A U-turn on Double Descent: Rethinking Parameter Counting in Statistical Learning
This work resolves tensions between double descent observations and statistical intuition for classical machine learning methods, which is important for researchers in statistical learning theory.
The paper challenges the claim that double descent phenomena extend beyond neural networks to classical statistical methods, showing that apparent double descent curves in these methods actually fold back into traditional U-shaped complexity-generalization curves when analyzed with a proper measure of effective parameters.
Conventional statistical wisdom established a well-understood relationship between model complexity and prediction error, typically presented as a U-shaped curve reflecting a transition between under- and overfitting regimes. However, motivated by the success of overparametrized neural networks, recent influential work has suggested this theory to be generally incomplete, introducing an additional regime that exhibits a second descent in test error as the parameter count p grows past sample size n - a phenomenon dubbed double descent. While most attention has naturally been given to the deep-learning setting, double descent was shown to emerge more generally across non-neural models: known cases include linear regression, trees, and boosting. In this work, we take a closer look at evidence surrounding these more classical statistical machine learning methods and challenge the claim that observed cases of double descent truly extend the limits of a traditional U-shaped complexity-generalization curve therein. We show that once careful consideration is given to what is being plotted on the x-axes of their double descent plots, it becomes apparent that there are implicitly multiple complexity axes along which the parameter count grows. We demonstrate that the second descent appears exactly (and only) when and where the transition between these underlying axes occurs, and that its location is thus not inherently tied to the interpolation threshold p=n. We then gain further insight by adopting a classical nonparametric statistics perspective. We interpret the investigated methods as smoothers and propose a generalized measure for the effective number of parameters they use on unseen examples, using which we find that their apparent double descent curves indeed fold back into more traditional convex shapes - providing a resolution to tensions between double descent and statistical intuition.