In pattern recognition, a random label Y is to be predicted based upon observing a random vector X valued in $\mathbb{R}^d$ with d> 1 by means of a classification rule with minimum probability of error.

As a consequence, empirical risk minimizers generally perform very poorly in extreme regions. It is the purpose of this paper to develop a general framework for classification in the extremes.

Representing the meaning of natural language in a mathematically grounded way is a scientific challenge that has received increasing attention withthe explosion of digital content and text data in the last decade.

Representing the meaning of natural language in a mathematically grounded way is a scientific challenge that has received increasing attention withthe explosion of digital content and text data in the last decade.

In pattern recognition, a random label Y is to be predicted based upon observing a random vector X valued in $\mathbb{R}^d$ with d> 1 by means of a classification rule with minimum probability of error.

The dominant approaches to text representation in natural language rely on learning embeddings on massive corpora which have convenient properties such as compositionality and distance preservation.

The dominant approaches to text representation in natural language rely on learning embeddings on massive corpora which have convenient properties such as compositionality and distance preservation.

This paper proposes a formalism for understanding and guaranteeing generalization in extreme regions of feature space. On the applied side, it is a very welcome and timely contribution, as it touches upon the safety and robustness of learning. On the methodological side, machine learning is bound to benefit from the years of experience of probabilists in extreme value theory. I only have minor comments. In light of Remark 2, one sees that we need to assume the /- case \alpha to be the same.

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.