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Deep ReLU networks trained with the square loss have been observed to performwell in classification tasks. We provide here a theoretical justification basedon analysis of the associated gradient flow. We show that convergence to asolution with the absolute minimum norm is expected when normalizationtechniques such as Batch Normalization (BN) or Weight Normalization (WN) areused together with Weight Decay (WD). The main property of the minimizers thatbounds their expected error is the norm: we prove that among all theclose-to-interpolating solutions, the ones associated with smaller Frobeniusnorms of the unnormalized weight matrices have better margin and better boundson the expected classification error. With BN but in the absence of WD, thedynamical system is singular. Implicit dynamical regularization -- that iszero-initial conditions biasing the dynamics towards high margin solutions --is also possible in the no-BN and no-WD case. The theory yields severalpredictions, including the role of BN and weight decay, aspects of Papyan, Hanand Donoho's Neural Collapse and the constraints induced by BN on the networkweights.

Explicit regularization and implicit bias in deep network classifiers trained with the square loss