GAFIN: Gradient direction-agnostic differential privacy control for federated learning via nussbaum-enhanced PID tracking

Handling uncertain gradient descent directions poses a critical challenge in privacy-preserving federated learning (FL) systems under Gaussian differential privacy (GDP). Classical Proportional-Integral-Derivative (PID) controllers, while effective in tracking target privacy budgets, may suffer from convergence failures when facing direction uncertainty caused by DP noise and heterogeneous client updates. To address this, we propose GAFIN, a novel gradient direction-agnostic control framework that integrates a Nussbaum-type function into a PID controller. The Nussbaum modulation dynamically adjusts the control gain to compensate for unknown or varying gradient directions, thereby enabling stable and robust tracking of the desired privacy budget. Extensive experiments across multiple benchmark datasets and privacy settings demonstrate that GAFIN preserves the fast convergence behavior of PID control while consistently accelerating convergence by 20.9% to 80.0% across MLP and CNN models on MNIST, FEMNIST, and CIFAR10, effectively balancing privacy protection and communication efficiency in FL systems under varying initial privacy budgets.