Box-constrained optimization for minimax supervised learning

Cyprien Gilet,Lionel Fillatre,Susana Barbosa

doi:10.1051/proc/202171109

Cyprien Gilet, Lionel Fillatre + Show 1 more

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https://doi.org/10.1051/proc/202171109

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Abstract

In this paper, we present the optimization procedure for computing the discrete boxconstrained minimax classifier introduced in [1, 2]. Our approach processes discrete or beforehand discretized features. A box-constrained region defines some bounds for each class proportion independently. The box-constrained minimax classifier is obtained from the computation of the least favorable prior which maximizes the minimum empirical risk of error over the box-constrained region. After studying the discrete empirical Bayes risk over the probabilistic simplex, we consider a projected subgradient algorithm which computes the prior maximizing this concave multivariate piecewise affine function over a polyhedral domain. The convergence of our algorithm is established.

Highlights

Supervised classification is becoming essential in several real applications such as medical diagnosis, condition monitoring, or fraud detection
We propose to use a projected subgradient algorithm based on [30] that follows the scheme π(n+1) = PU
This paper presents the optimization procedure for computing a box-constrained minimax classifier in the context of discrete or discretized features with multiple classes, a positive loss function, and some dependencies between the features

Summary

Introduction

Supervised classification is becoming essential in several real applications such as medical diagnosis, condition monitoring, or fraud detection. In such applications, we often have to face the following difficulties: imbalanced class proportions, prior probability shifts, presence of both numeric and categorical features (mixed attributes), and dependencies between some features. Given K ≥ 2 classes and a set S = {(Yi, Xi) , i ∈ I} of m labeled training samples, the objective in fitting a supervised classifier [3, 4] is to learn a decision rule δ : X → Y := {1, . ESAIM: PROCEEDINGS AND SURVEYS observed attributes, and such that δ minimizes the empirical risk of classification errors r(δ) = m L(Yi, δ(Xi)), (1). The risk of classification errors (1) can be written as (see [5])

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