Optimal Algorithms for Differentially Private Stochastic Monotone Variational Inequalities and Saddle-Point Problems

TL;DR

This paper introduces optimal differentially private algorithms for stochastic variational inequalities and saddle-point problems, providing theoretical guarantees and matching lower bounds, advancing privacy-preserving optimization methods.

Contribution

It proposes two novel algorithms, NSEG and NISPP, with proven optimal risk bounds and stability analysis for private stochastic variational inequality and saddle-point problems.

Findings

Algorithms achieve risk bounds vanishing with dataset size

Sampling with replacement algorithms are optimal

Lower bounds match the algorithms' risk bounds

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 $n^2$ for NSEG and $\tilde{O}(n^{3/2})$ for NISPP.