  While personalized recommendations systems have become increasingly popular,
ensuring user data protection remains a top concern in the development of these
learning systems. A common approach to enhancing privacy involves training
models using anonymous data rather than individual data. In this paper, we
explore a natural technique called \emph{look-alike clustering}, which involves
replacing sensitive features of individuals with the cluster's average values.
We provide a precise analysis of how training models using anonymous cluster
centers affects their generalization capabilities. We focus on an asymptotic
regime where the size of the training set grows in proportion to the features
dimension. Our analysis is based on the Convex Gaussian Minimax Theorem (CGMT)
and allows us to theoretically understand the role of different model
components on the generalization error. In addition, we demonstrate that in
certain high-dimensional regimes, training over anonymous cluster centers acts
as a regularization and improves generalization error of the trained models.
Finally, we corroborate our asymptotic theory with finite-sample numerical
experiments where we observe a perfect match when the sample size is only of
order of a few hundreds.
