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
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
arXiv:2610.16248v1 Announce Type: cross
Abstract: We study sampling from $\pi(x)\propto\mathrm{e}^{-V(x)}$ on $\mathbb{R}^d$, where $V$ is $\alpha$-strongly convex and $\beta$-smooth with condition n...
By Bo Yuan, Wei Guo, Yongxin Chen
arXiv:2608.25279v1 Announce Type: cross
Abstract: The OBABO and BAOAB schemes and the other standard Strang splittings of kinetic (underdamped) Langevin dynamics are widely used Markov chain Monte Ca...
By Nawaf Bou-Rabee
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:2606. 26164v1 Announce Type: new Abstract: Finding all modes of a multimodal black-box function is a fundamental challenge in optimization, Bayesian inference, and scientific computing.
By Ira Wolfson
arXiv:2609.40193v1 Announce Type: new
Abstract: We establish near-linear accuracy bounds for the classical Moreau--Yosida unadjusted Langevin algorithm (MYULA). The target is $\pi\propto e^{-f-g}$, w...
By Yuchen Xin, Zhihua Zhang
While entropy regularization is widely used to stabilize and accelerate Natural Policy Gradient methods, its ability to yield faster convergence rates for the unregularized objective remains underexplored. Existing analyses often rely on double-loop architectures and invoke a linear entropy penalty.
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.
The paper introduces an iterative neural variational solver for the committor function in under‑damped Langevin dynamics. By splitting the infinitesimal generator into self‑adjoint and skew‑adjoint parts, each subproblem gains a variational form that can be efficiently solved with neural networks, resembling a generalized Concus‑Golub‑Widlund splitting. The method is validated on benchmark systems such as a double‑well potential, the Muller‑Brown potential, and alanine dipeptide, demonstrating practical accuracy and extensibility for studying rare transition events across different damping coefficients.
By Tiejun Li, Zhaoqi Liu, Xiang Zhou
Finding all modes of a multimodal black-box function is a fundamental challenge in optimization, Bayesian inference, and scientific computing. Existing approaches -- basin-hopping, CMA-ES, multistart gradient descent -- operate sequentially and cannot exploit the massive parallelism of modern GPU hardware.
arXiv:2606. 32005v1 Announce Type: cross Abstract: Stochastic Gradient Descent ($\textsf{SGD}$) is one of the most classical optimization algorithms with favorable theoretical guarantees, yet the practical implementation of $\textsf{SGD}$ differs subtly from its well-known form and is often referred to as Shuffling Stochastic Gradient Descent ($\textsf{Shuffling SGD}$).
By Zijian Liu