Private information retrieval protocols guarantee that a user can privately and losslessly retrieve a single file from a database stored across multiple servers. In this work, we propose to simultaneously relax the conditions of perfect retrievability and privacy in order to obtain improved download rates in the single server scenario, i.e., all files are stored uncoded on a single server. In particular, we derive the optimal tradeoff between download rate, distortion, and information leakage when the file size is infinite and the information leakage is measured in terms of the average success probability for the server of correctly guessing the identity of the requested file. Moreover, we present a novel approach based on linear programming to construct schemes for a finite file size and an arbitrary number of files. When the database contains at most four bits, this approach can be leveraged to find provably optimal schemes.