  We propose a generalization of the lasso that allows the model coefficients
to vary as a function of a general set of modifying variables. These modifiers
might be variables such as gender, age or time. The paradigm is quite general,
with each lasso coefficient modified by a sparse linear function of the
modifying variables $Z$. The model is estimated in a hierarchical fashion to
control the degrees of freedom and avoid overfitting. The modifying variables
may be observed, observed only in the training set, or unobserved overall.
There are connections of our proposal to varying coefficient models and
high-dimensional interaction models. We present a computationally efficient
algorithm for its optimization, with exact screening rules to facilitate
application to large numbers of predictors. The method is illustrated on a
number of different simulated and real examples.
