  In many scenarios, one uses a large training set to train a model with the
goal of performing well on a smaller testing set with a different distribution.
Learning a weight for each data point of the training set is an appealing
solution, as it ideally allows one to automatically learn the importance of
each training point for generalization on the testing set. This task is usually
formalized as a bilevel optimization problem. Classical bilevel solvers are
based on a warm-start strategy where both the parameters of the models and the
data weights are learned at the same time. We show that this joint dynamic may
lead to sub-optimal solutions, for which the final data weights are very
sparse. This finding illustrates the difficulty of data reweighting and offers
a clue as to why this method is rarely used in practice.
