# On sample complexity for covariance estimation via the unadjusted Langevin algorithm

arXiv: https://arxiv.org/abs/2601.21717 · HTML: https://mlsanity.com/paper/2601.21717/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 7 of its 17 references that are in the map.
- First posted on arXiv 29 Jan 2026.

## Builds on (most cited first, 7 of 7)

- [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)
- [Non-asymptotic convergence analysis for the Unadjusted Langevin Algorithm](https://mlsanity.com/paper/1507.05021/) (2015)
- [High-dimensional Bayesian inference via the Unadjusted Langevin Algorithm](https://mlsanity.com/paper/1605.01559/) (2016)
- [Faster high-accuracy log-concave sampling via algorithmic warm starts](https://mlsanity.com/paper/2302.10249/) (2023)
- [Concentration of the Langevin Algorithm's Stationary Distribution](https://mlsanity.com/paper/2212.12629/) (2022)
- [Covariance estimation using Markov chain Monte Carlo](https://mlsanity.com/paper/2410.17147/) (2024)

## Built on by (most cited first, 1 of 1)

- [A Non-asymptotic Analysis for Learning and Applying a Preconditioner in MCMC](https://mlsanity.com/paper/2602.10714/) (2026)

## Abstract (arXiv)

We establish sample complexity guarantees for estimating the covariance matrix of a strongly log-concave smooth distribution using the unadjusted Langevin algorithm (ULA). We quantitatively compare our complexity estimates on single-chain ULA with embarrassingly parallel ULA and derive that the sample complexity of the single-chain approach is smaller than that of embarrassingly parallel ULA by a logarithmic factor in the dimension and the reciprocal of the prescribed precision, with the difference arising from effective bias reduction through burn-in. The key technical contribution is a concentration bound for the sample covariance matrix around its expectation, derived via a log-Sobolev inequality for the joint distribution of ULA iterates.