Hessian-Free High-Resolution Nesterov Acceleration For Sampling

Nesterov's Accelerated Gradient (NAG) for optimization has better performance than its continuous time limit (noiseless kinetic Langevin) when a finite step-size is employed (Shi et al., 2021) . This work explores the sampling counterpart of this phenonemon and proposes a diffusion process, whose discretizations can yield accelerated gradient-based MCMC methods. More precisely, we reformulate the optimizer of NAG for strongly convex functions (NAG-SC) as a Hessian-Free High-Resolution ODE, change its high-resolution coefficient to a hyperparameter, inject appropriate noise, and discretize the resulting diffusion process. The acceleration effect of the new hyperparameter is quantified and it is not an artificial one created by time-rescaling. Instead, acceleration beyond underdamped Langevin in W 2 distance is quantitatively established for log-stronglyconcave-and-smooth targets, at both the continuous dynamics level and the discrete algorithm level. Empirical experiments in both log-stronglyconcave and multi-modal cases also numerically demonstrate this acceleration.