Deep learning: a statistical viewpoint
Name
2103.09177.pdf
Description
Submitted version
Size
1.4 MB
Format
Adobe PDF
Checksum (MD5)
3ce9bccccf7624daacfad04b6cae32b2
Author(s)
Bartlett, Peter L
Montanari, Andrea
Rakhlin, Alexander
Date Issued
May 2021
Journal
Acta Numerica
Publisher
Cambridge University Press (CUP)
Citation
Bartlett, Peter L, Montanari, Andrea and Rakhlin, Alexander. 2021. "Deep learning: a statistical viewpoint." Acta Numerica, 30.
Version
Original manuscript
Abstract
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.
MIT Department
Massachusetts Institute of Technology. Department of Brain and Cognitive Sciences
Statistics and Data Science Center (Massachusetts Institute of Technology)
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Creative Commons Attribution-Noncommercial-Share Alike
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DOI of Published Version
https://doi.org/10.1017/s0962492921000027
