There is a gap between finding a first-order stationary point (FOSP) and a second-order stationary point (SOSP) under differential privacy constraints, and it remains unclear whether privately finding an SOSP is more challenging than finding an FOSP. Specifically, Ganesh et al. (2023) demonstrated that an $\alpha$-SOSP can be found with $\alpha=O(\frac{1}{n^{1/3}}+(\frac{\sqrt{d}}{n\epsilon})^{3/7})$, where $n$ is the dataset size, $d$ is the dimension, and $\epsilon$ is the differential privacy parameter. Building on the SpiderBoost algorithm framework, we propose a new approach that uses adaptive batch sizes and incorporates the binary tree mechanism. Our method improves the results for privately finding an SOSP, achieving $\alpha=O(\frac{1}{n^{1/3}}+(\frac{\sqrt{d}}{n\epsilon})^{1/2})$. This improved bound matches the state-of-the-art for finding an FOSP, suggesting that privately finding an SOSP may be achievable at no additional cost.

翻译：在差分隐私约束下，寻找一阶平稳点与二阶平稳点之间存在性能差距，且目前尚不清楚差分隐私下寻找二阶平稳点是否比寻找一阶平稳点更具挑战性。具体而言，Ganesh等人（2023）证明了可以在$\alpha=O(\frac{1}{n^{1/3}}+(\frac{\sqrt{d}}{n\epsilon})^{3/7})$的精度下找到$\alpha$-二阶平稳点，其中$n$为数据集大小，$d$为维度，$\epsilon$为差分隐私参数。基于SpiderBoost算法框架，我们提出了一种采用自适应批量大小并结合二叉树机制的新方法。我们的方法改进了差分隐私下寻找二阶平稳点的结果，实现了$\alpha=O(\frac{1}{n^{1/3}}+(\frac{\sqrt{d}}{n\epsilon})^{1/2})$的精度。这一改进后的边界与当前寻找一阶平稳点的最优结果相匹配，表明差分隐私下寻找二阶平稳点可能无需付出额外代价。