Curvature-Aware Clipping: Mitigating Bias in DPSGD via Approximate Second-Order Information

Differentially private stochastic gradient descent (DP-SGD) has become the standard framework for privacy-preserving machine learning, yet its reliance on a fixed gradient clipping threshold to limit sensitivity remains a significant practical limitation. Adaptive clipping algorithms such as AdaClip shift and scale the gradient prior to clipping and adding noise so that the clipped gradient yields a more informative descent direction. The shift and scaling parameters are selected adaptively based on the empirical mean and variance. However, in existing adaptive clipping algorithms, these empirical estimates have not been also used for momentum to accelerate training itself. On the other hand, DP-Adam is an algorithm that exploits Adam-like momentum updates based on the gradient mean and variance to accelerate training, but does not exploit these estimates for adaptive clipping. In this work, we propose Differentially Private Mechanism with Adaptive Clipping and Adaptive Momentum (DP-MacAdam), a novel algorithm that combines these two approaches so as to use the same mean and variance estimates for both clipping and momentum. We perform an analysis showing that DP-MacAdam estimates the gradient variances in a bias-free manner. In addition, we empirically evaluate the privacy and accuracy of DP-MacAdam, demonstrating that it achieves improved model utility compared to DP-SGD, AdaClip, and DP-Adam baselines, without requiring manual tuning of the clipping threshold.

Under a fixed privacy budget, the utility of differentially private (DP) training is ultimately determined by its optimization efficiency. Standard first-order DP optimizers such as DP-SGD rely solely on local gradients and ignore the underlying loss curvature. This geometric blindness causes severe zigzagging in ill-conditioned landscapes, squandering precious privacy budgets on inefficient iterations. Practitioners are thus trapped in a bind: either stop training prematurely or inject massive per-step noise, both of which critically compromise final model utility. Natural Gradient Descent (NGD) resolves this by preconditioning gradients with curvature, aligning updates with the loss geometry and extracting more efficient signal from every noisy step, offering a principled pathway to break the privacy-utility bottleneck. Despite its theoretical appeal, directly integrating NGD with DP introduces fundamental challenges: curvature estimation itself consumes prohibitive privacy budgets, isotropic DP operations conflict with the anisotropic scaling of NGD, and the inverse curvature catastrophically amplify parameter updates in flat directions, causing training instability. We propose DP-NGD, a practical framework that systematically addresses these obstacles by decoupling curvature estimation from private data, reconciling isotropic DP constraints with anisotropic second-order optimization via a whitened-space mechanism, and dynamically clamping the curvature to stabilize training. Extensive experiments on standard benchmarks demonstrate that DP-NGD achieves state-of-the-art accuracy, breaking through the utility ceilings of first-order baselines while delivering up to a $10\times$ convergence speedup under the same privacy budget.

Machine learning's reliance on sensitive data necessitates privacy-preserving techniques like Differentially Private Stochastic Gradient Descent (DPSGD). However, DPSGD suffers from substantial utility degradation and slow convergence due to gradient clipping and noise injection. Prior works have attempted to improve DPSGD from various perspectives; notably, the Differentially Private Selective Update and Release (DPSUR) algorithm has achieved remarkable model utility. However, the privacy accounting in DPSUR overlooks the variation in sampling probability introduced by the selective release mechanism, which compromises the rigor of its privacy guarantees. To address these limitations, we re-evaluate the privacy analysis of the selective release mechanism and propose a novel algorithm: Differentially Private Selective Release based on Clipped Gradients (DPSR-CG). Through a rigorous, newly derived privacy analysis and extensive experiments on multiple datasets (MNIST, CIFAR-10, IMDB, and FMNIST), we demonstrate that our DPSR-CG mechanism maintains strict privacy guarantees while achieving exceptional model performance.