Smoothed Picard Hamiltonian Monte Carlo

We develop a new low-accuracy sampler, called 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 α I preceq nabla2 V preceq β I, with condition number kappa := β/α, smoothed Picard HMC returns a sample with √ α,W2(·,π) ≤ varepsilon using widetilde O(kappa2 + kappa7/6 d1/6/varepsilon1/3) gradient queries. We also prove stronger Wq bounds, and then develop an algorithmic framework, the recursive warm start generator, to upgrade these Wq 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((kappa7/6 d1/6 + kappa1/2 d1/4)polylog(1/varepsilon)).