Regularized ERM on random subspaces

TL;DR

This paper extends empirical risk minimization to random subspaces, including non-smooth convex losses like hinge loss, demonstrating conditions where computational efficiency improves without sacrificing accuracy.

Contribution

It generalizes previous results to convex Lipschitz losses, providing new proofs and showing when computational gains do not compromise learning performance.

Findings

Computational efficiency can be improved without loss in accuracy in certain settings.

Extension of analysis to non-smooth convex loss functions like hinge loss.

New proof techniques using sub-gaussian inputs for fast rates.

Abstract

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.