Entropy hypocoercivity and query complexity for the bouncy particle sampler

We study convergence in relative entropy of the bouncy particle sampler with Gaussian velocity refreshment, for a target $e^{-U}$ on ${\mathbb{R}}^d$ with $U$ $m$-strongly convex and $L$-smooth, and condition number $\kappa =L\mathrm{/}m$. For initial laws with bounded density relative to equilibrium, we show that the relative entropy decays exponentially up to an arbitrary additive error $\delta$, with a rate that depends only logarithmically on the density bound and on $\delta$. From a Gaussian cold start and with a suitably chosen refreshment rate, the relative entropy falls below $\delta$ after time $\tilde{O}(\sqrt{d\mathrm{/}m})$. Combined with exact simulation by windowed thinning, this gives an expected gradient-query complexity of $\tilde{O}(d\sqrt{\kappa })$, improving the $\tilde{O}(d^2\sqrt{\kappa })$ bound based on $L^2$ hypocoercivity.