Conventional synchronous federated learning (SFL) frameworks suffer from performance degradation in heterogeneous systems due to imbalanced local data size and diverse computing power on the client side. To address this problem, asynchronous FL (AFL) and semi-asynchronous FL have been proposed to recover the performance loss by allowing asynchronous aggregation. However, asynchronous aggregation incurs a new problem of inconsistency between local updates and global updates. Motivated by the issues of conventional SFL and AFL, we first propose a time-driven SFL (T-SFL) framework for heterogeneous systems. The core idea of T-SFL is that the server aggregates the models from different clients, each with varying numbers of iterations, at regular time intervals. To evaluate the learning performance of T-SFL, we provide an upper bound on the global loss function. Further, we optimize the aggregation weights to minimize the developed upper bound. Then, we develop a discriminative model selection (DMS) algorithm that removes local models from clients whose number of iterations falls below a predetermined threshold. In particular, this algorithm ensures that each client's aggregation weight accurately reflects its true contribution to the global model update, thereby improving the efficiency and robustness of the system. To validate the effectiveness of T-SFL with the DMS algorithm, we conduct extensive experiments using several popular datasets including MNIST, Cifar-10, Fashion-MNIST, and SVHN. The experimental results demonstrate that T-SFL with the DMS algorithm can reduce the latency of conventional SFL by 50\%, while achieving an average 3\% improvement in learning accuracy over state-of-the-art AFL algorithms.

The directional mean shift (DMS) algorithm is a nonparametric method for pursuing local modes of densities defined by kernel density estimators on the unit hypersphere. In this paper, we show that any DMS iteration can be viewed as a generalized Expectation-Maximization (EM) algorithm; in particular, when the von Mises kernel is applied, it becomes an exact EM algorithm. Under the (generalized) EM framework, we provide a new proof for the ascending property of density estimates and demonstrate the global convergence of directional mean shift sequences. Finally, we give a new insight into the linear convergence of the DMS algorithm.