Hessian-Free High-Resolution Nesterov Acceleration for Sampling

We propose an accelerated-gradient-based MCMC method. It relies on a modification of the Nesterov's accelerated gradient method for strongly convex functions (NAG-SC): We reformulate NAG-SC as a Hessian-Free High-Resolution ODE, lift its high-resolution coefficient to be a hyperparameter, and then inject appropriate noise and discretize the resulting diffusion process. Accelerated sampling enabled by the new hyperparameter is theoretically quantified. At continuous-time level, for log-concave/log-strongly-concave target measures, exponential convergence in $\chi^2$ divergence/2-Wasserstein distance is proved, with a rate analogous to the state-of-the-art results of underdamped Langevin dynamics, plus an \textbf{additional acceleration}. At discrete algorithm level, a dedicated discretization algorithm is proposed to simulate the Hessian-Free High-Resolution SDE in a cost-efficient manner. For log-strong-concave target measures, the proposed algorithm achieves $\tilde{\mathcal{O}}(\frac{\sqrt{d}}{\epsilon})$ iteration complexity in 2-Wasserstein distance, same as underdamped Langevin dynamics, but with a reduced constant. Empirical experiments are conducted to numerically verify our theoretical results and demonstrate accelerated sampling in practice.