ABSTRACT In federated learning (FL)‐assisted Internet of Things (IoT) systems, FL trains models using datasets on various client devices without sending the datasets to a centralized server. This approach enhances the accuracy and reliability of models while preserving the privacy of client devices. However, FL implementations face challenges, such as single point of server failure and lack of incentives. To address the server failure issue, a backup server can be added. Meanwhile, each FL client has varying data quality and motivations to participate, leading to differences in the quality of local models uploaded to the server. To motivate clients to contribute more, we designed a novel incentive mechanism based on the Stackelberg game. This mechanism allocates rewards based on the quality of the models each client uploads, rather than the amount of data trained. We separately modelled the utilities of the server and the clients, allowing the server to rationally allocate rewards based on each client's contribution to model training. After analysing the utilities, we transform the game into two optimization problems and develop an algorithm whose per‐round complexity scales linearly with the number of clients under fixed numerical tolerances. The obtained equilibrium matches exhaustive search within numerical precision while significantly reducing computation.

We consider the problem of estimating a sparse Gaussian Graphical Model with a special graph topological structure and more than a million variables. Most previous scalable estimators still contain expensive calculation steps (e.g., matrix inversion or Hessian matrix calculation) and become infeasible in high-dimensional scenarios, where p (number of variables) is larger than n (number of samples). To overcome this challenge, we propose a novel method, called Fast and Scalable Inverse Covariance Estimator by Thresholding (FST). FST first obtains a graph structure by applying a generalized threshold to the sample covariance matrix. Then, it solves multiple block-wise subproblems via element-wise thresholding. By using matrix thresholding instead of matrix inversion as the computational bottleneck, FST reduces its computational complexity to a much lower order of magnitude (O(p2)). We show that FST obtains the same sharp convergence rate O(√(log max{p, n}/n) as other state-of-the-art methods. We validate the method empirically, on multiple simulated datasets and one real-world dataset, and show that FST is two times faster than the four baselines while achieving a lower error rate under both Frobenius-norm and max-norm.