# Finding Differentially Private Second Order Stationary Points in Stochastic Minimax Optimization

arXiv: https://arxiv.org/abs/2602.01339 · HTML: https://mlsanity.com/paper/2602.01339/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 19 of its 27 references that are in the map.
- First posted on arXiv 1 Feb 2026.

## Builds on (most cited first, 19 of 19)

- [Towards Deep Learning Models Resistant to Adversarial Attacks](https://mlsanity.com/paper/1706.06083/) (2017)
- [SPIDER: Near-Optimal Non-Convex Optimization via Stochastic Path Integrated Differential Estimator](https://mlsanity.com/paper/1807.01695/) (2018)
- [Escaping From Saddle Points --- Online Stochastic Gradient for Tensor Decomposition](https://mlsanity.com/paper/1503.02101/) (2015)
- [Global Optimality of Local Search for Low Rank Matrix Recovery](https://mlsanity.com/paper/1605.07221/) (2016)
- [Stochastic Recursive Gradient Descent Ascent for Stochastic Nonconvex-Strongly-Concave Minimax Problems](https://mlsanity.com/paper/2001.03724/) (2020)
- [Weakly-Convex Concave Min-Max Optimization: Provable Algorithms and Applications in Machine Learning](https://mlsanity.com/paper/1810.02060/) (2018)
- [Multi-Agent Reinforcement Learning via Double Averaging Primal-Dual Optimization](https://mlsanity.com/paper/1806.00877/) (2018)
- [Faster Rates of Convergence to Stationary Points in Differentially Private Optimization](https://mlsanity.com/paper/2206.00846/) (2022)
- [Linear Convergence of the Primal-Dual Gradient Method for Convex-Concave Saddle Point Problems without Strong Convexity](https://mlsanity.com/paper/1802.01504/) (2018)
- [Differentially Private SGDA for Minimax Problems](https://mlsanity.com/paper/2201.09046/) (2022)
- [Bring Your Own Algorithm for Optimal Differentially Private Stochastic Minimax Optimization](https://mlsanity.com/paper/2206.00363/) (2022)
- [Accelerating Inexact HyperGradient Descent for Bilevel Optimization](https://mlsanity.com/paper/2307.00126/) (2023)
- [Finding Second-Order Stationary Points in Nonconvex-Strongly-Concave Minimax Optimization](https://mlsanity.com/paper/2110.04814/) (2021)
- [Private (Stochastic) Non-Convex Optimization Revisited: Second-Order Stationary Points and Excess Risks](https://mlsanity.com/paper/2302.09699/) (2023)
- [Differentially Private Worst-group Risk Minimization](https://mlsanity.com/paper/2402.19437/) (2024)
- [Differentially Private Temporal Difference Learning with Stochastic Nonconvex-Strongly-Concave Optimization](https://mlsanity.com/paper/2201.10447/) (2022)
- [Improved Rates of Differentially Private Nonconvex-Strongly-Concave Minimax Optimization](https://mlsanity.com/paper/2503.18317/) (2025)
- [Second-Order Convergence in Private Stochastic Non-Convex Optimization](https://mlsanity.com/paper/2505.15647/) (2025)
- [Adaptive Batch Size for Privately Finding Second-Order Stationary Points](https://mlsanity.com/paper/2410.07502/) (2024)

## Abstract (arXiv)

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.