  The paradigm of Differentially Private SGD~(DP-SGD) can provide a theoretical
guarantee for training data in both centralized and federated settings.
However, the utility degradation caused by DP-SGD limits its wide application
in high-stakes tasks, such as medical image diagnosis. In addition to the
necessary perturbation, the convergence issue is attributed to the information
loss on the gradient clipping. In this work, we propose a general framework
PCDP-SGD, which aims to compress redundant gradient norms and preserve more
crucial top gradient components via projection operation before gradient
clipping. Additionally, we extend PCDP-SGD as a fundamental component in
differential privacy federated learning~(DPFL) for mitigating the data
heterogeneous challenge and achieving efficient communication. We prove that
pre-projection enhances the convergence of DP-SGD by reducing the dependence of
clipping error and bias to a fraction of the top gradient eigenspace, and in
theory, limits cross-client variance to improve the convergence under
heterogeneous federation. Experimental results demonstrate that PCDP-SGD
achieves higher accuracy compared with state-of-the-art DP-SGD variants in
computer vision tasks. Moreover, PCDP-SGD outperforms current federated
learning frameworks when DP is guaranteed on local training sets.
