DC-SGD: Differentially Private SGD with Dynamic Clipping through Gradient Norm Distribution Estimation

Differentially Private Stochastic Gradient Descent (DP-SGD) is a widely adopted technique for privacy-preserving deep learning. A critical challenge in DP-SGD is selecting the optimal clipping threshold C, which involves balancing the trade-off between clipping bias and noise magnitude, incurring substantial privacy and computing overhead during hyperparameter tuning. In this paper, we propose Dynamic Clipping DP-SGD (DC-SGD), a framework that leverages differentially private histograms to estimate gradient norm distributions and dynamically adjust the clipping threshold C. Our framework includes two novel mechanisms: DC-SGD-P and DC-SGD-E. DC-SGD-P adjusts the clipping threshold based on a percentile of gradient norms, while DC-SGD-E minimizes the expected squared error of gradients to optimize C. These dynamic adjustments significantly reduce the burden of hyperparameter tuning C. The extensive experiments on various deep learning tasks, including image classification and natural language processing, show that our proposed dynamic algorithms achieve up to 9 times acceleration on hyperparameter tuning than DP-SGD. And DC-SGD-E can achieve an accuracy improvement of 10.62% on CIFAR10 than DP-SGD under the same privacy budget of hyperparameter tuning. We conduct rigorous theoretical privacy and convergence analyses, showing that our methods seamlessly integrate with the Adam optimizer. Our results highlight the robust performance and efficiency of DC-SGD, offering a practical solution for differentially private deep learning with reduced computational overhead and enhanced privacy guarantees.