High-accuracy simulation of Picard HMC, part I: Gaussian cloud correction

We study the problem of sampling from a continuous density $\pi \propto \exp (-V)$ on ${\mathbb{R}}^d$, where $V\in C^2({\mathbb{R}}^d)$ has a $\beta$-Lipschitz gradient and $\pi$ satisfies a logarithmic Sobolev inequality with constant ${\alpha }^{-1}$, and write $\kappa =\beta \mathrm{/}\alpha$. We introduce the Gaussian cloud sampler, which achieves total variation accuracy $\epsilon$ using $\tilde{O}(\kappa d^{1\mathrm{/}5}\text{polylog}(1\mathrm{/}\epsilon ))$ gradient queries in expectation. The algorithm uses first-order rejection sampling (FORS) to correct the law of smoothed Picard HMC trajectories. To do so, we represent the iterates of the ideal Picard iteration via Gaussian clouds, whose centers are never evaluated, and we develop a suite of likelihood correction gadgets which only use samples from this indirect cloud representation.