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

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.

Development and use of feature models of anthropogenic objects in thematic processing of space imagery

The article describes the problem of automated recognition of the anthropogenic elements of landscape. The recognition is based on the aerospace data in the optical range of the spectrum and a feature model of an object, consisting of the geometric and reflectance characteristics. Using this model, we formed training samples for a convolutional neural network. There is a real example of the practical implementation of the model in identification of the aviation objects.