Triple descent and the two kinds of overfitting: where and why do they appear?

By Stéphane d'Ascoli, Levent Sagun, Giulio Biroli. 2020

91 citations recorded by the source corpus.

A recent line of research has highlighted the existence of a ‘double descent’ phenomenon in deep learning, whereby increasing the number of training examples N causes the generalization error of neural networks (NNs) to peak when N is of the same order as the number of parameters P. In earlier works, a similar phenomenon was shown to exist in simpler models such as linear regression, where the peak instead occurs when N is equal to the input dimension D. Since both peaks coincide with the interpolation threshold, they are often conflated in the literature. In this paper, we show that despite their apparent similarity, these two scenarios are inherently different. In fact, both peaks can co-exist when NNs are applied to noisy regression tasks. The relative size of the peaks is then governed by the degree of nonlinearity of the activation function. Building on recent developments in the analysis of random feature models, we provide a theoretical ground for this sample-wise triple descent. As shown previously, the nonlinear peak at N = P is a true divergence caused by the extreme sensitivity of the output function to both the noise corrupting the labels and the initialization of the random features (or the weights in NNs). This peak survives in the absence of noise, but can be suppressed by regularization. In contrast, the linear peak at N = D is solely due to overfitting the noise in the labels, and forms earlier during training. We show that this peak is implicitly regularized by the nonlinearity, which is why it only becomes salient at high noise and is weakly affected by explicit regularization. Throughout the paper, we compare analytical results obtained in the random feature model with the outcomes of numerical experiments involving deep NNs.

Subfields: Artificial Intelligence, Computational Mechanics, Statistical and Nonlinear Physics, Scaling & Learning Theory

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Citations in the corpus

Grokking: Generalization Beyond Overfitting on Small Algorithmic Datasets (2022, background): “A double descent of validation loss has been documented in some circumstances, but is considered unusual among practitioners Nakkiran et al. (2019); Belkin et al. (2018); d’Ascoli et al. (2020).”
Pixelated Butterfly: Simple and Efficient Sparse training for Neural Network Models (2021, background): “Deep Double Descent [Nakkiran et al., 2019, d’Ascoli et al., 2020] conjectures that the generalization error improves as the parameter count grows.”
Spectral bias and task-model alignment explain generalization in kernel regression and infinitely wide neural networks (2020, background): “In these models, one of the peaks (termed “ nonlinear ” in a previous study 43 ) happens when the number of samples reaches the number of features, and thus the number of parameters of the model, crossing the interpolation threshold.”