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arXiv 2602.01339cs.LG
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在随机博弈优化中寻找差分隐私的二阶 stationary 点

Finding Differentially Private Second Order Stationary Points in Stochastic Minimax Optimization

King Abdullah University of Science and Technology(国王阿卜杜勒-阿齐兹大学);Provable Responsible AI and Data Analytics (PRADA) Lab(可证责任AI和数据分析师实验室);State University of New York at Buffalo(纽约州立大学布法罗分校);Seoul National University(首尔国立大学)

Difei Xu, Youming Tao, Meng Ding, Chenglin Fan, Di Wang

AI总结:

本文提出了一种结合梯度下降-上升和方差减少的算法,在随机博弈优化中实现差分隐私的二阶 stationary 点。

摘要:

我们首次研究了在随机(非凸)博弈优化中寻找差分隐私(DP)二阶 stationary 点(SOSP)的问题。现有文献要么只关注博弈问题中的第一阶 stationary 点,要么只关注经典随机最小化问题中的 SOSP。本文首次对经验风险和总体风险进行了统一且详细的处理。具体来说,我们提出了一种纯第一阶方法,结合嵌套梯度下降-上升方案、SPIDER 风格的方差减少和高斯扰动以确保隐私。关键的技术设备是块状(q-周期)分析,它通过不求和整个迭代时间跨度来控制随机方差和隐私噪声的累积,从而实现了经验风险和总体风险的统一处理。在标准平滑性、Hessian- Lipschitz 性和强凹性假设下,我们建立了达到 (α,√(ρ_Φα)) 近似二阶 stationary 点的高概率保证,其中 α=O((√d/(nε))^(2/3)) 对于经验风险目标,和 O(1/n^(1/3)) + ( √d/(nε) )^(1/2) 对于总体目标,与已知的最佳隐私第一阶 stationary 的速率相匹配。

英文摘要:

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 $(α,\sqrt{ρ_Φα})$-approximate second-order stationary point with $α= \mathcal{O}( (\frac{\sqrt{d}}{n\varepsilon})^{2/3})$ for empirical risk objectives and $\mathcal{O}(\frac{1}{n^{1/3}} + (\frac{\sqrt{d}}{n\varepsilon})^{1/2})$ for population objectives, matching the best known rates for private first-order stationarity.

机构、总结、中文摘要均由AI分析而来,可能存在错误,请以论文正文为准。

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