  We analyze the connection between minimizers with good generalizing
properties and high local entropy regions of a threshold-linear classifier in
Gaussian mixtures with the mean squared error loss function. We show that there
exist configurations that achieve the Bayes-optimal generalization error, even
in the case of unbalanced clusters. We explore analytically the error-counting
loss landscape in the vicinity of a Bayes-optimal solution, and show that the
closer we get to such configurations, the higher the local entropy, implying
that the Bayes-optimal solution lays inside a wide flat region. We also
consider the algorithmically relevant case of targeting wide flat minima of the
(differentiable) mean squared error loss. Our analytical and numerical results
show not only that in the balanced case the dependence on the norm of the
weights is mild, but also, in the unbalanced case, that the performances can be
improved.
