A Scale Invariant Measure of Flatness for Deep Network Minima

It has been empirically observed that the flatness of minima obtained from training deep networks seems to correlate with better generalization. However, for deep networks with positively homogeneous activations, most measures of flatness are not invariant to rescaling of the network parameters. This means that the measure of flatness can be made as small or as large as possible through rescaling, rendering the quantitative measures meaningless. In this paper we show that for deep networks with positively homogenous activations, these rescalings constitute equivalence relations, and that these equivalence relations induce a quotient manifold structure in the parameter space. Using an appropriate Riemannian metric, we propose a Hessian-based measure for flatness that is invariant to rescaling and perform simulations to empirically verify our claim. Finally we perform experiments to verify that our flatness measure correlates with generalization by using minibatch stochastic gradient descent with different batch sizes to find deep network minima with different generalization properties.