Learning curves of generic features maps for realistic datasets with a teacher-student model*This article is an updated version of: Loureiro B, Gerbelot C, Cui H, Goldt S, Krzakala F, Mezard M and Zdeborová L 2021 Learning curves of generic features maps for realistic datasets with a teacher-student model Advances in Neural Information Processing Systems vol 34 ed M Ranzato, A Beygelzimer, Y Dauphin, P S Liang and J Wortman Vaughan (New York: Curran Associates) pp 18137–51. (1st November 2022) - Loureiro, Bruno; Gerbelot, Cédric; Cui, Hugo; Goldt, Sebastian; Krzakala, Florent; Mézard, Marc; Zdeborová, Lenka
Abstract: Teacher-student models provide a framework in which the typical-case performance of high-dimensional supervised learning can be described in closed form. The assumptions of Gaussian i.i.d. input data underlying the canonical teacher-student model may, however, be perceived as too restrictive to capture the behaviour of realistic data sets. In this paper, we introduce a Gaussian covariate generalisation of the model where the teacher and student can act on different spaces, generated with fixed, but generic feature maps. While still solvable in a closed form, this generalization is able to capture the learning curves for a broad range of realistic data sets, thus redeeming the potential of the teacher-student framework. Our contribution is then two-fold: first, we prove a rigorous formula for the asymptotic training loss and generalisation error. Second, we present a number of situations where the learning curve of the model captures the one of a realistic data set learned with kernel regression and classification, with out-of-the-box feature maps such as random projections or scattering transforms, or with pre-learned ones—such as the features learned by training multi-layer neural networks. We discuss both the power and the limitations of the framework.