Learning Kernel Based Halfspaces with the 0-1 Loss
Shai Shalev-Shwartz, Ohad Shamir and Karthik Sridharan
SIAM Journal on Computing 2011.

Abstract

We describe and analyze a new algorithm for agnostically learning kernel-based halfspaces with respect to the 0-1 loss function. Unlike most previous formulations which rely on surrogate convex loss functions (e.g. hinge-loss in SVM and log-loss in logistic regression), we provide nite time/sample guarantees with respect to the more natural 0-1 loss function. The proposed algorithm can learn kernel-based halfspaces in worst-case time poly(exp(L log(L=))), for any distribution, where L is a Lipschitz constant (which can be thought of as the reciprocal of the margin), and the learned classier is worse than the optimal halfspace by at most . We also prove a hardness result, showing that under a certain cryptographic assumption, no algorithm can learn kernel-based halfspaces in time polynomial in L

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