Title: Agnostic Personalized Federated Learning with Kernel Factorization. Abstract: Considering the futuristic scenarios of federated learning at a worldwide scale, it is highly probable that local participants can have their own personalized labels, which might not be compatible with each other even for the same class, and can be also possibly from a variety of multiple domains. Nevertheless, they should be benefited from others while selectively taking helpful knowledge. Toward such extreme scenarios of federated learning, however, most existing approaches are limited in that they often assume: (1) labeling schemes are all synchronized amongst clients; (2) the local data is from the same single dataset (domain). In this sense, we introduce an intensively realistic problem of federated learning, namely Agnostic Personalized Federated Learning (APFL), where any clients, regardless of what they have learned with their personalized labels, can collaboratively learn while benefiting each other. We then study two essential challenges of the agnostic personalized federated learning, which are (1) Label Heterogeneity where local clients learn from the same single domain but labeling schemes are not synchronized with each other and (2) Domain Heterogeneity where the clients learn from the different datasets which can be semantically similar or dissimilar for each other. To tackle these problems, we propose our novel method, namely Similarity Matching and Kernel Factorization (SimFed). Our method measures semantic similarity/dissimilarity between locally learned knowledge and matches/aggregates the relevant ones that are beneficial to each other. Furthermore, we factorize our model parameters into two basis vectors and the sparse masks to effectively capture permutation-robust representations and reduce information loss when aggregating the heterogeneous knowledge. We exhaustively validate our method on both single- and multi-domain datasets, showing that our method outperforms the current state-of-the-art federated learning methods.