Abstract--In some real-world applications, data samples are usually distributed on local devices, where federated learning (FL) techniques are proposed to coordinate decentralized clients without directly sharing users' private data. FL commonly follows the parameter server architecture and contains multiple personalization and aggregation procedures. In this paper, we focus on a special kind of Non-I.I.D. scene where clients own incomplete classes, i.e., each client can only access a partial set of the whole class set. The server aims to aggregate a complete classification model that could generalize to all classes, while the clients are inclined to improve the performance of distinguishing their observed classes. For better model aggregation, we point out that the standard softmax will encounter several problems caused by missing classes and propose "restricted softmax" as an alternative. For better model personalization, we point out that the hard-won personalized models are not well exploited and propose "inherited private model" to store the personalization experience. Our proposed algorithm named MAP could simultaneously achieve the aggregation and personalization goals in FL. Abundant experimental studies verify the superiorities of our algorithm. Index Terms--Federated learning, Non-I.I.D. data, label distribution shift, aggregation, personalization, incomplete classes, restricted softmax, inherited private model Due cost, data from individual participants can not be located to these advantages, FL has been successfully applied to on the same device in some real-world applications [3].