Accelerated High-Accuracy Sampling from a Warm Start via the Proximal Bouncy Particle Sampler

RESUMO

We study the problem of sampling from $\mu(\mathrm{d}x)\propto e^{-V(x)}\,\mathrm{d}x$ on $\mathbb{R}^d$, where $V$ is $\alpha$-strongly convex and $\beta$-smooth, and write $\kappa:=\beta/\alpha$. We design and analyze the Proximal Bouncy Particle Sampler (Proximal BPS), a new sampler that combines ideas from the proximal sampler and the bouncy particle sampler. From a warm start initialization with $ O(1) $ Rényi divergence w.r.t. $\mu$, Proximal BPS returns a sample whose law is $\varepsilon$-close to $\mu$ in total variation distance using $\widetilde O(\sqrt\kappa\,d^{1/4} \,\mathrm{polylog}(1/\varepsilon))$ gradient queries in expectation.