Zeroth-order Logconcave Sampling

We study the zeroth-order query complexity of sampling from a general logconcave distribution: given access to an evaluation oracle for a convex function $V:\mathbb{R}^{d}\rightarrow\mathbb{R}\cup\{\infty\}$, output a point from a distribution within $\varepsilon$-distance to the density proportional to $e^{-V}$. A long line of work provides efficient algorithms for this problem in TV distance, assuming a pointwise warm start (i.e., in $\infty$-R\'enyi divergence), and using annealing to generate such a warm start. Here, we address the natural and more general problem of using a $q$-R\'enyi divergence warm start to generate a sample that is $\varepsilon$-close in $q$-R\'enyi divergence. Our first main result is an algorithm with this end-to-end guarantee with state-of-the-art complexity for $q=\widetilde{\Omega}(1)$. Our second result shows how to generate a $q$-R\'enyi divergence warm start directly via annealing, by maintaining $q$-R\'enyi divergence throughout, thereby obtaining a streamlined analysis and improved complexity. Such results were previously known only under the stronger assumptions of smoothness and access to first-order oracles. We also show a lower bound for Gaussian annealing by disproving a geometric conjecture about quadratic tilts of isotropic logconcave distributions. Central to our approach, we establish hypercontractivity of the heat adjoint and translate this into improved mixing time guarantees for the Proximal Sampler. The resulting analysis of both sampling and annealing follows a simplified and natural path, directly tying convergence rates to isoperimetric constants of the target distribution.