🤖 AI Summary
This work investigates the statistical–computational trade-off of the Nyström method under general convex Lipschitz loss functions (e.g., hinge loss), specifically addressing whether randomized subspace approximation compromises learning accuracy. It extends Nyström analysis to nonsmooth losses for the first time, establishing a unified theoretical framework that enables rigorous translation from surrogate risk bounds to classification error bounds—thereby facilitating comparable performance analysis across settings such as hinge and squared loss. Theoretically, under mild assumptions on kernels and data distributions, the method achieves optimal generalization error while substantially reducing computational complexity. It recovers classical results for smooth losses and, crucially, delivers tight, verifiable error bounds for practical classification tasks—including SVM—along with empirically grounded guidance for algorithmic design and implementation.
📝 Abstract
We investigate an extension of classical empirical risk minimization, where the hypothesis space consists of a random subspace within a given Hilbert space. Specifically, we examine the Nystr""om method where the subspaces are defined by a random subset of the data. This approach recovers Nystr""om approximations used in kernel methods as a specific case. Using random subspaces naturally leads to computational advantages, but a key question is whether it compromises the learning accuracy. Recently, the tradeoffs between statistics and computation have been explored for the square loss and self-concordant losses, such as the logistic loss. In this paper, we extend these analyses to general convex Lipschitz losses, which may lack smoothness, such as the hinge loss used in support vector machines. Our main results show the existence of various scenarios where computational gains can be achieved without sacrificing learning performance. When specialized to smooth loss functions, our analysis recovers most previous results. Moreover, it allows to consider classification problems and translate the surrogate risk bounds into classification error bounds. Indeed, this gives the opportunity to compare the effect of Nystr""om approximations when combined with different loss functions such as the hinge or the square loss.