Margins, Kernels and Non-linear Smoothed Perceptrons
This work addresses kernel-based classification with theoretical guarantees, offering improved convergence rates and feasibility certificates, though it appears incremental as it builds on classical Perceptron and Von-Neumann algorithms.
The paper tackles the problem of finding non-linear classifiers in RKHS for both separable and non-separable cases, deriving an accelerated smoothed algorithm with a convergence rate of √(log n)/ρ for separable points and a primal-dual algorithm that halts in min{√n/|ρ|, √n/ε} iterations to provide a separator or certificate.
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 %use the Representer Theorem to rephrase it 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 $\tfrac{\sqrt {\log n}}ρ$ given $n$ separable points, which is strikingly similar to the classical kernelized Perceptron algorithm whose rate is $\tfrac1{ρ^2}$. 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 $\min\{\tfrac{\sqrt n}{|ρ|}, \tfrac{\sqrt n}ε\}$ iterations with a perfect separator in the RKHS if the primal is feasible or a dual $ε$-certificate of near-infeasibility.