Smoothed Picard Hamiltonian Monte Carlo

Abstract

We develop a new low-accuracy sampler, called \emph{smoothed Picard Hamiltonian Monte Carlo}, which combines Gaussian smoothing, Picard iteration, and higher-order discretization. For a log-concave target $π\propto \exp(-V)$ in dimension $d$ satisfying $0 \prec αI \preceq \nabla^2 V \preceq βI$, with condition number $κ:= β/α$, smoothed Picard HMC returns a sample with $\sqrt α\,W_2(\cdot,π) \le \varepsilon$ using $\widetilde O(κ^2 + κ^{7/6} d^{1/6}/\varepsilon^{1/3})$ gradient queries. We also prove stronger $W_q$ bounds, and then develop an algorithmic framework, the recursive warm start generator, to upgrade these $W_q$ bounds to stronger divergence guarantees. This produces a warm start for the proximal bouncy particle sampler, introduced in a companion work, leading to a high-accuracy log-concave sampler with complexity $\widetilde O((κ^{7/6} d^{1/6} + κ^{1/2} d^{1/4})\mathrm{polylog}(1/\varepsilon))$.