A Scale Invariant Flatness Measure for Deep Network Minima

TL;DR

This paper introduces a rescaling-invariant flatness measure for deep networks with homogeneous activations, addressing the issue of non-invariance in existing measures and confirming that large-batch SGD minima are sharper.

Contribution

It proposes a novel Hessian-based flatness measure invariant to rescaling, leveraging a quotient manifold structure in parameter space.

Findings

The new measure is invariant under rescaling transformations.

Large-batch SGD minima are sharper than small-batch SGD minima.

Rescaling invariance improves the meaningfulness of flatness measures.

Abstract

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 sharpness/flatness are not invariant to rescaling of the network parameters, corresponding to the same function. This means that the measure of flatness/sharpness 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 this manifold structure and an appropriate metric, we propose a Hessian-based measure for flatness that is invariant to rescaling. We use this new measure to confirm the proposition that Large-Batch SGD minima are indeed sharper than Small-Batch SGD minima.