Adaptive Batch Size for Privately Finding Second-Order Stationary Points

TL;DR

This paper introduces an adaptive batch size method using the binary tree mechanism to improve the private detection of second-order stationary points, matching the best results for first-order points and addressing previous methodological issues.

Contribution

It presents a novel adaptive batch size approach that corrects prior analysis errors and enhances the guarantees for privately finding second-order stationary points.

Findings

Achieves improved privacy guarantees for SOSP detection.

Matches state-of-the-art results for FOSP detection.

Addresses previous methodological flaws in saddle point escape procedures.

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 $\alpha$-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. 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 $\alpha=O(\frac{1}{n^{1/3}}+(\frac{\sqrt{d}}{n\epsilon})^{1/2})$. This improved bound matches the state-of-the-art for finding a FOSP, suggesting that privately finding an SOSP may be achievable at no additional cost.