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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 to peak when $N$ is ofthe same order as the number of parameters $P$. In earlier works, a similarphenomenon 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 oftenconflated in the litterature. In this paper, we show that despite theirapparent similarity, these two scenarios are inherently different. In fact,both peaks can co-exist when neural networks are applied to noisy regressiontasks. The relative size of the peaks is then governed by the degree ofnonlinearity of the activation function. Building on recent developments in theanalysis of random feature models, we provide a theoretical ground for thissample-wise triple descent. As shown previously, the nonlinear peak at$N\!=\!P$ is a true divergence caused by the extreme sensitivity of the outputfunction to both the noise corrupting the labels and the initialization of therandom features (or the weights in neural networks). This peak survives in theabsence of noise, but can be suppressed by regularization. In contrast, thelinear 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 implicitlyregularized by the nonlinearity, which is why it only becomes salient at highnoise and is weakly affected by explicit regularization. Throughout the paper,we compare analytical results obtained in the random feature model with theoutcomes of numerical experiments involving deep neural networks.

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