  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
when all files are stored uncoded on a single server. Information leakage is
measured in terms of the average success probability for the server of
correctly guessing the identity of the desired file. The main findings are: i)
The derivation of the optimal tradeoff between download rate, distortion, and
information leakage when the file size is infinite. Closed-form expressions of
the optimal tradeoff for the special cases of "no-leakage" and "no-privacy" are
also given. ii) A novel approach based on linear programming (LP) to construct
schemes for a finite file size and an arbitrary number of files. The proposed
LP approach can be leveraged to find provably optimal schemes with
corresponding closed-form expressions for the rate-distortion-leakage tradeoff
when the database contains at most four bits.
  Finally, for a database that contains 320 bits, we compare two construction
methods based on the LP approach with a nonconstructive scheme downloading
subsets of files using a finite-length lossy compressor based on random coding.
