High-dimensional analysis of double descent for linear regression with random projections

We consider linear regression problems with a varying number of random projections, where we provably exhibit a double descent curve for a fixed prediction problem, with a high-dimensional analysis based on random matrix theory. We first consider the ridge regression estimator and re-interpret earlier results using classical notions from non-parametric statistics, namely degrees of freedom, also known as effective dimensionality. In particular, we show that the random design performance of ridge regression with a specific regularization parameter matches the classical bias and variance expressions coming from the easier fixed design analysis but for another larger implicit regularization parameter. We then compute asymptotic equivalents of the generalization performance (in terms of bias and variance) of the minimum norm least-squares fit with random projections, providing simple expressions for the double descent phenomenon.