Finding Differentially Private Second Order Stationary Points in Stochastic Minimax Optimization

We provide the first study of the problem of finding differentially private (DP) second-order stationary points (SOSP) in stochastic (non-convex) minimax optimization. Existing literature either focuses only on first-order stationary points for minimax problems or on SOSP for classical stochastic minimization problems. This work provides, for the first time, a unified and detailed treatment of both empirical and population risks. Specifically, we propose a purely first-order method that combines a nested gradient descent--ascent scheme with SPIDER-style variance reduction and Gaussian perturbations to ensure privacy. A key technical device is a block-wise (q-period) analysis that controls the accumulation of stochastic variance and privacy noise without summing over the full iteration horizon, yielding a unified treatment of both empirical-risk and population formulations. Under standard smoothness, Hessian-Lipschitzness, and strong concavity assumptions, we establish high-probability guarantees for reaching an (α,√ρ_Phi α)-approximate second-order stationary point with α = O( (frac√dnvarepsilon)2/3) for empirical risk objectives and O(frac1n1/3 + (frac√dnvarepsilon)1/2) for population objectives, matching the best known rates for private first-order stationarity.