Algorithmic Aspects of the Log-Laplace Transform and a Non-Euclidean Proximal Sampler

• URL: http://arxiv.org/abs/2302.06085v1
• Date: Mon, 13 Feb 2023 04:08:41 GMT
• Title: Algorithmic Aspects of the Log-Laplace Transform and a Non-Euclidean Proximal Sampler
• Authors: Sivakanth Gopi, Yin Tat Lee, Daogao Liu, Ruoqi Shen, Kevin Tian
• Abstract summary: 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 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.
• Score: 23.166642097170545
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• 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.

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