Hidden unit specialization in layered neural networks: ReLU vs. sigmoidal activation

Elisa Oostwal, Michiel Straat, Michael Biehl*

*Corresponding author for this work

Research output: Contribution to journalArticleAcademicpeer-review

31 Citations (Scopus)
115 Downloads (Pure)


By applying concepts from the statistical physics of learning, we study layered neural networks of rectified linear units (ReLU). The comparison with conventional, sigmoidal activation functions is in the center of interest. We compute typical learning curves for large shallow networks with K hidden units in matching student teacher scenarios. The systems undergo phase transitions, i.e. 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 hidden layers in the limit K→∞. Our findings are also confirmed in Monte Carlo simulations of the training processes.
Original languageEnglish
Article number125517
JournalPhysica A: Statistical Mechanics and its Applications
Early online dateNov-2020
Publication statusPublished - Feb-2021


  • Neural networks
  • Machine learning
  • Statistical physics

Cite this