Hidden Unit Specialization in Layered Neural Networks: ReLU vs. Sigmoidal Activation

Abstract

We study layered neural networks of rectified linear units (ReLU) in a modelling framework for stochastic training processes. The comparison with sigmoidal activation functions is in the center of interest. We compute typical learning curves for shallow networks with K hidden units in matching student teacher scenarios. The systems exhibit sudden changes of the generalization performance via the process of hidden unit specialization at critical sizes of the training set. Surprisingly, our results show that the training behavior of ReLU networks is qualitatively different from that of networks with sigmoidal activations. In networks with K ≥ 3 sigmoidal hidden units, the transition is discontinuous: Specialized network configurations co-exist and compete with states of poor performance even for very large training sets. On the contrary, the use of ReLU activations results in continuous transitions for all K. For large enough training sets, two competing, differently specialized states display similar generalization abilities, which coincide exactly for large networks in the limit K → ∞. * (who contributed equally)This work was supported through the Northern Netherlands Region of Smart Factories (RoSF) consortium (see: http://www.rosf.nl).E. Oostwal, M. Straat, and M. Biehl are with the Bernoulli Institute for Mathematics,

Dynamics of coordinate ascent variational inference: A case study in 2D Ising models

“…2). Phase change behavior of dynamical systems using the sigmoid and RELU activation functions are known in the literature in the context of generalization performance of deep neural networks C ¸akmak andOpper [2020],Oostwal et al [2019]. In this section we present a complete proof of the bifurcation analysis of non-linear dynamical systems involving sigmoid activation function despite its connections with C ¸akmak andOpper [2020],Oostwal et al [2019].…”