Fast Kernel Methods for Generic Lipschitz Losses via $p$-Sparsified Sketches

TL;DR

This paper introduces a novel sparsified sketching technique for kernel methods with Lipschitz losses, achieving significant computational efficiency while maintaining theoretical accuracy, and demonstrates its effectiveness through empirical results.

Contribution

It presents a new sparsified Gaussian and Rademacher sketching method with an efficient decomposition trick, providing theoretical guarantees and practical improvements over existing sketching approaches.

Findings

The method achieves faster computation and reduced memory usage.

Theoretical excess risk bounds are established for various kernel problems.

Empirical results show superior performance compared to state-of-the-art sketching methods.

Abstract

Kernel methods are learning algorithms that enjoy solid theoretical foundations while suffering from important computational limitations. Sketching, which consists in looking for solutions among a subspace of reduced dimension, is a well studied approach to alleviate these computational burdens. However, statistically-accurate sketches, such as the Gaussian one, usually contain few null entries, such that their application to kernel methods and their non-sparse Gram matrices remains slow in practice. In this paper, we show that sparsified Gaussian (and Rademacher) sketches still produce theoretically-valid approximations while allowing for important time and space savings thanks to an efficient \emph{decomposition trick}. To support our method, we derive excess risk bounds for both single and multiple output kernel problems, with generic Lipschitz losses, hereby providing new guarantees for a wide range of applications, from robust regression to multiple quantile regression. Our theoretical results are complemented with experiments showing the empirical superiority of our approach over SOTA sketching methods.