  In multi-task learning several related tasks are considered simultaneously,
with the hope that by an appropriate sharing of information across tasks, each
task may benefit from the others. In the context of learning linear functions
for supervised classification or regression, this can be achieved by including
a priori information about the weight vectors associated with the tasks, and
how they are expected to be related to each other. In this paper, we assume
that tasks are clustered into groups, which are unknown beforehand, and that
tasks within a group have similar weight vectors. We design a new spectral norm
that encodes this a priori assumption, without the prior knowledge of the
partition of tasks into groups, resulting in a new convex optimization
formulation for multi-task learning. We show in simulations on synthetic
examples and on the IEDB MHC-I binding dataset, that our approach outperforms
well-known convex methods for multi-task learning, as well as related non
convex methods dedicated to the same problem.
