  We consider the problem of learning a non-negative linear classifier with a
$1$-norm of at most $k$, and a fixed threshold, under the hinge-loss. This
problem generalizes the problem of learning a $k$-monotone disjunction. We
prove that we can learn efficiently in this setting, at a rate which is linear
in both $k$ and the size of the threshold, and that this is the best possible
rate. We provide an efficient online learning algorithm that achieves the
optimal rate, and show that in the batch case, empirical risk minimization
achieves this rate as well. The rates we show are tighter than the uniform
convergence rate, which grows with $k^2$.
