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.

翻译：核方法是具有坚实理论基础的学习算法，但面临重要的计算限制。草图化是一种通过降低维度子空间寻找解的方法，已被广泛研究以缓解这些计算负担。然而，统计上精确的草图（如高斯草图）通常包含少量零元素，导致其应用于核方法及其非稀疏格拉姆矩阵时仍较慢。本文证明，稀疏化高斯（及拉德马赫）草图仍能产生理论上有效的近似，同时通过高效的分解技巧实现显著的时间和空间节省。为支持该方法，我们针对单输出和多输出核问题推导出过量风险界，适用于一般利普希茨损失，从而为从鲁棒回归到多分位数回归的广泛应用提供新保证。理论结果通过实验得到补充，表明我们的方法在经验上优于最先进的草图化方法。