On Connectivity of Solutions in Deep Learning: The Role of Over-parameterization and Feature Quality

It has been empirically observed that, in deep neural networks, the solutions found by stochastic gradient descent from different random initializations can be often connected by a path with low loss. Recent works have shed light on this intriguing phenomenon by assuming either the over-parameterization of the network or the dropout stability of the solutions. In this paper, we reconcile these two views and present a novel condition for ensuring the connectivity of two arbitrary points in parameter space. This condition is provably milder than dropout stability, and it provides a connection between the problem of finding low-loss paths and the memorization capacity of neural nets. This last point brings about a trade-off between the quality of features at each layer and the over-parameterization of the network. As an extreme example of this trade-off, we show that (i) if subsets of features at each layer are linearly separable, then almost no over-parameterization is needed, and (ii) under generic assumptions on the features at each layer, it suffices that the last two hidden layers have $\Omega(\sqrt{N})$ neurons, $N$ being the number of samples. Finally, we provide experimental evidence demonstrating that the presented condition is satisfied in practical settings even when dropout stability does not hold.