Two Models of Double Descent for Weak Features

The “double descent” risk curve was proposed to qualitatively describe the out-of-sample prediction accuracy of variably parameterized machine learning models. This article provides a precise mathematical analysis for the shape of this curve in two simple data models with the least squares/least norm predictor. Specifically, it is shown that the risk peaks when the number of features p is close to the sample size n but also that the risk sometimes decreases toward its minimum as p increases beyond n. This behavior parallels some key patterns observed in large models, including modern neural networks, and is contrasted with that of “prescient” models that select features in an a priori optimal order.