Zeroth-order log-concave sampling

We study the zeroth-order query complexity of log-concave sampling, specifically uniform sampling from convex bodies using membership oracles. We propose a simple variant of the proximal sampler that achieves the query complexity with matched Rényi orders between the initial warmness and output guarantee. Specifically, for any ε>0 and q≥2, the sampler, initialized at π₀, outputs a sample whose law is ε-close in q-Rényi divergence to $π$, the uniform distribution over a convex body in Rᵈ, using O(qMqq/(q-1)d²\,πlog1/ε) membership queries, where Mq=π₀/dπ_Lq(π). We further introduce a simple annealing scheme that produces a warm start in q-Rényi divergence (i.e., Mq=O(1)) using O(qd²R3/2\,π1/4) queries, where R²=E_π[|·|²]. This interpolates between known complexities for warm-start generation in total variation and Rényi-infinity divergence. To relay a Rényi warmness across the annealing scheme, we establish hypercontractivity under simultaneous heat flow and translate it into an improved mixing guarantee for the proximal sampler under a logarithmic Sobolev inequality. These results extend naturally to general log-concave distributions accessible via evaluation oracles, incurring additional quadratic queries.