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We investigate the problem of semantic private information retrieval (semantic PIR). In semantic PIR, a user retrieves a message out of K independent messages stored in N replicated and non-colluding databases without revealing the identity of the desired message to any individual database. The messages come with different semantics, i.e., the messages are allowed to have non-uniform a priori probabilities denoted by (p i > 0, i ∈ [K]), which are a proxy for their respective popularity of retrieval, and arbitrary message sizes). This is a generalization of the classical private information retrieval (PIR) problem, where messages are assumed to have equal a priori probabilities and equal message sizes. We derive the semantic PIR capacity for general K, N . The results show that the semantic PIR capacity depends on the number of databases N , the number of messages K, the a priori probability distribution of messages p i , and the message sizes L i . We present two achievable semantic PIR schemes: The first one is a deterministic scheme which is based on message asymmetry. This scheme employs non-uniform subpacketization. The second scheme is probabilistic and is based on choosing one query set out of multiple options at random to retrieve the required message without the need for exponential subpacketization. We derive necessary and sufficient conditions for the semantic PIR capacity to exceed the classical PIR capacity with equal priors and sizes. Our results show that the semantic PIR capacity can be larger than the classical PIR capacity when longer messages have higher popularities. However, when messages are equal-length, the non-uniform priors cannot be exploited to improve the retrieval rate over the classical PIR capacity.

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We investigate the problem of private read update write (PRUW) with heterogeneous storage constrained databases in federated submodel learning (FSL). In FSL a machine learning (ML) model is divided into multiple submodels based on different types of data used to train it. A given user downloads, updates and uploads the updates back to a single submodel of interest, based on the type of user's local data. With PRUW, the process of reading (downloading) and writing (uploading) is carried out such that information theoretic privacy of the updating submodel index and the values of updates is guaranteed. We consider the practical scenario where the submodels are stored in databases with arbitrary (heterogeneous) storage constraints, and provide a PRUW scheme with a storage mechanism that utilizes submodel partitioning and encoding to minimize the communication cost.