Margins, Kernels and Non-linear Smoothed Perceptrons

TL;DR

This paper develops accelerated algorithms for finding non-linear classifiers in RKHS, providing convergence guarantees and certificates of non-existence, extending classical perceptron and Von-Neumann methods.

Contribution

It introduces a smoothed, accelerated algorithm with improved convergence rates for kernelized perceptrons and extends Gordan's theorem to RKHSs for primal-dual certification.

Findings

Achieves convergence rate of √(log n)/ρ for separable data.

Provides primal and dual certificates for feasibility and near-infeasibility.

Extends classical perceptron analysis to non-linear RKHS settings.

Abstract

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 ($\rho$) 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 $\tfrac{\sqrt {\log n}}{\rho}$ given $n$ separable points, which is strikingly similar to the classical kernelized Perceptron algorithm whose rate is $\tfrac1{\rho^2}$. 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 $\min\{\tfrac{\sqrt n}{|\rho|}, \tfrac{\sqrt n}{\epsilon}\}$ iterations with a perfect separator in the RKHS if the primal is feasible or a dual $\epsilon$-certificate of near-infeasibility.