  Private information retrieval (PIR) protocols ensure that a user can download
a file from a database without revealing any information on the identity of the
requested file to the servers storing the database. While existing protocols
strictly impose that no information is leaked on the file's identity, this work
initiates the study of the tradeoffs that can be achieved by relaxing the
perfect privacy requirement. We refer to such protocols as weakly-private
information retrieval (WPIR) protocols. In particular, for the case of multiple
noncolluding replicated servers, we study how the download rate, the upload
cost, and the access complexity can be improved when relaxing the full privacy
constraint. To quantify the information leakage on the requested file's
identity we consider mutual information (MI), worst-case information leakage,
and maximal leakage (MaxL). We present two WPIR schemes, denoted by Scheme A
and Scheme B, based on two recent PIR protocols and show that the download rate
of the former can be optimized by solving a convex optimization problem. We
also show that Scheme A achieves an improved download rate compared to the
recently proposed scheme by Samy et al. under the so-called $\epsilon$-privacy
metric. Additionally, a family of schemes based on partitioning is presented.
Moreover, we provide an information-theoretic converse bound for the maximum
possible download rate for the MI and MaxL privacy metrics under a practical
restriction on the alphabet size of queries and answers. For two servers and
two files, the bound is tight under the MaxL metric, which settles the WPIR
capacity in this particular case. Finally, we compare the performance of the
proposed schemes and their gap to the converse bound.
