---
author:
- 'Sébastien Gadat, Sébastien Gerchinovitz, and Clément Marteau'
bibliography:
- 'references.bib'
title: Optimal functional supervised classification with separation condition
---

Introduction {#sec:intro}
============

The binary supervised classification problem is perhaps one of the most common tasks in statistics and machine learning. Even so, this problem still fosters new theoretical and applied questions because of the large variety of the data encountered so far. We refer the reader to [@DGL96] and [@BBG2005] and to the references therein for a comprehensive introduction to binary supervised classification. This problem unfolds as follows. The learner has access to $n$ independent copies $(X_1,Y_1),\ldots, (X_n,Y_n)$ of a pair $(X,Y)$, where $X$ lies in a measurable space $\mathcal{H}$ and $Y \in \{0,1\}$. The goal of the learner is to predict the label $Y$ after observing the new input $X$, with the help of the sample $\mathcal{S}_n := (X_i,Y_i)_{1 {\leqslant}i {\leqslant}n}$ to learn the unknown joint distribution ${\mathbb{P}}_{X,Y}$ of the pair $(X,Y)$.

In some standard situations, $X$ lies in the simplest possible Hilbert space: $\mathcal{H} = \mathbb{R}^d$, which corresponds to the finite-dimensional binary classification problem. This setting has been extensively studied so far. Popular classification procedures that are now theoretically well understood include the ERM method 