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August 2022 Improved bounds for discretization of Langevin diffusions: Near-optimal rates without convexity
Wenlong Mou, Nicolas Flammarion, Martin J. Wainwright, Peter L. Bartlett
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Bernoulli 28(3): 1577-1601 (August 2022). DOI: 10.3150/21-BEJ1343

Abstract

Discretizations of the Langevin diffusion have been proven very useful for developing and analyzing algorithms for sampling and stochastic optimization. We present an improved non-asymptotic analysis of the Euler-Maruyama discretization of the Langevin diffusion. Our analysis does not require global contractivity, and yields polynomial dependence on the time horizon. Compared to existing approaches, we make an additional smoothness assumption, and improve the existing rate in discretization step size from O(η) to O(η2) in terms of the KL divergence. This result matches the correct order for numerical SDEs, without suffering from exponential time dependence. When applied to MCMC, this result simultaneously improves on the analyses of a range of sampling algorithms that are based on Dalalyan’s approach.

Acknowledgements

This work was partially supported by Office of Naval Research Grant ONR-N00014-18-1-2640 to MJW, and National Science Foundation Grants IIS-1619362 to PLB and CCF-1909365 and DMS-2023505 to MJW and PLB. We thank Xiang Cheng, Valentin De Bortolo, Yi-An Ma, and Andre Wibisono for helpful discussions, as well as the referees and associate editor for their comments and suggestions that helped to improve the paper.

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Wenlong Mou. Nicolas Flammarion. Martin J. Wainwright. Peter L. Bartlett. "Improved bounds for discretization of Langevin diffusions: Near-optimal rates without convexity." Bernoulli 28 (3) 1577 - 1601, August 2022. https://doi.org/10.3150/21-BEJ1343

Information

Received: 1 February 2020; Published: August 2022
First available in Project Euclid: 25 April 2022

Digital Object Identifier: 10.3150/21-BEJ1343

Keywords: Euler-Maruyama discretization , KL divergence , Langevin diffusion , Markov chain Monte Carlo , non-asymptotic bound

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Vol.28 • No. 3 • August 2022
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