Convergence rates of least squares regression estimators with heavy-tailed errors

We study the performance of the Least Squares Estimator (LSE) in a general nonparametric regression model, when the errors are independent of the covariates but may only have a $p$-th moment ($pgeq 1$). In such a heavy-tailed regression setting, we show that if the model satisfies a standard `entropy condition with exponent $alpha in (0,2)$, then the $L_2$ loss of the LSE converges at a rate begin{align*} mathcal{O}_{mathbf{P}}big(n^{-frac{1}{2+alpha}} vee n^{-frac{1}{2}+frac{1}{2p}}big). end{align*} Such a rate cannot be improved under the entropy condition alone. This rate quantifies both some positive and negative aspects of the LSE in a heavy-tailed regression setting. On the positive side, as long as the errors have $pgeq 1+2/alpha$ moments, the $L_2$ loss of the LSE converges at the same rate as if the errors are Gaussian. On the negative side, if $p<1+2/alpha$, there are (many) hard models at any entropy level $alpha$ for which the $L_2$ loss of the LSE converges at a strictly slower rate than other robust estimators. The validity of the above rate relies crucially on the independence of the covariates and the errors. In fact, the $L_2$ loss of the LSE can converge arbitrarily slowly when the independence fails. The key technical ingredient is a new multiplier inequality that gives sharp bounds for the `multiplier empirical process associated with the LSE. We further give an application to the sparse linear regression model with heavy-tailed covariates and errors to demonstrate the scope of this new inequality.