Margins, Kernels and Non-linear Smoothed Perceptrons
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We focus on the problem of finding a non-linear classification function that lies in a Reproducing Kernel Hilbert Space (RKHS) both from the primal point of view (finding a perfect separator when one exists) and the dual point of view (giving a certificate of non-existence), with special focus on generalizations of two classical schemes - the Perceptron (primal) and Von-Neumann (dual) algorithms. We cast our problem as one of maximizing the regularized normalized hard-margin (ρ) in an RKHS and in terms of a Mahalanobis dot-product/semi-norm associated with the kernel's (normalized and signed) Gram matrix. We derive an accelerated smoothed algorithm with a convergence rate of √( n)ρ given n separable points, which is strikingly similar to the classical kernelized Perceptron algorithm whose rate is 1ρ^2. When no such classifier exists, we prove a version of Gordan's separation theorem for RKHSs, and give a reinterpretation of negative margins. This allows us to give guarantees for a primal-dual algorithm that halts in {√(n)|ρ|, √(n)ϵ} iterations with a perfect separator in the RKHS if the primal is feasible or a dual ϵ-certificate of near-infeasibility.
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