  We study a natural extension of classical empirical risk minimization, where
the hypothesis space is a random subspace of a given space. In particular, we
consider possibly data dependent subspaces spanned by a random subset of the
data, recovering as a special case Nystrom approaches for kernel methods.
Considering random subspaces naturally leads to computational savings, but the
question is whether the corresponding learning accuracy is degraded. These
statistical-computational tradeoffs have been recently explored for the least
squares loss and self-concordant loss functions, such as the logistic loss.
Here, we work to extend these results to convex Lipschitz loss functions, that
might not be smooth, such as the hinge loss used in support vector machines.
This unified analysis requires developing new proofs, that use different
technical tools, such as sub-gaussian inputs, to achieve fast rates. Our main
results show the existence of different settings, depending on how hard the
learning problem is, for which computational efficiency can be improved with no
loss in performance.
