  Large and complex datasets are often collected from several, possibly
heterogeneous sources. Collaborative learning methods improve efficiency by
leveraging commonalities across datasets while accounting for possible
differences among them. Here we study collaborative linear regression and
contextual bandits, where each instance's associated parameters are equal to a
global parameter plus a sparse instance-specific term. We propose a novel
two-stage estimator called MOLAR that leverages this structure by first
constructing an entry-wise median of the instances' linear regression
estimates, and then shrinking the instance-specific estimates towards the
median. MOLAR improves the dependence of the estimation error on the data
dimension, compared to independent least squares estimates. We then apply MOLAR
to develop methods for sparsely heterogeneous collaborative contextual bandits,
which lead to improved regret guarantees compared to independent bandit
methods. We further show that our methods are minimax optimal by providing a
number of lower bounds. Finally, we support the efficiency of our methods by
performing experiments on both synthetic data and the PISA dataset on student
educational outcomes from heterogeneous countries.
