A Brief Prehistory of Double Descent
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In their thought-provoking paper [1], Belkin et al. illustrate and discuss the shape of risk curves in the context of modern high-complexity learners. Given a fixed training sample size n, such curves show the risk of a learner as a function of some (approximate) measure of its complexity N. With N the number of features, these curves are also referred to as feature curves. A salient observation in [1] is that these curves can display, what they call, double descent: with increasing N, the risk initially decreases, attains a minimum, and then increases until N equals n, where the training data is fitted perfectly. Increasing N even further, the risk decreases a second and final time, creating a peak at N=n. This twofold descent may come as a surprise, but as opposed to what [1] reports, it has not been overlooked historically. Our letter draws attention to some original, earlier findings, of interest to contemporary machine learning.
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