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With the proliferation of smart devices having built-in sensors, Internet connectivity, and programmable computation capability in the era of Internet of things (IoT), tremendous data is being generated at the network edge. Federated learning is capable of analyzing the large amount of data from a distributed set of smart devices without requiring them to upload their data to a central place. However, the commonly-used federated learning algorithm is based on stochastic gradient descent (SGD) and not suitable for resource-constrained IoT environments due to its high communication resource requirement. Moreover, the privacy of sensitive data on smart devices has become a key concern and needs to be protected rigorously. This paper proposes a novel federated learning framework called DP-PASGD for training a machine learning model efficiently from the data stored across resource-constrained smart devices in IoT while guaranteeing differential privacy. The optimal schematic design of DP-PASGD that maximizes the learning performance while satisfying the limits on resource cost and privacy loss is formulated as an optimization problem, and an approximate solution method based on the convergence analysis of DP-PASGD is developed to solve the optimization problem efficiently. Numerical results based on real-world datasets verify the effectiveness of the proposed DP-PASGD scheme.