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

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 $\kappa$, in dimension $d$, to accuracy $\epsilon$ in Rényi divergence, in $\tilde{O}({\kappa }^{2/3}{d}^{1/3}\mathrm{polylog}(1/\epsilon ))$ 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 $\tilde{O}(\kappa d^{1/2}\mathrm{polylog}(1/\epsilon ))$ 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.