# A Non-asymptotic Analysis for Learning and Applying a Preconditioner in MCMC

arXiv: https://arxiv.org/abs/2602.10714 · HTML: https://mlsanity.com/paper/2602.10714/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- Builds on 12 of its 27 references that are in the map.
- First posted on arXiv 11 Feb 2026.

## Builds on (most cited first, 12 of 12)

- [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)
- [User-friendly guarantees for the Langevin Monte Carlo with inaccurate gradient](https://mlsanity.com/paper/1710.00095/) (2017)
- [Structured Logconcave Sampling with a Restricted Gaussian Oracle](https://mlsanity.com/paper/2010.03106/) (2020)
- [Minimax Mixing Time of the Metropolis-Adjusted Langevin Algorithm for Log-Concave Sampling](https://mlsanity.com/paper/2109.13055/) (2021)
- [Optimal Preconditioning and Fisher Adaptive Langevin Sampling](https://mlsanity.com/paper/2305.14442/) (2023)
- [Fast Convergence of Phi-Divergence Along the Unadjusted Langevin Algorithm and Proximal Sampler](https://mlsanity.com/paper/2410.10699/) (2024)
- [Unadjusted Hamiltonian MCMC with Stratified Monte Carlo Time Integration](https://mlsanity.com/paper/2211.11003/) (2022)
- [Air Markov Chain Monte Carlo](https://mlsanity.com/paper/1801.09309/) (2018)
- [Characterizing Dependence of Samples along the Langevin Dynamics and Algorithms via Contraction of Phi-Mutual Information](https://mlsanity.com/paper/2402.17067/) (2024)
- [Covariance estimation using Markov chain Monte Carlo](https://mlsanity.com/paper/2410.17147/) (2024)
- [On sample complexity for covariance estimation via the unadjusted Langevin algorithm](https://mlsanity.com/paper/2601.21717/) (2026)

## Built on by (most cited first, 2 of 2)

- [Gradient-free Riemannian Langevin Sampler](https://mlsanity.com/paper/2607.07519/) (2026)
- [Efficient Mass Matrix Estimation with Gaussian Cooling](https://mlsanity.com/paper/2610.04806/) (2026)

## Abstract (arXiv)

Preconditioning is a common method applied to modify Markov chain Monte Carlo algorithms with the goal of making them more efficient. In practice it is often extremely effective, even when the preconditioner is learned from the chain. We analyse and compare the finite-time computational costs of schemes which learn a preconditioner based on the target covariance or the expected Hessian of the target potential with that of a corresponding scheme that does not use preconditioning. We apply our results to various algorithms including the Unadjusted Langevin Algorithm (ULA) and the proximal sampler for an appropriately regular target, establishing non-asymptotic guarantees for versions of these algorithms that learn and use preconditioners. To do so, we establish non-asymptotic guarantees on the time taken to collect N approximately independent samples from the target for schemes that learn their preconditioners under the assumption that the underlying Markov chain satisfies a contraction in the Wasserstein-2 distance. This approximate independence condition, that we formalize, allows us to bridge the non-asymptotic bounds of modern MCMC theory and classical heuristics of effective sample size and mixing time, and is needed to amortise the costs of learning a preconditioner across the samples it will help to produce.