We consider here the class of supervised learning algorithms known as Empirical Risk Minimization (ERM). The classical theory by Vapnik and others characterize universal consistency of ERM in the classical regime in which the architecture of the learning network is fixed and n, the number of training examples, goes to infinity. We do not have a similar general theory for the modern regime of interpolating regressors and overparameterized deep networks, in which d > *n* as *n* goes to infinity.

In this note I propose the outline of such a theory based on the specific notion of CVloo stability of the learning algorithm with respect to perturbations of the training set. The theory shows that for interpolating regressors and separating classifiers (either kernel machines or deep RELU networks)

- minimizing CVloo stability minimizes the expected error
- minimum norm solutions are the most stable solutions

The hope is that this approach may lead to a unified theory encompassing both the modern regime and the classical one.

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