Quantifying the accuracy of approximate diffusions and Markov chains

TL;DR

This paper provides theoretical bounds on the error introduced by approximating diffusions and Markov chains, analyzing the trade-offs between computational efficiency and statistical accuracy in sampling algorithms.

Contribution

It develops general bounds on Wasserstein distance for approximate diffusions and applies these to analyze the accuracy of Langevin dynamics and zig-zag sampling.

Findings

Error bounds are tight in Gaussian cases.

Approximate Langevin dynamics can outperform exact methods in constrained settings.

Quantitative analysis of approximation errors in sampling algorithms.

Abstract

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.