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

• URL: http://arxiv.org/abs/2102.09671v1
• Date: Thu, 18 Feb 2021 23:44:08 GMT
• Title: On Connectivity of Solutions in Deep Learning: The Role of Over-parameterization and Feature Quality
• Authors: Quynh Nguyen, Pierre Brechet, Marco Mondelli
• Abstract summary: We 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.
• Score: 21.13299067136635
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: 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.

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