Part of Advances in Neural Information Processing Systems 32 (NeurIPS 2019)
Ji Xu, Daniel J. Hsu
We study least squares linear regression over N uncorrelated Gaussian features that are selected in order of decreasing variance. When the number of selected features p is at most the sample size n, the estimator under consideration coincides with the principal component regression estimator; when p>n, the estimator is the least ℓ2 norm solution over the selected features. We give an average-case analysis of the out-of-sample prediction error as p,n,N→∞ with p/N→α and n/N→β, for some constants α∈[0,1] and β∈(0,1). In this average-case setting, the prediction error exhibits a double descent'' shape as a function of p. We also establish conditions under which the minimum risk is achieved in the interpolating (p>n) regime.