Learning curves for the multi-class teacher–student perceptron

Author:

Cornacchia Elisabetta,Mignacco FrancescaORCID,Veiga RodrigoORCID,Gerbelot Cédric,Loureiro Bruno,Zdeborová Lenka

Abstract

Abstract One of the most classical results in high-dimensional learning theory provides a closed-form expression for the generalisation error of binary classification with a single-layer teacher–student perceptron on i.i.d. Gaussian inputs. Both Bayes-optimal (BO) estimation and empirical risk minimisation (ERM) were extensively analysed in this setting. At the same time, a considerable part of modern machine learning practice concerns multi-class classification. Yet, an analogous analysis for the multi-class teacher–student perceptron was missing. In this manuscript we fill this gap by deriving and evaluating asymptotic expressions for the BO and ERM generalisation errors in the high-dimensional regime. For Gaussian teacher, we investigate the performance of ERM with both cross-entropy and square losses, and explore the role of ridge regularisation in approaching Bayes-optimality. In particular, we observe that regularised cross-entropy minimisation yields close-to-optimal accuracy. Instead, for Rademacher teacher we show that a first-order phase transition arises in the BO performance.

Funder

Coordenação de Aperfeiçoamento de Pessoal de Nível Superior

H2020 European Research Council

Publisher

IOP Publishing

Subject

Artificial Intelligence,Human-Computer Interaction,Software

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