  The properties of flat minima in the empirical risk landscape of neural
networks have been debated for some time. Increasing evidence suggests they
possess better generalization capabilities with respect to sharp ones. First,
we discuss Gaussian mixture classification models and show analytically that
there exist Bayes optimal pointwise estimators which correspond to minimizers
belonging to wide flat regions. These estimators can be found by applying
maximum flatness algorithms either directly on the classifier (which is norm
independent) or on the differentiable loss function used in learning. Next, we
extend the analysis to the deep learning scenario by extensive numerical
validations. Using two algorithms, Entropy-SGD and Replicated-SGD, that
explicitly include in the optimization objective a non-local flatness measure
known as local entropy, we consistently improve the generalization error for
common architectures (e.g. ResNet, EfficientNet). An easy to compute flatness
measure shows a clear correlation with test accuracy.
