On the robustness of the minimim ℓ₂ interpolator

We analyse the interpolator with minimal ℓ₂-norm β in a general high dimensional linear regression framework where (Formula Presented) with X a random n × p matrix with independent N(0,Σ) rows. We prove that, with high probability, without assumption on the noise vector (Formula Presented), the ellipsoid risk (Formula Presented) is bounded from above by (Formula Presented), where c is an absolute constant and, for any (Formula Presented) is the tail sum of the eigenvalues of Σ. These bounds show a transition in the rates. For high signal to noise ratios, the rates (Formula Presented) broadly improve the existing ones. For low signal to noise ratio, we also provide lower bound holding with large probability. General lower bounds are proved under minor restrictions on the noise ξ (see Theorem 1). Under assumptions on the sprectrum of Σ, this lower bound is of order (Formula Presented) matching the upper bound. Consequently, in the large noise regime, we are able to precisely track the ellipsoid risk with large probability. These results give new insight when the interpolation can be harmless in high dimensions.