A quantum extension of SVM-perf for training nonlinear SVMs in almost linear time

TL;DR

This paper introduces a quantum algorithm that significantly accelerates training nonlinear SVMs, achieving near-linear time complexity and extending quantum speedups to more practical soft-margin models.

Contribution

It presents a quantum extension of the SVM-perf algorithm that scales linearly with training data size for nonlinear SVMs, unlike previous quantum methods.

Findings

Classical simulation shows practical potential for the quantum algorithm.

Algorithm achieves near-linear scaling in training data size.

Extends quantum speedup to soft-margin nonlinear SVMs.

Abstract

We propose a quantum algorithm for training nonlinear support vector machines (SVM) for feature space learning where classical input data is encoded in the amplitudes of quantum states. Based on the classical SVM-perf algorithm of Joachims, our algorithm has a running time which scales linearly in the number of training examples $m$ (up to polylogarithmic factors) and applies to the standard soft-margin $\ell_1$-SVM model. In contrast, while classical SVM-perf has demonstrated impressive performance on both linear and nonlinear SVMs, its efficiency is guaranteed only in certain cases: it achieves linear $m$ scaling only for linear SVMs, where classification is performed in the original input data space, or for the special cases of low-rank or shift-invariant kernels. Similarly, previously proposed quantum algorithms either have super-linear scaling in $m$, or else apply to different SVM models such as the hard-margin or least squares $\ell_2$-SVM which lack certain desirable properties of the soft-margin $\ell_1$-SVM model. We classically simulate our algorithm and give evidence that it can perform well in practice, and not only for asymptotically large data sets.