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：发展适用于非欧几里得几何的高效采样算法一直是极具挑战性的课题，因为欧几里得框架下成功的离散化技术难以直接推广到更一般的场景。我们构建了近期[LST21]提出的近端采样器的非欧几里得类比方法，该方法通过密度函数的对数-拉普拉斯变换（LLT）自然引入正则化机制。我们证明了LLT具有强凸-光滑对偶性及等周不等式等数学新性质（兼具算法内涵），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]相当的混合时间。作为主要应用，我们展示了所提出的热启动采样器能够将$\ell_p$范数和Schatten-$p$范数（$p \in [1, 2]$）下差分隐私凸优化的预言复杂度降低至与欧几里得场景[GLL22]匹配的水平，同时保持最优的过风险界[GLLST23]。我们相信对LLT的研究作为设计采样器工具的概念验证具有广阔前景，并为未来探索指明了方向。