Adaptive Batch Size for Privately Finding Second-Order Stationary Points
Daogao Liu, Kunal Talwar
TL;DR
This work addresses privately finding second-order stationary points (SOSP) for non-convex objectives under differential privacy. Building on SpiderBoost, it introduces adaptive batch sizes and a tree mechanism to produce accurate gradient and Hessian estimates while preserving privacy, and it adds a Hessian-driven saddle-point escape procedure. The main result shows an $\alpha$-SOSP with $\alpha=\tilde{O}\left(\frac{1}{n^{1/3}}+(\frac{\sqrt{d}}{n\varepsilon})^{1/2}\right)$, matching the state-of-the-art for private FOSP and suggesting SOSP can be achieved at no extra private cost under standard assumptions. This advances private non-convex optimization by resolving a gap in prior guarantees and highlighting the effectiveness of adaptive batching and the tree mechanism in DP settings, with implications for private optimization in high-dimensional models.
Abstract
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) claimed that an $α$-SOSP can be found with $α=O(\frac{1}{n^{1/3}}+(\frac{\sqrt{d}}{nε})^{3/7})$, where $n$ is the dataset size, $d$ is the dimension, and $ε$ is the differential privacy parameter. However, a recent analysis revealed an issue in their saddle point escape procedure, leading to weaker guarantees. Building on the SpiderBoost algorithm framework, we propose a new approach that uses adaptive batch sizes and incorporates the binary tree mechanism. Our method not only corrects this issue but also improves the results for privately finding an SOSP, achieving $α=O(\frac{1}{n^{1/3}}+(\frac{\sqrt{d}}{nε})^{1/2})$. This improved bound matches the state-of-the-art for finding a FOSP, suggesting that privately finding an SOSP may be achievable at no additional cost.