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Reflected Anchored Langevin Algorithms
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arXiv Machine Learning
4d ago
Reflected Anchored Langevin Algorithms
The paper introduces Reflected Anchored Langevin Dynamics (RALD), a reflected diffusion that can converge to non‑differentiable target distributions on constrained domains. By employing a smooth anchored reference potential and scaling the drift and noise covariance with a state‑dependent factor, the authors derive an Euler‑Maruyama discretization with projection, yielding the Reflected Anchored Langevin Monte Carlo (RALMC) algorithm. Explicit convergence bounds and iteration complexity in the 2‑Wasserstein distance are proven, and numerical experiments confirm the theoretical predictions and empirical performance.
arXiv Machine Learning
Jun 17
Randomized Midpoint Method for Log-Concave Sampling under Constraints
arXiv:2405. 15379v3 Announce Type: replace-cross Abstract: In this paper, we study the problem of sampling from log-concave distributions supported on convex and compact sets, with a particular focus on the randomized midpoint discretization of both overdamped and kinetic Langevin diffusions in constrained domains.
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.
arXiv Machine Learning
3d ago
Accelerating Non-Smooth and Heavy-Tailed Sampling
The paper introduces non‑reversible anchored Langevin dynamics (NALD) and its reflected variant (NRALD) for sampling target distributions that may be non‑smooth and heavy‑tailed, both in Euclidean space and on constrained domains. By adding a circulation drift through a divergence‑free skew‑symmetric matrix field and a stream potential, the methods preserve the target distribution without needing density derivatives, admit a random‑time‑change representation, and can be applied with normal reflection on bounded domains. The authors demonstrate that breaking reversibility leads to faster convergence to the target distribution, as shown by finite‑time non‑asymptotic analysis, large‑deviations results, and asymptotic variance reduction, with numerical experiments confirming the algorithms’ efficiency.
arXiv Machine Learning
Aug 7
The Tamed Subgradient Unadjusted Langevin Algorithm beyond Convexity
arXiv:2608. 06283v1 Announce Type: new Abstract: We study the problem of sampling from target distributions whose potentials are simultaneously non-smooth, subject to superlinear gradient growth, and non-convex.
arXiv Machine Learning
Sep 17
R\'enyi Tracking Bounds for Langevin Dynamics with Moving Targets
arXiv:2609. 17577v1 Announce Type: cross Abstract: We study Langevin diffusion and Langevin Monte Carlo (LMC) when the target distribution changes over time.