🤖 AI Summary
Efficient sampling under non-Euclidean geometries—specifically ℓₚ and Schatten-𝑝 norms for 𝑝 ∈ [1,2]—remains challenging due to the failure of standard Euclidean discretization techniques.
Method: This paper introduces a novel proximal sampling framework based on the Log-Laplace Transform (LLT), overcoming the geometric limitations of Euclidean methods. It establishes, for the first time, the strong convexity–smoothness duality of LLT and integrates isoperimetric inequality analysis to design a sampler.
Contribution/Results: The resulting algorithm achieves the same mixing time as state-of-the-art Euclidean proximal samplers under non-Euclidean geometries. Theoretically, it attains optimal oracle complexity O(1/ε) for differentially private convex optimization while preserving state-of-the-art excess risk bounds. This work unifies and extends proximal sampling theory to ℓₚ and Schatten-𝑝 geometries, delivering tight computational complexity guarantees for non-Euclidean private optimization.
📝 Abstract
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.
