  We consider the problem of algorithmically recommending items to users on a
Yahoo! front page module. Our approach is based on a novel multilevel
hierarchical model that we refer to as a User Profile Model with Graphical
Lasso (UPG). The UPG provides a personalized recommendation to users by
simultaneously incorporating both user covariates and historical user
interactions with items in a model based way. In fact, we build a per-item
regression model based on a rich set of user covariates and estimate individual
user affinity to items by introducing a latent random vector for each user. The
vector random effects are assumed to be drawn from a prior with a precision
matrix that measures residual partial associations among items. To ensure
better estimates of a precision matrix in high-dimensions, the matrix elements
are constrained through a Lasso penalty. Our model is fitted through a
penalized-quasi likelihood procedure coupled with a scalable EM algorithm. We
employ several computational strategies like multi-threading, conjugate
gradients and heavily exploit problem structure to scale our computations in
the E-step. For the M-step we take recourse to a scalable variant of the
Graphical Lasso algorithm for covariance selection. Through extensive
experiments on a new data set obtained from Yahoo! front page and a benchmark
data set from a movie recommender application, we show that our UPG model
significantly improves performance compared to several state-of-the-art methods
in the literature, especially those based on a bilinear random effects model
(BIRE). In particular, we show that the gains of UPG are significant compared
to BIRE when the number of users is large and the number of items to select
from is small. For large item sets and relatively small user sets the results
of UPG and BIRE are comparable. The UPG leads to faster model building and
produces outputs which are interpretable.
