Gradient perturbation, widely used for differentially private optimization, injects noise at every iterative update to guarantee differential privacy. Previous work first determines the noise level that can satisfy the privacy requirement and then analyzes the utility of noisy gradient updates as in non-private case. In this paper, we explore how the privacy noise affects the optimization property. We show that for differentially private convex optimization, the utility guarantee of both DP-GD and DP-SGD is determined by an \emph{expected curvature} rather than the minimum curvature. The \emph{expected curvature} represents the average curvature over the optimization path, which is usually much larger than the minimum curvature and hence can help us achieve a significantly improved utility guarantee. By using the \emph{expected curvature}, our theory justifies the advantage of gradient perturbation over other perturbation methods and closes the gap between theory and practice. Extensive experiments on real world datasets corroborate our theoretical findings.

本文探讨梯度扰动在差分私有性上优劣的影响。我们发现在不同凸优化问题中，期望曲率可更好地决定噪声扰动的实际效果，而不是最小曲率。进一步实验表明使用高级组合方法的梯度扰动比其他扰动方法表现更好。

梯度扰动在差分隐私凸优化中的价值被低估了