$p$-Sparsified Sketches for Fast Multiple Output Kernel Methods

Kernel methods are learning algorithms that enjoy solid theoretical foundations while suffering from important computational limitations. Sketching, that consists in looking for solutions among a subspace of reduced dimension, is a widely studied approach to alleviate this numerical burden. However, fast sketching strategies, such as non-adaptive subsampling, significantly degrade the guarantees of the algorithms, while theoretically-accurate sketches, such as the Gaussian one, turn out to remain relatively slow in practice. In this paper, we introduce the $p$-sparsified sketches, that combine the benefits from both approaches to achieve a good tradeoff between statistical accuracy and computational efficiency. To support our method, we derive excess risk bounds for both single and multiple output problems, with generic Lipschitz losses, providing new guarantees for a wide range of applications, from robust regression to multiple quantile regression. We also provide empirical evidences of the superiority of our sketches over recent SOTA approaches.