# Reflected Proximal Dynamics for Accelerated High-Accuracy Log-Concave Sampling

arXiv: https://arxiv.org/abs/2610.16248 · HTML: https://mlsanity.com/paper/2610.16248/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 30 of its 42 references that are in the map.
- First posted on arXiv 12 Oct 2026.

## Builds on (most cited first, 30 of 30)

- [Rapid Convergence of the Unadjusted Langevin Algorithm: Isoperimetry Suffices](https://mlsanity.com/paper/1903.08568/) (2019)
- [Theoretical guarantees for approximate sampling from smooth and log-concave densities](https://mlsanity.com/paper/1412.7392/) (2014)
- [Underdamped Langevin MCMC: A non-asymptotic analysis](https://mlsanity.com/paper/1707.03663/) (2017)
- [Analysis of Langevin Monte Carlo via convex optimization](https://mlsanity.com/paper/1802.09188/) (2018)
- [Structured Logconcave Sampling with a Restricted Gaussian Oracle](https://mlsanity.com/paper/2010.03106/) (2020)
- [Efficient Bayesian computation by proximal Markov chain Monte Carlo: when Langevin meets Moreau](https://mlsanity.com/paper/1612.07471/) (2016)
- [The Randomized Midpoint Method for Log-Concave Sampling](https://mlsanity.com/paper/1909.05503/) (2019)
- [Improved analysis for a proximal algorithm for sampling](https://mlsanity.com/paper/2202.06386/) (2022)
- [On sampling from a log-concave density using kinetic Langevin diffusions](https://mlsanity.com/paper/1807.09382/) (2018)
- [Hamiltonian Monte Carlo for Hierarchical Models](https://mlsanity.com/paper/1312.0906/) (2013)
- [Faster high-accuracy log-concave sampling via algorithmic warm starts](https://mlsanity.com/paper/2302.10249/) (2023)
- [Primal Dual Interpretation of the Proximal Stochastic Gradient Langevin Algorithm](https://mlsanity.com/paper/2006.09270/) (2020)
- [Improved dimension dependence of a proximal algorithm for sampling](https://mlsanity.com/paper/2302.10081/) (2023)
- [Improved Discretization Analysis for Underdamped Langevin Monte Carlo](https://mlsanity.com/paper/2302.08049/) (2023)
- [The Bouncy Particle Sampler: A Non-Reversible Rejection-Free Markov Chain Monte Carlo Method](https://mlsanity.com/paper/1510.02451/) (2015)
- [A Proximal Algorithm for Sampling from Non-smooth Potentials](https://mlsanity.com/paper/2110.04597/) (2021)
- [A Proximal Algorithm for Sampling](https://mlsanity.com/paper/2202.13975/) (2022)
- [In-and-Out: Algorithmic Diffusion for Sampling Convex Bodies](https://mlsanity.com/paper/2405.01425/) (2024)
- [High-accuracy sampling for diffusion models and log-concave distributions](https://mlsanity.com/paper/2602.01338/) (2026)
- [Mixing Time of the Proximal Sampler in Relative Fisher Information via Strong Data Processing Inequality](https://mlsanity.com/paper/2502.05623/) (2025)
- [Fast Convergence of Phi-Divergence Along the Unadjusted Langevin Algorithm and Proximal Sampler](https://mlsanity.com/paper/2410.10699/) (2024)
- [The Boomerang Sampler](https://mlsanity.com/paper/2006.13777/) (2020)
- [Algorithmic Aspects of the Log-Laplace Transform and a Non-Euclidean Proximal Sampler](https://mlsanity.com/paper/2302.06085/) (2023)
- [Piecewise-Deterministic Markov Chain Monte Carlo](https://mlsanity.com/paper/1707.05296/) (2017)
- [Proximal Oracles for Optimization and Sampling](https://mlsanity.com/paper/2404.02239/) (2024)
- [A Separation in Heavy-Tailed Sampling: Gaussian vs. Stable Oracles for Proximal Samplers](https://mlsanity.com/paper/2405.16736/) (2024)
- [Zeroth-order Logconcave Sampling](https://mlsanity.com/paper/2507.18021/) (2025)
- [Exact simulation of diffusions and improved algorithms for log-concave sampling](https://mlsanity.com/paper/2608.05022/) (2026)
- [Accelerated High-Accuracy Sampling from a Warm Start via the Proximal Bouncy Particle Sampler](https://mlsanity.com/paper/2609.06905/) (2026)
- [Smoothed Picard Hamiltonian Monte Carlo](https://mlsanity.com/paper/2609.06906/) (2026)

## Abstract (arXiv)

We study sampling from π(x)proptoe-V(x) on ℝd, where V is α-strongly convex and β-smooth with condition number kappa=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 textitreflected 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 chi2 divergence, both methods achieve total variation error varepsilon using widetilde O(√kappad1/4) 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 widetilde O(kappa6/7d1/4) expected oracle queries. In both regimes, the dependence on the target accuracy varepsilon is polylog(1/varepsilon), making reflected proximal dynamics a promising framework for accelerated high-accuracy log-concave sampling.