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Deep learning: a statistical viewpoint

Published online by Cambridge University Press:  04 August 2021

Peter L. Bartlett ,
Andrea Montanari  and
Alexander Rakhlin
Peter L. Bartlett
Affiliation:
Departments of Statistics and EECS, University of California, Berkeley, CA 94720, USA E-mail: peter@berkeley.edu
Andrea Montanari
Affiliation:
Departments of EE and Statistics, Stanford University, Stanford, CA 94304, USA E-mail: montanar@stanford.edu
Alexander Rakhlin
Affiliation:
Department of Brain & Cognitive Sciences and Statistics and Data Science Center, Massachusetts Institute of Technology, Cambridge, MA 02139, USA E-mail: rakhlin@mit.edu
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Abstract

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The remarkable practical success of deep learning has revealed some major surprises from a theoretical perspective. In particular, simple gradient methods easily find near-optimal solutions to non-convex optimization problems, and despite giving a near-perfect fit to training data without any explicit effort to control model complexity, these methods exhibit excellent predictive accuracy. We conjecture that specific principles underlie these phenomena: that overparametrization allows gradient methods to find interpolating solutions, that these methods implicitly impose regularization, and that overparametrization leads to benign overfitting, that is, accurate predictions despite overfitting training data. In this article, we survey recent progress in statistical learning theory that provides examples illustrating these principles in simpler settings. We first review classical uniform convergence results and why they fall short of explaining aspects of the behaviour of deep learning methods. We give examples of implicit regularization in simple settings, where gradient methods lead to minimal norm functions that perfectly fit the training data. Then we review prediction methods that exhibit benign overfitting, focusing on regression problems with quadratic loss. For these methods, we can decompose the prediction rule into a simple component that is useful for prediction and a spiky component that is useful for overfitting but, in a favourable setting, does not harm prediction accuracy. We focus specifically on the linear regime for neural networks, where the network can be approximated by a linear model. In this regime, we demonstrate the success of gradient flow, and we consider benign overfitting with two-layer networks, giving an exact asymptotic analysis that precisely demonstrates the impact of overparametrization. We conclude by highlighting the key challenges that arise in extending these insights to realistic deep learning settings.

Information

Type
Research Article
Information
Acta Numerica , Volume 30 , May 2021 , pp. 87 - 201
Creative Commons
Creative Common License - CCCreative Common License - BY
This is an Open Access article, distributed under the terms of the Creative Commons Attribution licence (https://creativecommons.org/licenses/by/4.0/), which permits unrestricted re-use, distribution, and reproduction in any medium, provided the original work is properly cited.
Copyright
© The Author(s), 2021. Published by Cambridge University Press

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