  Deep ReLU networks trained with the square loss have been observed to perform
well in classification tasks. We provide here a theoretical justification based
on analysis of the associated gradient flow. We show that convergence to a
solution with the absolute minimum norm is expected when normalization
techniques such as Batch Normalization (BN) or Weight Normalization (WN) are
used together with Weight Decay (WD). The main property of the minimizers that
bounds their expected error is the norm: we prove that among all the
close-to-interpolating solutions, the ones associated with smaller Frobenius
norms of the unnormalized weight matrices have better margin and better bounds
on the expected classification error. With BN but in the absence of WD, the
dynamical system is singular. Implicit dynamical regularization -- that is
zero-initial conditions biasing the dynamics towards high margin solutions --
is also possible in the no-BN and no-WD case. The theory yields several
predictions, including the role of BN and weight decay, aspects of Papyan, Han
and Donoho's Neural Collapse and the constraints induced by BN on the network
weights.
