We propose new Markov chain Monte Carlo algorithms to sample a uniform distribution on a convex body $K$. Our algorithms are based on the proximal sampler, which uses Gibbs sampling on an augmented distribution and assumes access to the so-called restricted Gaussian oracle (RGO). The key contribution of this work is an efficient implementation of the RGO for uniform sampling on convex $K$ that goes beyond the membership-oracle model used in many classical and modern uniform samplers, and instead leverages richer oracle access commonly assumed in convex optimization. We implement the RGO via rejection sampling and access to either a projection oracle or a separation oracle on $K$. In both oracle models, we provide non-asymptotic complexity guarantees for obtaining unbiased samples, with accuracy quantified in Rényi divergence and $χ^2$-divergence, and we support these theoretical guarantees with numerical experiments.

翻译：本文提出了一种新的马尔可夫链蒙特卡洛算法，用于对凸体 $K$ 上的均匀分布进行采样。我们的算法基于近端采样器，该采样器在增广分布上使用吉布斯采样，并假设能够访问所谓的受限高斯预言机（RGO）。本工作的核心贡献是为凸体 $K$ 上的均匀采样提供了一种高效的 RGO 实现方法，它超越了众多经典和现代均匀采样器中使用的成员预言机模型，转而利用了凸优化中通常假设的更丰富的预言机访问能力。我们通过拒绝采样并结合对 $K$ 的投影预言机或分离预言机的访问来实现 RGO。在这两种预言机模型中，我们为获得无偏样本提供了非渐近复杂度保证，其精度通过 Rényi 散度和 $χ^2$-散度进行量化，并通过数值实验验证了这些理论保证。