On the robustness of the minimum $\ell_2$ interpolator

We analyse the interpolator with minimal $\ell_2$-norm $\hat{\beta}$ in a general high dimensional linear regression framework where $\mathbb Y=\mathbb X\beta^*+\xi$ where $\mathbb X$ is a random $n\times p$ matrix with independent $\mathcal N(0,\Sigma)$ rows and without assumption on the noise vector $\xi\in \mathbb R^n$. We prove that, with high probability, the prediction loss of this estimator is bounded from above by $(\|\beta^*\|^2_2r_{cn}(\Sigma)\vee \|\xi\|^2)/n$, where $r_{k}(\Sigma)=\sum_{i\geq k}\lambda_i(\Sigma)$ are the rests of the sum of eigenvalues of $\Sigma$. These bounds show a transition in the rates. For high signal to noise ratios, the rates $\|\beta^*\|^2_2r_{cn}(\Sigma)/n$ broadly improve the existing ones. For low signal to noise ratio, we also provide lower bound holding with large probability. Under assumptions on the sprectrum of $\Sigma$, this lower bound is of order $\| \xi\|_2^2/n$, matching the upper bound. Consequently, in the large noise regime, we are able to precisely track the prediction error with large probability. This results give new insight when the interpolation can be harmless in high dimensions.