On Binary Classification in Extreme Regions
In pattern recognition, a random label Y is to be predicted based upon observing a random vector X valued in R d with d ≥ 1 by means of a classification rule with minimum probability of error. In a wide variety of applications, ranging from finance/insurance to environmental sciences through teletraffic data analysis for instance, extreme (i.e. very large) observations X are of crucial importance, while contributing in a negligible manner to the (empirical) error however, simply because of their rarity. 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. Precisely, under non-parametric heavy-tail assumptions for the class distributions, we prove that a natural and asymptotic notion of risk, accounting for predictive performance in extreme regions of the input space, can be defined and show that minimizers of an empirical version of a non-asymptotic approximant of this dedicated risk, based on a fraction of the largest observations, lead to classification rules with good generalization capacity, by means of maximal deviation inequalities in low probability regions. Beyond theoretical results, numerical experiments are presented in order to illustrate the relevance of the approach developed.
Summary This paper introduces theoretical analysis of learning a classifier in an extreme region, where the norm of input vector is very large. The authors describe a new definition of natural and asymptotic notion of risk for predictive performance in extreme regions. The theoretical properties of definition are fully investigated and empirical performance was validated by synthetic datasets.  Strengths [Motivation] Applicable well to clinical data analysis. Sometimes, large feature values are very rarely observed, but its meaning is significantly important.  [Theoretical analysis] The authors presented some interesting theoretical results on learning a classifier in extreme regions, though it’s difficult to follow every step in the proof.  Weaknesses [Positive orthant assumption] I’m not quite convinced to analyze a classifier by restricting an input space to the positive orthant. We could normalize the training data to fit them in the positive orthant, but it’s rarely conducted. Moreover, in section 3, a classifier is defined on the unit sphere. All theories in section 2 could be generalized to the unit sphere?  [Fixing alpha] It seems to be too restrictive to fix alpha=1. It could be possible to generalize the theory to smaller alpha values?  [Lack of comparison] There exist many similar works to learn a classifier with heavy-tail distributions. This paper doesn’t compare any previous works and discussion on the difference between the proposed algorithm and existing ones. Moreover, the proposed algorithm is just learning a classifier by sub-sampling instances with large values.  [Lack of real-world experiments] Since this paper doesn’t contain any results on real-world datasets, it’s very difficult to understand a practical importance of the proposed method.  == Updated review == I’ve read authors’ response on all reviewers’ comments. My original concerns are fully addressed, though I still believe that empirical justification is relatively weaker than the novelty of theoretical analysis. During discussion, I’ve changed my original score to 6 (weak accept).