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Recent extensive numerical experiments in high scale machine learning haveallowed to uncover a quite counterintuitive phase transition, as a function ofthe ratio between the sample size and the number of parameters in the model. Asthe number of parameters $p$ approaches the sample size $n$, the generalisationerror increases, but surprisingly, it starts decreasing again past thethreshold $p=n$. This phenomenon, brought to the theoretical communityattention in \cite{belkin2019reconciling}, has been thoroughly investigatedlately, more specifically for simpler models than deep neural networks, such asthe linear model when the parameter is taken to be the minimum norm solution tothe least-squares problem, firstly in the asymptotic regime when $p$ and $n$tend to infinity, see e.g. \cite{hastie2019surprises}, and recently in thefinite dimensional regime and more specifically for linear models\cite{bartlett2020benign}, \cite{tsigler2020benign},\cite{lecue2022geometrical}. In the present paper, we propose a finite sampleanalysis of non-linear models of \textit{ridge} type, where we investigate the\textit{overparametrised regime} of the double descent phenomenon for both the\textit{estimation problem} and the \textit{prediction} problem. Our resultsprovide a precise analysis of the distance of the best estimator from the trueparameter as well as a generalisation bound which complements recent works of\cite{bartlett2020benign} and \cite{chinot2020benign}. Our analysis is based ontools closely related to the continuous Newton method\cite{neuberger2007continuous} and a refined quantitative analysis of theperformance in prediction of the minimum $\ell_2$-norm solution.

A finite sample analysis of the benign overfitting phenomenon for ridge function estimation