R\'enyi-infinity constrained sampling with $d^3$ membership queries

Uniform sampling over a convex body is a fundamental algorithmic problem, yet the convergence in KL or R\'enyi divergence of most samplers remains poorly understood. In this work, we propose a constrained proximal sampler, a principled and simple algorithm that possesses elegant convergence guarantees. Leveraging the uniform ergodicity of this sampler, we show that it converges in the R\'enyi-infinity divergence ($\mathcal R_\infty$) with no query complexity overhead when starting from a warm start. This is the strongest of commonly considered performance metrics, implying rates in $\{\mathcal R_q, \mathsf{KL}\}$ convergence as special cases. By applying this sampler within an annealing scheme, we propose an algorithm which can approximately sample $\varepsilon$-close to the uniform distribution on convex bodies in $\mathcal R_\infty$-divergence with $\widetilde{\mathcal{O}}(d^3\, \text{polylog} \frac{1}{\varepsilon})$ query complexity. This improves on all prior results in $\{\mathcal R_q, \mathsf{KL}\}$-divergences, without resorting to any algorithmic modifications or post-processing of the sample. It also matches the prior best known complexity in total variation distance.