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A common strategy to train deep neural networks (DNNs) is to use very largearchitectures and to train them until they (almost) achieve zero trainingerror. Empirically observed good generalization performance on test data, evenin the presence of lots of label noise, corroborate such a procedure. On theother hand, in statistical learning theory it is known that over-fitting modelsmay lead to poor generalization properties, occurring in e.g. empirical riskminimization (ERM) over too large hypotheses classes. Inspired by thiscontradictory behavior, so-called interpolation methods have recently receivedmuch attention, leading to consistent and optimally learning methods for somelocal averaging schemes with zero training error. However, there is notheoretical analysis of interpolating ERM-like methods so far. We take a stepin this direction by showing that for certain, large hypotheses classes, someinterpolating ERMs enjoy very good statistical guarantees while others fail inthe worst sense. Moreover, we show that the same phenomenon occurs for DNNswith zero training error and sufficiently large architectures.

Empirical Risk Minimization in the Interpolating Regime with Application to Neural Network Learning