# Unbiased Estimation using Underdamped Langevin Dynamics

arXiv: https://arxiv.org/abs/2206.07202 · HTML: https://mlsanity.com/paper/2206.07202/

## Key facts

- Cited by 3 papers in the map (data through 14 Oct 2026).
- Builds on 16 of its 44 references that are in the map.
- First posted on arXiv 14 Jun 2022.

## Builds on (most cited first, 16 of 16)

- [A Mean Field View of the Landscape of Two-Layers Neural Networks](https://mlsanity.com/paper/1804.06561/) (2018)
- [On the Global Convergence of Gradient Descent for Over-parameterized Models using Optimal Transport](https://mlsanity.com/paper/1805.09545/) (2018)
- [Theoretical guarantees for approximate sampling from smooth and log-concave densities](https://mlsanity.com/paper/1412.7392/) (2014)
- [Underdamped Langevin MCMC: A non-asymptotic analysis](https://mlsanity.com/paper/1707.03663/) (2017)
- [Coupling and Convergence for Hamiltonian Monte Carlo](https://mlsanity.com/paper/1805.00452/) (2018)
- [Fractional Underdamped Langevin Dynamics: Retargeting SGD with Momentum under Heavy-Tailed Gradient Noise](https://mlsanity.com/paper/2002.05685/) (2020)
- [Unbiased Markov chain Monte Carlo with couplings](https://mlsanity.com/paper/1708.03625/) (2017)
- [Ergodicity of the underdamped mean-field Langevin dynamics](https://mlsanity.com/paper/2007.14660/) (2020)
- [Maximal couplings of the Metropolis-Hastings algorithm](https://mlsanity.com/paper/2010.08573/) (2020)
- [Unbiased Smoothing using Particle Independent Metropolis-Hastings](https://mlsanity.com/paper/1902.01781/) (2019)
- [Unbiased Estimation of the Gradient of the Log-Likelihood in Inverse Problems](https://mlsanity.com/paper/2003.04896/) (2020)
- [Unbiased Estimation using a Class of Diffusion Processes](https://mlsanity.com/paper/2203.03013/) (2022)
- [On Unbiased Estimation for Discretized Models](https://mlsanity.com/paper/2102.12230/) (2021)
- [On Unbiased Score Estimation for Partially Observed Diffusions](https://mlsanity.com/paper/2105.04912/) (2021)
- [Unbiased Estimation of the Hessian for Partially Observed Diffusions](https://mlsanity.com/paper/2109.02371/) (2021)
- [Unbiased Parameter Inference for a Class of Partially Observed Levy-Process Models](https://mlsanity.com/paper/2112.13874/) (2021)

## Built on by (most cited first, 3 of 3)

- [Reflection coupling for unadjusted generalized Hamiltonian Monte Carlo in the nonconvex stochastic gradient case](https://mlsanity.com/paper/2310.18774/) (2023)
- [Kinetic Langevin Splitting Schemes for Constrained Sampling](https://mlsanity.com/paper/2603.23397/) (2026)
- [Score-Based Martingale Posteriors for Deep Neural Networks](https://mlsanity.com/paper/2606.15725/) (2026)

## Abstract (arXiv)

In this work we consider the unbiased estimation of expectations w.r.t.~probability measures that have non-negative Lebesgue density, and which are known point-wise up-to a normalizing constant. We focus upon developing an unbiased method via the underdamped Langevin dynamics, which has proven to be popular of late due to applications in statistics and machine learning. Specifically in continuous-time, the dynamics can be constructed so that as the time goes to infinity they admit the probability of interest as a stationary measure. In many cases, time-discretized versions of the underdamped Langevin dynamics are used in practice which are run only with a fixed number of iterations. We develop a novel scheme based upon doubly randomized estimation as in citeubgrad,discmodel, which requires access only to time-discretized versions of the dynamics. The proposed scheme aims to remove the dicretization bias and the bias resulting from running the dynamics for a finite number of iterations. We prove, under standard assumptions, that our estimator is of finite variance and either has finite expected cost, or has finite cost with a high probability. To illustrate our theoretical findings we provide numerical experiments which verify our theory, which include challenging examples from Bayesian statistics and statistical physics.