Optimal algorithms for differentially private stochastic monotone variational inequalities and saddle-point problems

• 成果类型: Article
• 署名作者: Boob, Digvijay; Guzman, Cristobal
• 署名单位: Southern Methodist University; Pontificia Universidad Catolica de Chile
• 刊物名称: MATHEMATICAL PROGRAMMING
• ISSN/ISSBN: 0025-5610
• DOI: 10.1007/s10107-023-01953-5
• 发表日期: 2024
• 页码: 255-297
• 关键词: operators; STABILITY

摘要

In this work, we conduct the first systematic study of stochastic variational inequality (SVI) and stochastic saddle point (SSP) problems under the constraint of differential privacy (DP). We propose two algorithms: Noisy Stochastic Extragradient (NSEG) and Noisy Inexact Stochastic Proximal Point (NISPP). We show that a stochastic approximation variant of these algorithms attains risk bounds vanishing as a function of the dataset size, with respect to the strong gap function; and a sampling with replacement variant achieves optimal risk bounds with respect to a weak gap function. We also show lower bounds of the same order on weak gap function. Hence, our algorithms are optimal. Key to our analysis is the investigation of algorithmic stability bounds, both of which are new even in the nonprivate case. The dependence of the running time of the sampling with replacement algorithms, with respect to the dataset size n, is n(2) for NSEG and O (n(3/2)) for NISPP.