Optimal algorithms for differentially private stochastic monotone variational inequalities and saddle-point problems
首次系统研究差分隐私约束下的随机变分不等式和随机鞍点问题，提出两种最优算法并证明其风险界随数据集大小递减，运行时间分别为n²和Õ(n^{3/2})。
Abstract 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 $$n^2$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mi>n</mml:mi> <mml:mn>2</mml:mn> </mml:msup> </mml:math> for NSEG and $${\widetilde{O}}(n^{3/2})$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mover> <mml:mi>O</mml:mi> <mml:mo>~</mml:mo> </mml:mover> <mml:mrow> <mml:mo>(</mml:mo> <mml:msup> <mml:mi>n</mml:mi> <mml:mrow> <mml:mn>3</mml:mn> <mml:mo>/</mml:mo> <mml:mn>2</mml:mn> </mml:mrow> </mml:msup> <mml:mo>)</mml:mo> </mml:mrow> </mml:mrow> </mml:math> for NISPP.