The success of deep learning has revealed the application potential of neural networks across the sciences and opened up fundamental theoretical problems. In particular, the fact that learning algorithms based on simple variants of gradient methods are able to find near-optimal minima of highly nonconvex loss functions is an unexpected feature of neural networks which needs to be understood in depth. Such algorithms are able to fit the data almost perfectly, even in the presence of noise, and yet they have excellent predictive capabilities. Several empirical results have shown a reproducible correlation between the so-called flatness of the minima achieved by the algorithms and the generalization performance. At the same time, statistical physics results have shown that in nonconvex networks a multitude of narrow minima may coexist with a much smaller number of wide flat minima, which generalize well. Here we show that wide flat minima arise from the coalescence of minima that correspond to high-margin classifications. Despite being exponentially rare compared to zero-margin solutions, high-margin minima tend to concentrate in particular regions. These minima are in turn surrounded by other solutions of smaller and smaller margin, leading to dense regions of solutions over long distances. Our analysis also provides an alternative analytical method for estimating when flat minima appear and when algorithms begin to find solutions, as the number of model parameters varies.
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