  Search engines answer users' queries by listing relevant items (e.g.
documents, songs, products, web pages, ...). These engines rely on algorithms
that learn to rank items so as to present an ordered list maximizing the
probability that it contains relevant item. The main challenge in the design of
learning-to-rank algorithms stems from the fact that queries often have
different meanings for different users. In absence of any contextual
information about the query, one often has to adhere to the {\it diversity}
principle, i.e., to return a list covering the various possible topics or
meanings of the query. To formalize this learning-to-rank problem, we propose a
natural model where (i) items are categorized into topics, (ii) users find
items relevant only if they match the topic of their query, and (iii) the
engine is not aware of the topic of an arriving query, nor of the frequency at
which queries related to various topics arrive, nor of the topic-dependent
click-through-rates of the items. For this problem, we devise LDR (Learning
Diverse Rankings), an algorithm that efficiently learns the optimal list based
on users' feedback only. We show that after $T$ queries, the regret of LDR
scales as $O((N-L)\log(T))$ where $N$ is the number of all items. We further
establish that this scaling cannot be improved, i.e., LDR is order optimal.
Finally, using numerical experiments on both artificial and real-world data, we
illustrate the superiority of LDR compared to existing learning-to-rank
algorithms.
