arXiv Machine Learning

Delocalization of bias in unadjusted Hamiltonian Monte Carlo and underdamped Langevin

arXiv:2607. 15208v1 Announce Type: cross Abstract: Unadjusted samplers such as unadjusted Hamiltonian Monte Carlo and underdamped Langevin are well-known to be biased.

arXiv Machine Learning
Aug 27

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

The paper introduces Hessian-free high-resolution (HFHR) dynamics, an extension of underdamped Langevin dynamics that incorporates reversible position diffusion for sampling in machine learning. It provides an explicit quantitative contraction rate under a position Poincaré inequality, weighted Hessian and Laplacian bounds, and a compact Sobolev embedding, even when the potential is non‑convex. For the HFHR Monte Carlo algorithm, a path‑space Girsanov argument yields a non‑asymptotic convergence bound and an explicit iteration complexity in total variation distance, improving on previous HFHR results and demonstrating benefits of a positive diffusion parameter through numerical experiments.

By Wujun Lv, Xiaoyu Wang, Yingli Wang, Lingjiong Zhu
arXiv Statistics ML
Sep 25

Global Convergence of Third-Order Langevin Dynamics for Non-Convex Optimization via Simulated Annealing

The paper establishes global convergence guarantees for third‑order Langevin dynamics applied to non‑convex optimization via simulated annealing with fixed friction and decreasing noise. It shows that, under dissipativity, regularity, and low‑temperature functional‑inequality assumptions, a logarithmic cooling schedule drives objective values to the global minimum with a barrier‑controlled kinetic rate, and that polynomially decreasing step sizes preserve this rate for both exact‑force‑integral and midpoint three‑stage discretizations. Numerical experiments on a double‑well problem and high‑dimensional neural‑network objectives demonstrate that third‑order Langevin schemes outperform overdamped Langevin dynamics and, in some cases, the one‑gradient UBU integrator in terms of terminal‑success point estimates and post‑quench test accuracy.

By Yingli Wang, Kelvin Shuangjian Zhang, Lingjiong Zhu
arXiv Statistics ML
Aug 26

A Non-asymptotic Analysis for Learning and Applying a Preconditioner in MCMC

The paper presents a non‑asymptotic analysis of Markov chain Monte Carlo (MCMC) algorithms that learn and apply a preconditioner based on either the target covariance or the expected Hessian of the target potential. It compares the finite‑time computational costs of these preconditioned schemes with unpreconditioned counterparts, providing guarantees for algorithms such as the Unadjusted Langevin Algorithm (ULA) and the proximal sampler. The analysis relies on a contraction assumption in the Wasserstein‑2 distance to formalize approximate independence and bridge modern MCMC theory with classical effective sample size heuristics.

By Max Hird, Florian Maire, Jeffrey Negrea
Hugging Face Trending Papers
Aug 6

The Tamed Subgradient Unadjusted Langevin Algorithm beyond Convexity

We study the problem of sampling from target distributions whose potentials are simultaneously non-smooth, subject to superlinear gradient growth, and non-convex. We introduce the Subgradient Tamed Unadjusted Langevin Algorithm (SG-TULA), a discretisation of the Langevin diffusion that operates directly on subgradients, without relying on computationally demanding smoothing procedures.