In the classic regression problem, the value of a real-valued random variable $Y$ is to be predicted based on the observation of a random vector $X$, taking its values in $\mathbb{R}^d$ with $d\geq 1$ say. The statistical learning problem consists in building a predictive function $\hat{f}:\mathbb{R}^d\to \mathbb{R}$ based on independent copies of the pair $(X,Y)$ so that $Y$ is approximated by $\hat{f}(X)$ with minimum error in the mean-squared sense. Motivated by various applications, ranging from environmental sciences to finance or insurance, special attention is paid here to the case of extreme (i.e. very large) observations $X$. Because of their rarity, they contribute in a negligible manner to the (empirical) error and the predictive performance of empirical quadratic risk minimizers can be consequently very poor in extreme regions. In this paper, we develop a general framework for regression in the extremes. It is assumed that $X$'s conditional distribution given $Y$ belongs to a non parametric class of heavy-tailed probability distributions. It is then shown that an asymptotic notion of risk can be tailored to summarize appropriately predictive performance in extreme regions of the input space. It is also proved that minimization of an empirical and non asymptotic version of this 'extreme risk', based on a fraction of the largest observations solely, yields regression functions with good generalization capacity. In addition, numerical results providing strong empirical evidence of the relevance of the approach proposed are displayed.

翻译：在经典回归问题中，需基于随机向量$X$的观测值预测实值随机变量$Y$的取值，其中$X$取值于$\mathbb{R}^d$空间（$d\geq 1$）。统计学习任务旨在通过$(X,Y)$的独立同分布样本构建预测函数$\hat{f}:\mathbb{R}^d\to \mathbb{R}$，使得在均方误差意义下，$\hat{f}(X)$能最优逼近$Y$。受环境科学、金融及保险等领域应用的驱动，本文重点关注极端（即极大）观测值$X$的情形。由于极端值具有稀缺性，其对（经验）误差的贡献可忽略不计，因此经验二次风险最小化器在极端区域的预测性能可能极差。本文构建了一个适用于极端情况的回归通用框架。假设给定$Y$时$X$的条件分布属于重尾概率分布的非参数类别，进而证明了可通过定制渐近风险概念来恰当总结输入空间极端区域的预测性能。同时证实，仅基于最大观测值子集的经验非渐近版本的“极端风险”最小化，能够生成具有良好泛化能力的回归函数。最后，数值实验为所提方法的相关性提供了强有力的经验证据。