Tamim El Ahmad

Tamim El Ahmad, Pierre Laforgue, Florence d'Alché-Buc

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.

Tamim El Ahmad, Luc Brogat-Motte, Pierre Laforgue et al.

Leveraging the kernel trick in both the input and output spaces, surrogate kernel methods are a flexible and theoretically grounded solution to structured output prediction. If they provide state-of-the-art performance on complex data sets of moderate size (e.g., in chemoinformatics), these approaches however fail to scale. We propose to equip surrogate kernel methods with sketching-based approximations, applied to both the input and output feature maps. We prove excess risk bounds on the original structured prediction problem, showing how to attain close-to-optimal rates with a reduced sketch size that depends on the eigendecay of the input/output covariance operators. From a computational perspective, we show that the two approximations have distinct but complementary impacts: sketching the input kernel mostly reduces training time, while sketching the output kernel decreases the inference time. Empirically, our approach is shown to scale, achieving state-of-the-art performance on benchmark data sets where non-sketched methods are intractable.

Tamim El Ahmad, Junjie Yang, Pierre Laforgue et al.

By leveraging the kernel trick in the output space, kernel-induced losses provide a principled way to define structured output prediction tasks for a wide variety of output modalities. In particular, they have been successfully used in the context of surrogate non-parametric regression, where the kernel trick is typically exploited in the input space as well. However, when inputs are images or texts, more expressive models such as deep neural networks seem more suited than non-parametric methods. In this work, we tackle the question of how to train neural networks to solve structured output prediction tasks, while still benefiting from the versatility and relevance of kernel-induced losses. We design a novel family of deep neural architectures, whose last layer predicts in a data-dependent finite-dimensional subspace of the infinite-dimensional output feature space deriving from the kernel-induced loss. This subspace is chosen as the span of the eigenfunctions of a randomly-approximated version of the empirical kernel covariance operator. Interestingly, this approach unlocks the use of gradient descent algorithms (and consequently of any neural architecture) for structured prediction. Experiments on synthetic tasks as well as real-world supervised graph prediction problems show the relevance of our method.