Rényi-infinity constrained sampling with $d^3$ membership queries
Yunbum Kook, Matthew S. Zhang
TL;DR
The paper investigates uniform sampling from a convex body ${\mathcal K}$ presented via a membership oracle, aiming for convergence guarantees in the strong Rényi-$\infty$ metric. It introduces the constrained ${\mathsf{Proximal\ sampler}}$, which realizes uniform or truncated Gaussian sampling with ${\mathcal R}_\infty$ convergence from an ${\mathcal O}(1)$ warm-start, and shows that annealing this sampler via ${\mathsf{Proximal\ Gaussian\ cooling}}$ achieves ${\varepsilon}$-closeness to the uniform distribution in ${\mathcal R}_\infty$ with ${\tilde{O}}(d^3 \mathrm{polylog}(1/\varepsilon))$ membership queries. The work links warm-start analysis and boosting from ${\mathcal R}_q$ or TV to ${\mathcal R}_\infty$, leveraging log-Sobolev inequalities and uniform ergodicity to avoid overhead and post-processing. Compared to prior TV-based approaches, this yields stronger guarantees without algorithmic modifications and matches the best known total-variation complexity in the stronger ${\mathcal R}_\infty$ metric. The results provide a principled, simple framework for constrained sampling with strong privacy-related divergences, opening avenues for broader applicability to other constrained distributions and more general annealing strategies.
Abstract
Uniform sampling over a convex body is a fundamental algorithmic problem, yet the convergence in KL or Rényi 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 Rényi-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.