# Langevin Quasi-Monte Carlo

arXiv: https://arxiv.org/abs/2309.12664 · HTML: https://mlsanity.com/paper/2309.12664/

## Key facts

- Cited by 4 papers in the map (data through 14 Oct 2026).
- Builds on 17 of its 48 references that are in the map.
- First posted on arXiv 22 Sep 2023.

## Builds on (most cited first, 17 of 17)

- [Non-convex learning via Stochastic Gradient Langevin Dynamics: a nonasymptotic analysis](https://mlsanity.com/paper/1702.03849/) (2017)
- [Underdamped Langevin MCMC: A non-asymptotic analysis](https://mlsanity.com/paper/1707.03663/) (2017)
- [User-friendly guarantees for the Langevin Monte Carlo with inaccurate gradient](https://mlsanity.com/paper/1710.00095/) (2017)
- [Global Convergence of Langevin Dynamics Based Algorithms for Nonconvex Optimization](https://mlsanity.com/paper/1707.06618/) (2017)
- [Consistency and fluctuations for stochastic gradient Langevin dynamics](https://mlsanity.com/paper/1409.0578/) (2014)
- [The promises and pitfalls of Stochastic Gradient Langevin Dynamics](https://mlsanity.com/paper/1811.10072/) (2018)
- [Global Non-convex Optimization with Discretized Diffusions](https://mlsanity.com/paper/1810.12361/) (2018)
- [On the Convergence of Stochastic Gradient MCMC Algorithms with High-Order Integrators](https://mlsanity.com/paper/1610.06665/) (2016)
- [Stochastic Runge-Kutta Accelerates Langevin Monte Carlo and Beyond](https://mlsanity.com/paper/1906.07868/) (2019)
- [Control Variates for Stochastic Gradient MCMC](https://mlsanity.com/paper/1706.05439/) (2017)
- [Quasi-Monte Carlo Variational Inference](https://mlsanity.com/paper/1807.01604/) (2018)
- [Bridging the Gap between Stochastic Gradient MCMC and Stochastic Optimization](https://mlsanity.com/paper/1512.07962/) (2015)
- [Sampling with Mirrored Stein Operators](https://mlsanity.com/paper/2106.12506/) (2021)
- [Higher-order Quasi-Monte Carlo Training of Deep Neural Networks](https://mlsanity.com/paper/2009.02713/) (2020)
- [Bootstrap inference after using multiple queries for model selection](https://mlsanity.com/paper/1612.07811/) (2016)
- [Policy Learning and Evaluation with Randomized Quasi-Monte Carlo](https://mlsanity.com/paper/2202.07808/) (2022)
- [Structured Monte Carlo Sampling for Nonisotropic Distributions via Determinantal Point Processes](https://mlsanity.com/paper/1905.12667/) (2019)

## Built on by (most cited first, 4 of 4)

- [Combining Normalizing Flows and Quasi-Monte Carlo](https://mlsanity.com/paper/2401.05934/) (2024)
- [Prediction-Enhanced Monte Carlo: A Machine Learning View on Control Variate](https://mlsanity.com/paper/2412.11257/) (2024)
- [Transport Quasi-Monte Carlo](https://mlsanity.com/paper/2412.16416/) (2024)
- [Lattice-based Deep Neural Networks: Regularity and Tailored Regularization](https://mlsanity.com/paper/2603.02809/) (2026)

## Abstract (arXiv)

Langevin Monte Carlo (LMC) and its stochastic gradient versions are powerful algorithms for sampling from complex high-dimensional distributions. To sample from a distribution with density π(θ)propto exp(-U(θ)) , LMC iteratively generates the next sample by taking a step in the gradient direction nabla U with added Gaussian perturbations. Expectations w.r.t. the target distribution π are estimated by averaging over LMC samples. In ordinary Monte Carlo, it is well known that the estimation error can be substantially reduced by replacing independent random samples by quasi-random samples like low-discrepancy sequences. In this work, we show that the estimation error of LMC can also be reduced by using quasi-random samples. Specifically, we propose to use completely uniformly distributed (CUD) sequences with certain low-discrepancy property to generate the Gaussian perturbations. Under smoothness and convexity conditions, we prove that LMC with a low-discrepancy CUD sequence achieves smaller error than standard LMC. The theoretical analysis is supported by compelling numerical experiments, which demonstrate the effectiveness of our approach.