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 sampling with replacement variants of these algorithms attain the optimal risk for DP-SVI and DP-SSP. Key to our analysis is the investigation of algorithmic stability bounds, both of which are new even in the nonprivate case, together with a novel "stability implies generalization" result for the gap functions for SVI and SSP problems. The dependence of the running time of these algorithms, with respect to the dataset size $n$, is $n^2$ for NSEG and $\widetilde{O}(n^{3/2})$ for NISPP.
翻译:在这项工作中,我们首次系统研究差异性差异性不平等问题和差异性隐私限制(DP)下的问题。我们提出了两种算法:噪音性蒸汽外移(NSEG)和噪音性不异性蒸汽近效点(NISPP),我们显示,这些算法的抽样和替代变种为DP-SVI和DP-SSP提供了最佳风险。我们分析的关键是调查算法稳定性界限,两者即使在非私人案件中都是新的,以及新颖的SVI和SSP问题差距功能的“稳定性意味着普遍化”结果。这些算法运行时间对数据集规模美元的依赖性是NSEG2美元,而NEPP的运行时间是$2美元,NEPP的运行时间是$2美元。