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arXiv 2102.08127stat.MLcond-mat.dis-nncs.LGmath.PRmath.STstat.TH
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基于师生模型的真实数据集通用特征映射的学习曲线

Learning curves of generic features maps for realistic datasets with a teacher-student model

École Fédérale Polytechnique de Lausanne (EPFL)(洛桑联邦理工学院);Université PSL(巴黎文理研究大学);CNRS(法国国家科学研究中心);Sorbonne Université(索邦大学);Université Paris-Diderot(巴黎第七大学);International School of Advanced Studies (SISSA)(国际高等研究院)

Bruno Loureiro, Cédric Gerbelot, Hugo Cui, Sebastian Goldt, Florent Krzakala, Marc Mézard, Lenka Zdeborová

摘要:

教师-学生模型提供了一个框架,在这个框架中,高维监督学习的典型案例表现可以以封闭的形式描述。高斯i.i.d.然而,作为规范教师-学生模型基础的输入数据可能被认为限制性太大,无法捕捉实际数据集的行为。在本文中,我们介绍了一个高斯协变量推广的模型,教师和学生可以在不同的空间,产生固定的,但通用的特征映射。虽然在封闭形式中仍然是可解的,但这种泛化能够捕获广泛的现实数据集的学习曲线,从而弥补了教师-学生框架的潜力。我们的贡献是双重的:首先,我们证明了渐近训练损失和泛化误差的严格公式。其次,我们提出了一些情况,其中模型的学习曲线捕获了通过内核回归和分类学习的现实数据集,具有开箱即用的特征映射,如随机投影或散射变换,或预先学习的特征-例如通过训练多层神经网络学习的特征。我们讨论的权力和框架的局限性。

英文摘要:

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.

机构、总结、中文摘要均由AI分析而来,可能存在错误,请以论文正文为准。

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