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但存在显著计算局限性的学习算法。通过对降维子空间中的解进行搜索的草图技术，是缓解此类计算负担的重要研究途径。然而，高斯草图等具有高统计精度的草图方法通常包含极少的零元素，导致其在核方法及非稀疏Gram矩阵中的应用仍然面临速度瓶颈。本文证明，稀疏化高斯（及Rademacher）草图不仅能产生理论有效的近似，更可通过高效的\textit{分解技巧}实现显著的时间与空间节省。为支撑该方法，我们针对单输出与多输出核问题推导了通用Lipschitz损失下的超额风险界，从而为从鲁棒回归到多分位数回归的广泛实际应用提供了新理论保障。通过实验验证，我们的方法在经验性能上全面超越当前最优的草图技术。