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，在所有经典场景（强对数凹、对数凹、对数索博列夫不等式（LSI）、庞加莱不等式）以及具有半光滑或复合势能的更一般场景中，均实现了更优的复杂度界。该算法基于~citet{lee2021structured}引入的近端采样器，其性能由受限高斯预言机（RGO）——近端采样器中的关键步骤——所决定。本文的主要贡献在于通过近似拒绝采样实现了RGO的非精确版本。为约束RGO的非精确性，我们针对高斯分布上的半光滑函数建立了新的集中不等式，扩展了经典的Lipschitz函数集中不等式。将我们的RGO实现应用于近端采样器后，在几乎所有场景中均达到了当前最优的复杂度界。例如，对于强对数凹分布，本方法在无需热启动的情况下获得了复杂度界$\tilde\mathcal{O}(\kappa d^{1/2})$，优于MALA的极小极大界。对于满足LSI的分布，我们的复杂度界为$\tilde \mathcal{O}(\hat \kappa d^{1/2})$（其中$\hat \kappa$为光滑度与LSI常数之比），优于所有现有结果。