Resolving the Mixing Time of the Langevin Algorithm to its Stationary Distribution for Log-Concave Sampling

Jason Altschuler, Kunal Talwar
Proceedings of Thirty Sixth Conference on Learning Theory, PMLR 195:2509-2510, 2023.

Abstract

Sampling from a high-dimensional distribution is a fundamental task in statistics, engineering, and the sciences. A canonical approach is the Langevin Algorithm, i.e., the Markov chain for the discretized Langevin Diffusion. This is the sampling analog of Gradient Descent. Despite being studied for several decades in multiple communities, tight mixing bounds for this algorithm remain unresolved even in the seemingly simple setting of log-concave distributions over a bounded domain. This paper completely characterizes the mixing time of the Langevin Algorithm to its stationary distribution in this setting (and others). This mixing result can be combined with any bound on the discretization bias in order to sample from the stationary distribution of the Langevin Diffusion. In this way, we disentangle the study of the mixing and bias of the Langevin Algorithm.Our key insight is to analyze the Langevin Algorithm’s convergence by using a new Lyapunov function: the shifted divergence, a quantity studied in the differential privacy literature. Briefly, this Lyapunov function is a version of the Renyi divergence that is smoothed in optimal transport distance, and we use the amount of smoothing to measure the Langevin Algorithm’s progress. In addition to giving a short, simple proof of optimal mixing bounds, this analysis approach also has several additional appealing properties. First, our approach removes all unnecessary assumptions required by other sampling analyses. Second, our approach unifies many settings: it extends if the Langevin Algorithm uses projections, stochastic mini-batch gradients, or strongly convex potentials (whereby our mixing time improves exponentially). Third, our approach unifies many metrics: it proves mixing in the stringent notion of Renyi divergence, which immediately implies mixing in all common metrics via standard comparison inequalities. Fourth, our approach exploits convexity only through the contractivity of a gradient step—reminiscent of how convexity is used in textbook proofs of Gradient Descent.

Cite this Paper

BibTeX
@InProceedings{pmlr-v195-altschuler23a, title = {Resolving the Mixing Time of the Langevin Algorithm to its Stationary Distribution for Log-Concave Sampling}, author = {Altschuler, Jason and Talwar, Kunal}, booktitle = {Proceedings of Thirty Sixth Conference on Learning Theory}, pages = {2509--2510}, year = {2023}, editor = {Neu, Gergely and Rosasco, Lorenzo}, volume = {195}, series = {Proceedings of Machine Learning Research}, month = {12--15 Jul}, publisher = {PMLR}, pdf = {https://proceedings.mlr.press/v195/altschuler23a/altschuler23a.pdf}, url = {https://proceedings.mlr.press/v195/altschuler23a.html}, abstract = {Sampling from a high-dimensional distribution is a fundamental task in statistics, engineering, and the sciences. A canonical approach is the Langevin Algorithm, i.e., the Markov chain for the discretized Langevin Diffusion. This is the sampling analog of Gradient Descent. Despite being studied for several decades in multiple communities, tight mixing bounds for this algorithm remain unresolved even in the seemingly simple setting of log-concave distributions over a bounded domain. This paper completely characterizes the mixing time of the Langevin Algorithm to its stationary distribution in this setting (and others). This mixing result can be combined with any bound on the discretization bias in order to sample from the stationary distribution of the Langevin Diffusion. In this way, we disentangle the study of the mixing and bias of the Langevin Algorithm.Our key insight is to analyze the Langevin Algorithm’s convergence by using a new Lyapunov function: the shifted divergence, a quantity studied in the differential privacy literature. Briefly, this Lyapunov function is a version of the Renyi divergence that is smoothed in optimal transport distance, and we use the amount of smoothing to measure the Langevin Algorithm’s progress. In addition to giving a short, simple proof of optimal mixing bounds, this analysis approach also has several additional appealing properties. First, our approach removes all unnecessary assumptions required by other sampling analyses. Second, our approach unifies many settings: it extends if the Langevin Algorithm uses projections, stochastic mini-batch gradients, or strongly convex potentials (whereby our mixing time improves exponentially). Third, our approach unifies many metrics: it proves mixing in the stringent notion of Renyi divergence, which immediately implies mixing in all common metrics via standard comparison inequalities. Fourth, our approach exploits convexity only through the contractivity of a gradient step—reminiscent of how convexity is used in textbook proofs of Gradient Descent. } }
Endnote
%0 Conference Paper %T Resolving the Mixing Time of the Langevin Algorithm to its Stationary Distribution for Log-Concave Sampling %A Jason Altschuler %A Kunal Talwar %B Proceedings of Thirty Sixth Conference on Learning Theory %C Proceedings of Machine Learning Research %D 2023 %E Gergely Neu %E Lorenzo Rosasco %F pmlr-v195-altschuler23a %I PMLR %P 2509--2510 %U https://proceedings.mlr.press/v195/altschuler23a.html %V 195 %X Sampling from a high-dimensional distribution is a fundamental task in statistics, engineering, and the sciences. A canonical approach is the Langevin Algorithm, i.e., the Markov chain for the discretized Langevin Diffusion. This is the sampling analog of Gradient Descent. Despite being studied for several decades in multiple communities, tight mixing bounds for this algorithm remain unresolved even in the seemingly simple setting of log-concave distributions over a bounded domain. This paper completely characterizes the mixing time of the Langevin Algorithm to its stationary distribution in this setting (and others). This mixing result can be combined with any bound on the discretization bias in order to sample from the stationary distribution of the Langevin Diffusion. In this way, we disentangle the study of the mixing and bias of the Langevin Algorithm.Our key insight is to analyze the Langevin Algorithm’s convergence by using a new Lyapunov function: the shifted divergence, a quantity studied in the differential privacy literature. Briefly, this Lyapunov function is a version of the Renyi divergence that is smoothed in optimal transport distance, and we use the amount of smoothing to measure the Langevin Algorithm’s progress. In addition to giving a short, simple proof of optimal mixing bounds, this analysis approach also has several additional appealing properties. First, our approach removes all unnecessary assumptions required by other sampling analyses. Second, our approach unifies many settings: it extends if the Langevin Algorithm uses projections, stochastic mini-batch gradients, or strongly convex potentials (whereby our mixing time improves exponentially). Third, our approach unifies many metrics: it proves mixing in the stringent notion of Renyi divergence, which immediately implies mixing in all common metrics via standard comparison inequalities. Fourth, our approach exploits convexity only through the contractivity of a gradient step—reminiscent of how convexity is used in textbook proofs of Gradient Descent.
APA
Altschuler, J. & Talwar, K.. (2023). Resolving the Mixing Time of the Langevin Algorithm to its Stationary Distribution for Log-Concave Sampling. Proceedings of Thirty Sixth Conference on Learning Theory, in Proceedings of Machine Learning Research 195:2509-2510 Available from https://proceedings.mlr.press/v195/altschuler23a.html.

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