
Double Trouble in Double Descent : Bias and Variance(s) in the Lazy Regime
Deep neural networks can achieve remarkable generalization performances ...
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The Neural Tangent Kernel in High Dimensions: Triple Descent and a MultiScale Theory of Generalization
Modern deep learning models employ considerably more parameters than req...
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Benign Overfitting and Noisy Features
Modern machine learning often operates in the regime where the number of...
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Surprises in HighDimensional Ridgeless Least Squares Interpolation
Interpolators  estimators that achieve zero training error  have att...
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The Interpolation Phase Transition in Neural Networks: Memorization and Generalization under Lazy Training
Modern neural networks are often operated in a strongly overparametrized...
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Multiple Descent: Design Your Own Generalization Curve
This paper explores the generalization loss of linear regression in vari...
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Beyond Occam's Razor in System Identification: DoubleDescent when Modeling Dynamics
System identification aims to build models of dynamical systems from dat...
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The generalization error of random features regression: Precise asymptotics and double descent curve
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 socalled `double descent' curve. As the model complexity increases, the generalization error follows the usual Ushaped curve at the beginning, first decreasing and then peaking around the interpolation threshold (when the model achieves vanishing training error). However, it descends again as model complexity exceeds this threshold. The global minimum of the generalization error is found in this overparametrized regime, often when the number of parameters is much larger than the number of samples. Far from being a peculiar property of deep neural networks, elements of this behavior have been demonstrated in much simpler settings, including linear regression with random covariates. In this paper we consider the problem of learning an unknown function over the ddimensional sphere S^d1, from n i.i.d. samples ( x_i, y_i) ∈ S^d1× R, i < n. We perform ridge regression on N random features of the form σ( w_a^ T x), a < N. This can be equivalently described as a twolayers neural network with random firstlayer weights. We compute the precise asymptotics of the generalization error, in the limit N, n, d →∞ with N/d and n/d fixed. This provides the first analytically tractable model that captures all the features of the double descent phenomenon.
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