# Quantifying the accuracy of approximate diffusions and Markov chains

arXiv: https://arxiv.org/abs/1605.06420 · HTML: https://mlsanity.com/paper/1605.06420/

## Key facts

- Cited by 7 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 20 May 2016.

## Built on by (most cited first, 7 of 7)

- [Stationary Density Estimation of Ito Diffusions Using Deep Learning](https://mlsanity.com/paper/2109.03992/) (2021)
- [Error Bounds of the Invariant Statistics in Machine Learning of Ergodic Ito Diffusions](https://mlsanity.com/paper/2105.10102/) (2021)
- [Convergence of the Inexact Langevin Algorithm in KL Divergence with Application to Score-based Generative Models](https://mlsanity.com/paper/2211.01512/) (2022)
- [Simultaneous Approximation of the Score Function and Its Derivatives by Deep Neural Networks](https://mlsanity.com/paper/2512.23643/) (2025)
- [Dimension-Free Multimodal Sampling via Preconditioned Annealed Langevin Dynamics](https://mlsanity.com/paper/2602.01449/) (2026)
- [Stochastic Gradient Piecewise Deterministic Monte Carlo Samplers](https://mlsanity.com/paper/2406.19051/) (2024)
- [Scalable Monte Carlo for Bayesian Learning](https://mlsanity.com/paper/2407.12751/) (2024)

## Abstract (arXiv)

Markov chains and diffusion processes are indispensable tools in machine learning and statistics that are used for inference, sampling, and modeling. With the growth of large-scale datasets, the computational cost associated with simulating these stochastic processes can be considerable, and many algorithms have been proposed to approximate the underlying Markov chain or diffusion. A fundamental question is how the computational savings trade off against the statistical error incurred due to approximations. This paper develops general results that address this question. We bound the Wasserstein distance between the equilibrium distributions of two diffusions as a function of their mixing rates and the deviation in their drifts. We show that this error bound is tight in simple Gaussian settings. Our general result on continuous diffusions can be discretized to provide insights into the computational-statistical trade-off of Markov chains. As an illustration, we apply our framework to derive finite-sample error bounds of approximate unadjusted Langevin dynamics. We characterize computation-constrained settings where, by using fast-to-compute approximate gradients in the Langevin dynamics, we obtain more accurate samples compared to using the exact gradients. Finally, as an additional application of our approach, we quantify the accuracy of approximate zig-zag sampling. Our theoretical analyses are supported by simulation experiments.