  In this paper we analyze a budgeted learning setting, in which the learner
can only choose and observe a small subset of the attributes of each training
example. We develop efficient algorithms for ridge and lasso linear regression,
which utilize the geometry of the data by a novel data-dependent sampling
scheme. When the learner has prior knowledge on the second moments of the
attributes, the optimal sampling probabilities can be calculated precisely, and
result in data-dependent improvements factors for the excess risk over the
state-of-the-art that may be as large as $O(\sqrt{d})$, where $d$ is the
problem's dimension. Moreover, under reasonable assumptions our algorithms can
use less attributes than full-information algorithms, which is the main concern
in budgeted learning settings. To the best of our knowledge, these are the
first algorithms able to do so in our setting. Where no such prior knowledge is
available, we develop a simple estimation technique that given a sufficient
amount of training examples, achieves similar improvements. We complement our
theoretical analysis with experiments on several data sets which support our
claims.
