Spectral Multiplicity Entails Sample-wise Multiple Descent

In this paper, we study the generalization risk of ridge and ridgeless linear regression. We assume that the data features follow a multivariate normal distribution and that the spectrum of the covariance matrix consists of a given set of eigenvalues of proportionally growing multiplicity. We characterize the limiting bias and variance when the dimension and the number of training samples tend to infinity proportionally. Exact formulae for the bias and variance are derived using the random matrix theory and convex Gaussian min-max theorem. Based on these formulae, we study the sample-wise multiple descent phenomenon of the generalization risk curve, i.e., with more data, the generalization risk can be non-monotone, and specifically, can increase and then decrease multiple times with more training data samples. We prove that sample-wise multiple descent occurs when the spectrum of the covariance matrix is highly ill-conditioned. We also present numerical results to confirm the values of the bias and variance predicted by our theory and illustrate the multiple descent of the generalization risk curve. Moreover, we theoretically show that the ridge estimator with optimal regularization can result in a monotone generalization risk curve and thereby eliminate multiple descent under some assumptions.