Exact minimax risk for linear least squares, and the lower tail of sample covariance matrices

The first part of this paper is devoted to the decision-theoretic analysis of random-design linear prediction with square loss. It is known that, under boundedness constraints on the response (and thus regression coefficients), the minimax excess risk scales as $C\sigma^2d/n$ up to constants, where $d$ is the model dimension, $n$ the sample size, and $\sigma^2$ the noise parameter. Here, we study the expected excess risk with respect to the full linear class. We show that the ordinary least squares (OLS) estimator is minimax optimal in the well-specified case, for every distribution of covariates and noise level. Further, we express the minimax risk in terms of the distribution of statistical leverage scores of individual samples. We deduce a precise minimax lower bound of $\sigma^2d/(n-d+1)$, valid for any distribution of covariates, which nearly matches the risk of OLS for Gaussian covariates. We then obtain nonasymptotic upper bounds on the minimax risk for covariates that satisfy a "small ball"-type regularity condition, which scale as $(1+o(1))\sigma^2d/n$ as $d=o(n)$, both in the well-specified and misspecified cases. Our main technical contribution is the study of the lower tail of the smallest singular value of empirical covariance matrices around $0$. We establish a general lower bound on this lower tail, together with a matching upper bound under a necessary regularity condition. Our proof relies on the PAC-Bayesian technique for controlling empirical processes, and extends an analysis of Oliveira (2016) devoted to a different part of the lower tail. Equivalently, our upper bound shows that the operator norm of the inverse sample covariance matrix has bounded $L^q$ norm up to $q\asymp n$, and this exponent is unimprovable. Finally, we show that the regularity condition on the design naturally holds for independent coordinates.