  Recent increase in online privacy concerns prompts the following question:
can a recommender system be accurate if users do not entrust it with their
private data? To answer this, we study the problem of learning item-clusters
under local differential privacy, a powerful, formal notion of data privacy. We
develop bounds on the sample-complexity of learning item-clusters from
privatized user inputs. Significantly, our results identify a sample-complexity
separation between learning in an information-rich and an information-scarce
regime, thereby highlighting the interaction between privacy and the amount of
information (ratings) available to each user.
  In the information-rich regime, where each user rates at least a constant
fraction of items, a spectral clustering approach is shown to achieve a
sample-complexity lower bound derived from a simple information-theoretic
argument based on Fano's inequality. However, the information-scarce regime,
where each user rates only a vanishing fraction of items, is found to require a
fundamentally different approach both for lower bounds and algorithms. To this
end, we develop new techniques for bounding mutual information under a notion
of channel-mismatch, and also propose a new algorithm, MaxSense, and show that
it achieves optimal sample-complexity in this setting.
  The techniques we develop for bounding mutual information may be of broader
interest. To illustrate this, we show their applicability to $(i)$ learning
based on 1-bit sketches, and $(ii)$ adaptive learning, where queries can be
adapted based on answers to past queries.
