# The Generalization Error of Random Features Regression: Precise Asymptotics and the Double Descent Curve

@article{Mei2019TheGE,
title={The Generalization Error of Random Features Regression: Precise Asymptotics and the Double Descent Curve},
author={Song Mei and A. Montanari},
journal={arXiv: Statistics Theory},
year={2019}
}
• Published 2019
• Mathematics
• arXiv: Statistics Theory
Deep learning methods operate in regimes that defy the traditional statistical mindset. The neural network architectures often contain more parameters than training samples, and are so rich that they can interpolate the observed labels, even if the latter are replaced by pure noise. Despite their huge complexity, the same architectures achieve small generalization error on real data. This phenomenon has been rationalized in terms of a so-called double descent' curve. As the model complexity… Expand
220 Citations

#### Paper Mentions

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#### References

SHOWING 1-10 OF 86 REFERENCES
Toward Moderate Overparameterization: Global Convergence Guarantees for Training Shallow Neural Networks
• Computer Science, Mathematics
• IEEE Journal on Selected Areas in Information Theory
• 2020
Focusing on shallow neural nets and smooth activations, it is shown that (stochastic) gradient descent when initialized at random converges at a geometric rate to a nearby global optima as soon as the square-root of the number of network parameters exceeds the size of the training data. Expand
High-dimensional dynamics of generalization error in neural networks
• Computer Science, Mathematics
• Neural Networks
• 2020
It is found that the dynamics of gradient descent learning naturally protect against overtraining and overfitting in large networks, and standard application of theories such as Rademacher complexity are inaccurate in predicting the generalization performance of deep neural networks. Expand
To understand deep learning we need to understand kernel learning
• Computer Science, Mathematics
• ICML
• 2018
It is argued that progress on understanding deep learning will be difficult until more tractable "shallow" kernel methods are better understood, and a need for new theoretical ideas for understanding properties of classical kernel methods. Expand
Reconciling modern machine learning and the bias-variance trade-off
• Computer Science, Mathematics
• ArXiv
• 2018
A new "double descent" risk curve is exhibited that extends the traditional U-shaped bias-variance curve beyond the point of interpolation and shows that the risk of suitably chosen interpolating predictors from these models can, in fact, be decreasing as the model complexity increases, often below the risk achieved using non-interpolating models. Expand
On Lazy Training in Differentiable Programming
• Computer Science
• NeurIPS
• 2019
This work shows that this "lazy training" phenomenon is not specific to over-parameterized neural networks, and is due to a choice of scaling that makes the model behave as its linearization around the initialization, thus yielding a model equivalent to learning with positive-definite kernels. Expand
Harmless Interpolation of Noisy Data in Regression
• Computer Science
• IEEE Journal on Selected Areas in Information Theory
• 2020
It is shown that the fundamental generalization (mean-squared) error of any interpolating solution in the presence of noise decays to zero with the number of features, and overparameterization can be beneficial in ensuring harmless interpolation of noise. Expand
Understanding deep learning requires rethinking generalization
• Computer Science
• ICLR
• 2017
These experiments establish that state-of-the-art convolutional networks for image classification trained with stochastic gradient methods easily fit a random labeling of the training data, and confirm that simple depth two neural networks already have perfect finite sample expressivity. Expand
A Convergence Theory for Deep Learning via Over-Parameterization
• Computer Science, Mathematics
• ICML
• 2019
This work proves why stochastic gradient descent can find global minima on the training objective of DNNs in $\textit{polynomial time}$ and implies an equivalence between over-parameterized neural networks and neural tangent kernel (NTK) in the finite (and polynomial) width setting. Expand
Learning and Generalization in Overparameterized Neural Networks, Going Beyond Two Layers
• Computer Science, Mathematics
• NeurIPS
• 2019
It is proved that overparameterized neural networks can learn some notable concept classes, including two and three-layer networks with fewer parameters and smooth activations, and SGD (stochastic gradient descent) or its variants in polynomial time using polynomially many samples. Expand
Gradient Descent Provably Optimizes Over-parameterized Neural Networks
• Computer Science, Mathematics
• ICLR
• 2019
Over-parameterization and random initialization jointly restrict every weight vector to be close to its initialization for all iterations, which allows a strong convexity-like property to show that gradient descent converges at a global linear rate to the global optimum. Expand