# High-accuracy sampling from constrained spaces with the Metropolis-adjusted Preconditioned Langevin Algorithm

arXiv: https://arxiv.org/abs/2412.18701 · HTML: https://mlsanity.com/paper/2412.18701/

## Key facts

- Cited by 4 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 24 Dec 2024.

## Built on by (most cited first, 4 of 4)

- [Mirror Mean-Field Langevin Dynamics](https://mlsanity.com/paper/2505.02621/) (2025)
- [Estimating Rare Events in Language Models with Proper Evaluation](https://mlsanity.com/paper/2607.18454/) (2026)
- [Functional Stochastic Localization](https://mlsanity.com/paper/2602.03999/) (2026)
- [Time-Inhomogeneous Preconditioned Langevin Dynamics](https://mlsanity.com/paper/2605.06091/) (2026)

## Abstract (arXiv)

In this work, we propose a first-order sampling method called the Metropolis-adjusted Preconditioned Langevin Algorithm for approximate sampling from a target distribution whose support is a proper convex subset of ℝd. Our proposed method is the result of applying a Metropolis-Hastings filter to the Markov chain formed by a single step of the preconditioned Langevin algorithm with a metric mathscrG, and is motivated by the natural gradient descent algorithm for optimisation. We derive non-asymptotic upper bounds for the mixing time of this method for sampling from target distributions whose potentials are bounded relative to mathscrG, and for exponential distributions restricted to the support. Our analysis suggests that if mathscrG satisfies stronger notions of self-concordance introduced in Kook and Vempala (2024), then these mixing time upper bounds have a strictly better dependence on the dimension than when is merely self-concordant. We also provide numerical experiments that demonstrates the practicality of our proposed method. Our method is a high-accuracy sampler due to the polylogarithmic dependence on the error tolerance in our mixing time upper bounds.