|Title||Fisher-Rao Metric, Geometry, and Complexity of Neural Networks|
|Publication Type||Journal Article|
|Year of Publication||2017|
|Authors||Liang, T, Poggio, TA, Rakhlin, A, Stokes, J|
|Keywords||capacity control, deep learning, Fisher-Rao metric, generalization error, information geometry, Invariance, natural gradient, ReLU activation, statistical learning theory|
We study the relationship between geometry and capacity measures for deep neural networks from an invariance viewpoint. We introduce a new notion of capacity — the Fisher-Rao norm — that possesses desirable in- variance properties and is motivated by Information Geometry. We discover an analytical characterization of the new capacity measure, through which we establish norm-comparison inequalities and further show that the new measure serves as an umbrella for several existing norm-based complexity measures. We discuss upper bounds on the generalization error induced by the proposed measure. Extensive numerical experiments on CIFAR-10 support our theoretical findings. Our theoretical analysis rests on a key structural lemma about partial derivatives of multi-layer rectifier networks.
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