Beyond Counting Linear Regions of Neural Networks, Simple Linear Regions Dominate!

FENGLEI FAN, Wei Huang, Lecheng Ruan, Tieyong Zeng, Huan Xiong, Fei Wang

Abstract:

Functions represented by a neural network with the widely-used ReLU activation are piecewise linear functions over linear regions (polytopes). Figuring out the properties of such polytopes is of fundamental importance for the development of neural networks.
So far, either theoretical or empirical studies on polytopes stay at the level of counting their number. Despite successes in explaining the power of depth and so on, counting the number of polytopes puts all polytopes on an equal booting, which is essentially an incomplete characterization of polytopes. Beyond counting, here we study the shapes of polytopes via the number of simplices obtained by triangulations of polytopes. First, we demonstrate the properties of the number of simplices in triangulations of polytopes, and compute the upper and lower bounds of the maximal number of simplices that a network can generate. Next, by computing and analyzing the histogram of simplices across polytopes, we find that a ReLU network has surprisingly uniform and simple polytopes, although these polytopes theoretically can be rather diverse and complicated. This finding is a novel implicit bias that concretely reveals what kind of simple functions a network learns and sheds light on why deep learning does not overfit. Lastly, we establish a theorem to illustrate why polytopes produced by a deep network are simple and uniform. The core idea of the proof is counter-intuitive: adding depth probably does not create a more complicated polytope. We hope our work can inspire more research into investigating polytopes of a ReLU neural network, thereby upgrading the knowledge of neural networks to a new level.

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