Second-order federated learning (FL) algorithms offer faster convergence than their first-order counterparts by leveraging curvature information. However, they are hindered by high computational and storage costs, particularly for large-scale models. Furthermore, the communication overhead associated with large models and digital transmission exacerbates these challenges, causing communication bottlenecks. In this work, we propose a scalable second-order FL algorithm using a sparse Hessian estimate and leveraging over-the-air aggregation, making it feasible for larger models. Our simulation results demonstrate more than $67\%$ of communication resources and energy savings compared to other first and second-order baselines.

Federated learning is a machine learning approach where multiple devices collaboratively learn with the help of a parameter server by sharing only their local updates. While gradient-based optimization techniques are widely adopted in this domain, the curvature information that second-order methods exhibit is crucial to guide and speed up the convergence. This paper introduces a scalable second-order method, allowing the adoption of curvature information in federated large models. Our method, coined Fed-Sophia, combines a weighted moving average of the gradient with a clipping operation to find the descent direction. In addition to that, a lightweight estimation of the Hessian's diagonal is used to incorporate the curvature information. Numerical evaluation shows the superiority, robustness, and scalability of the proposed Fed-Sophia scheme compared to first and second-order baselines.