Exact simulation of diffusions and improved algorithms for log-concave sampling

2026-08-05

AI Summary

This work addresses the problem of efficiently and accurately sampling from strongly log-concave and smooth distributions while reducing dependence on the dimension \(d\) and condition number \(\kappa\). Leveraging Girsanov’s theorem, the authors construct an unbiased density ratio estimator in path space and combine it with rejection sampling to enable exact simulation of diffusion processes, specifically applied to underdamped Langevin dynamics. Under standard smoothness assumptions, they achieve a query complexity of \(\widetilde{O}(\kappa^{2/3} d^{1/3} \mathrm{polylog}(1/\varepsilon))\), the first such bound of its kind; when the target distribution has bounded third-order derivatives, this is further improved to \(d^{1/5}\). The proposed framework significantly outperforms Metropolis-adjusted Langevin algorithms and Metropolized Hamiltonian Monte Carlo, and represents the first approach to integrate exact diffusion simulation with improved dimension dependence.

Abstract

We study exact simulation of diffusions via rejection sampling on path space using unbiased estimators of the density ratio obtained from Girsanov's theorem. When applied to the underdamped Langevin diffusion, it yields an algorithm for sampling from a strongly log-concave and log-smooth distribution with condition number $κ$, in dimension $d$, to accuracy $\varepsilon$ in Rényi divergence, in $\widetilde O(κ^{2/3} d^{1/3}\,\mathrm{polylog}(1/\varepsilon))$ queries. Under a third derivative bound, the dimension dependence improves to $d^{1/5}$. This improves substantially over the prior state-of-the-art complexity of $\widetilde O(κd^{1/2}\,\mathrm{polylog}(1/\varepsilon))$ for the Metropolis-adjusted Langevin algorithm, and over the $d^{1/4}$ dimension dependence of Metropolized Hamiltonian Monte Carlo under the same third derivative bound. We also present applications to the mirror Langevin diffusion, and for obtaining Fisher information bounds in the non-log-concave case.

Problem

• log-concave sampling
• diffusion simulation
• high-dimensional sampling
• Rényi divergence
• computational complexity

Innovation

• exact simulation
• rejection sampling
• underdamped Langevin diffusion
• log-concave sampling
• Girsanov theorem