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The geometrical features of the (non-convex) loss landscape of neural networkmodels are crucial in ensuring successful optimization and, most importantly,the capability to generalize well. While minimizers' flatness consistentlycorrelates with good generalization, there has been little rigorous work inexploring the condition of existence of such minimizers, even in toy models.Here we consider a simple neural network model, the symmetric perceptron, withbinary weights. Phrasing the learning problem as a constraint satisfactionproblem, the analogous of a flat minimizer becomes a large and dense cluster ofsolutions, while the narrowest minimizers are isolated solutions. We performthe first steps toward the rigorous proof of the existence of a dense clusterin certain regimes of the parameters, by computing the first and second momentupper bounds for the existence of pairs of arbitrarily close solutions.Moreover, we present a non rigorous derivation of the same bounds for sets of$y$ solutions at fixed pairwise distances.

Clustering of solutions in the symmetric binary perceptron