We propose a sampling algorithm that achieves superior complexity bounds in all the classical settings (strongly log-concave, log-concave, Logarithmic-Sobolev inequality (LSI), Poincar\'e inequality) as well as more general settings with semi-smooth or composite potentials. Our algorithm is based on the proximal sampler introduced in~\citet{lee2021structured}. The performance of this proximal sampler is determined by that of the restricted Gaussian oracle (RGO), a key step in the proximal sampler. The main contribution of this work is an inexact realization of RGO based on approximate rejection sampling. To bound the inexactness of RGO, we establish a new concentration inequality for semi-smooth functions over Gaussian distributions, extending the well-known concentration inequality for Lipschitz functions. Applying our RGO implementation to the proximal sampler, we achieve state-of-the-art complexity bounds in almost all settings. For instance, for strongly log-concave distributions, our method has complexity bound $\tilde\mathcal{O}(\kappa d^{1/2})$ without warm start, better than the minimax bound for MALA. For distributions satisfying the LSI, our bound is $\tilde \mathcal{O}(\hat \kappa d^{1/2})$ where $\hat \kappa$ is the ratio between smoothness and the LSI constant, better than all existing bounds.

翻译：我们提出了一种采样算法，它在所有经典设置（强对数凹、对数凹、对数索博列夫不等式（LSI）、庞加莱不等式）以及更一般的半光滑或复合势函数设置中均实现了更优的复杂度界。该算法基于~\citet{lee2021structured} 提出的近端采样器。此近端采样器的性能由限制性高斯预言机（RGO）决定，而 RGO 是近端采样器中的关键步骤。本文的主要贡献是基于近似拒绝采样对 RGO 的非精确实现。为了界定 RGO 的非精确性，我们建立了一个适用于高斯分布上半光滑函数的新浓度不等式，扩展了著名的 Lipschitz 函数浓度不等式。将我们的 RGO 实现应用于近端采样器，我们在几乎所有设置中都达到了最先进的复杂度界。例如，对于强对数凹分布，我们的方法在无预热启动时的复杂度界为 $\tilde\mathcal{O}(\kappa d^{1/2})$，优于 MALA 的极小极大界。对于满足 LSI 的分布，我们的界为 $\tilde \mathcal{O}(\hat \kappa d^{1/2})$，其中 $\hat \kappa$ 是光滑性与 LSI 常数之比，优于所有现有界。