Reflected Proximal Dynamics for Accelerated High-Accuracy Log-Concave Sampling
Abstract
We study sampling from π(x)∝e−V(x) on ℝd, where V is α-strongly convex and β-smooth with condition number κ=β/α. For this problem, the proximal sampler preserves the target exactly and recovers overdamped Langevin dynamics in the small-step limit, but its Gaussian resampling step discards directional information. Underdamped Langevin dynamics retains such information through momentum to accelerate exploration, motivating a proximal construction that combines target preservation with directional persistence. We introduce \textit{reflected proximal dynamics}, which combines proximal sampling with an exact reflection of the auxiliary variable, conditional kinetic motion, and controlled refreshment. We develop two realizations: a flux-resampled underdamped half-turn and a persistent-momentum quarter-turn. From a warm start with constant χ2 divergence, both methods achieve total variation error ε using Õ(√κd1/4) expected oracle queries. We further remove the supplied warm-start assumption by combining either sampler with existing posterior samplers in a proximal initialization stage with an optimized coarse step size. Accounting for both initialization and sampling yields Õ(κ6/7d1/4) expected oracle queries. In both regimes, the dependence on the target accuracy ε is polylog(1/ε), making reflected proximal dynamics a promising framework for accelerated high-accuracy log-concave sampling.
Related
- Accelerated High-Accuracy Sampling from a Warm Start via the Proximal Bouncy Particle Sampler
- Averaged proximal reflected gradient method for monotone variational inequalities
- Neural Actor--Critic Learning with Minimax Critics and Reflected Langevin Sampling
- Wasserstein accelerated proximal gradient algorithms
- Accelerated Proximal Gradient Method for Non-Strongly Convex Time-Varying Optimization
BibTeX
@misc{yuan2026reflected,
title = {{Reflected Proximal Dynamics for Accelerated High-Accuracy Log-Concave Sampling}},
author = {Bo Yuan and Wei Guo and Yongxin Chen},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.16248},
eprint = {2610.16248},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.16248},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Reflected Proximal Dynamics for Accelerated High-Accuracy Log-Concave Sampling
AU - Bo Yuan
AU - Wei Guo
AU - Yongxin Chen
PY - 2026
DA - 2026/10/12
PB - arXiv
DO - 10.48550/arxiv.2610.16248
UR - https://arxiv.org/abs/2610.16248
AB - We study sampling from π(x)∝e^−V(x) on ℝ^d, where V is α-strongly convex and β-smooth with condition number κ=β/α. For this problem, the proximal sampler preserves the target exactly and recovers overdamped Langevin dynamics in the small-step limit, but its Gaussian resampling step discards directional information. Underdamped Langevin dynamics retains such information through momentum to accelerate exploration, motivating a proximal construction that combines target preservation with directional persistence. We introduce \textit{reflected proximal dynamics}, which combines proximal sampling with an exact reflection of the auxiliary variable, conditional kinetic motion, and controlled refreshment. We develop two realizations: a flux-resampled underdamped half-turn and a persistent-momentum quarter-turn. From a warm start with constant χ^2 divergence, both methods achieve total variation error ε using Õ(√κd^1/4) expected oracle queries. We further remove the supplied warm-start assumption by combining either sampler with existing posterior samplers in a proximal initialization stage with an optimized coarse step size. Accounting for both initialization and sampling yields Õ(κ^6/7d^1/4) expected oracle queries. In both regimes, the dependence on the target accuracy ε is polylog(1/ε), making reflected proximal dynamics a promising framework for accelerated high-accuracy log-concave sampling.
ER -