# Sampling as optimization in the space of measures: The Langevin dynamics as a composite optimization problem

arXiv: https://arxiv.org/abs/1802.08089 · HTML: https://mlsanity.com/paper/1802.08089/

## Key facts

- Cited by 123 papers in the map (data through 7 Oct 2026).
- First posted on arXiv 22 Feb 2018.

## Built on by (most cited first, 30 of 123)

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- [Birth-death dynamics for sampling: Global convergence, approximations and their asymptotics](https://mlsanity.com/paper/2211.00450/) (2022)
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- [A Proximal Algorithm for Sampling](https://mlsanity.com/paper/2202.13975/) (2022)
- [Resolving the Mixing Time of the Langevin Algorithm to its Stationary Distribution for Log-Concave Sampling](https://mlsanity.com/paper/2210.08448/) (2022)
- [Particle-based Variational Inference with Preconditioned Functional Gradient Flow](https://mlsanity.com/paper/2211.13954/) (2022)
- [Sampling via Gradient Flows in the Space of Probability Measures](https://mlsanity.com/paper/2310.03597/) (2023)
- [Algorithms for mean-field variational inference via polyhedral optimization in the Wasserstein space](https://mlsanity.com/paper/2312.02849/) (2023)
- [Mean-field Variational Inference via Wasserstein Gradient Flow](https://mlsanity.com/paper/2207.08074/) (2022)
- [A Unified and General Framework for Continual Learning](https://mlsanity.com/paper/2403.13249/) (2024)
- [The query complexity of sampling from strongly log-concave distributions in one dimension](https://mlsanity.com/paper/2105.14163/) (2021)
- [Regularized Stein Variational Gradient Flow](https://mlsanity.com/paper/2211.07861/) (2022)

## Abstract (arXiv)

We study sampling as optimization in the space of measures. We focus on gradient flow-based optimization with the Langevin dynamics as a case study. We investigate the source of the bias of the unadjusted Langevin algorithm (ULA) in discrete time, and consider how to remove or reduce the bias. We point out the difficulty is that the heat flow is exactly solvable, but neither its forward nor backward method is implementable in general, except for Gaussian data. We propose the symmetrized Langevin algorithm (SLA), which should have a smaller bias than ULA, at the price of implementing a proximal gradient step in space. We show SLA is in fact consistent for Gaussian target measure, whereas ULA is not. We also illustrate various algorithms explicitly for Gaussian target measure, including gradient descent, proximal gradient, and Forward-Backward, and show they are all consistent.