Establishing an $\Omega (\sqrt{d})$ complexity lower bound for PDMP samplers and how to break it: a sub-$\sqrt{d}$ algorithm for Gaussian-tailed targets

Abstract

Despite the theoretical appeal of their non-reversibility, to date, no Piecewise Deterministic Markov Process (PDMP) samplers have been developed that scale better than $\mathcal{O}(\sqrt{d})$ in computational complexity with respect to the target dimension $d$. We prove that this is a fundamental limitation by establishing an $\Omega (\sqrt{d})$ lower bound on the algorithmic complexity of PDMP samplers in a standard setup. By relaxing the assumption that the target density must remain invariant at all continuous times, we then demonstrate how to bypass this barrier. Specifically, we introduce a novel PDMP sampling scheme and show that it achieves an empirical complexity of $\mathcal{O}(d^{\alpha })$, where $\alpha \in [0.2,0.3]$ for Gaussian-tailed targets. In addition, this PDMP scheme is locally adaptive in both trajectory length and distance between velocity updates.