULD endpoint. Optimizing the iteration complexity bound yields a positive, accuracy-dependent position-diffusion parameter at finite accuracy, while its leading high-accuracy order coincides with that of the optimized ULD endpoint. Our iteration complexity bound improves upon the existing work on HFHR algorithms. Numerical experiments including Bayesian learning problems on real data are provided to illustrate the effect of positive α and its benefit.