The development of efficient sampling algorithms catering to non-Euclidean geometries has been a challenging endeavor, as discretization techniques which succeed in the Euclidean setting do not readily carry over to more general settings. We develop a non-Euclidean analog of the recent proximal sampler of [LST21], which naturally induces regularization by an object known as the log-Laplace transform (LLT) of a density. We prove new mathematical properties (with an algorithmic flavor) of the LLT, such as strong convexity-smoothness duality and an isoperimetric inequality, which are used to prove a mixing time on our proximal sampler matching [LST21] under a warm start. As our main application, we show our warm-started sampler improves the value oracle complexity of differentially private convex optimization in $\ell_p$ and Schatten-$p$ norms for $p \in [1, 2]$ to match the Euclidean setting [GLL22], while retaining state-of-the-art excess risk bounds [GLLST23]. We find our investigation of the LLT to be a promising proof-of-concept of its utility as a tool for designing samplers, and outline directions for future exploration.

翻译：针对非欧几里得几何的高效采样算法开发一直是一项具有挑战性的工作，因为欧几里得框架中成功的离散化技术难以直接推广至更一般的场景。我们构建了近期[LST21]提出的近端采样器的非欧几里得类比，该采样器通过称为密度对数-拉普拉斯变换（LLT）的物体自然引入正则化。我们证明了LLT的若干新数学性质（具有算法化特征），包括强凸性-光滑性对偶性以及等周不等式，并利用这些性质证明了在热启动条件下，我们的近端采样器具有与[LST21]一致的混合时间。作为主要应用，我们展示：对于$p \in [1, 2]$的$\ell_p$范数和Schatten-$p$范数，该热启动采样器将差分隐私凸优化的值预言复杂度降低至与欧几里得设置[GLL22]匹配的水平，同时保持最先进的过风险界[GLLST23]。我们相信对LLT的研究作为采样器设计工具具有良好前景，并概述了未来探索方向。