When Are Bias-Free ReLU Networks Effectively Linear Networks?

Yedi Zhang, Andrew M Saxe, Peter E. Latham

Keywords: ReLU network, linear network, gradient flow, implicit bias

TL;DR: We show that two-layer bias-free ReLU networks cannot express nonlinear odd functions and have the same learning dynamics as linear networks under symmetry conditions on data.

Abstract: We investigate the implications of removing bias in ReLU networks regarding their expressivity and learning dynamics. We first show that two-layer bias-free ReLU networks have limited expressivity: the only odd function two-layer bias-free ReLU networks can express is a linear one. We then show that, under symmetry conditions on the data, these networks have the same learning dynamics as linear networks. This enables us to give analytical time-course solutions to certain two-layer bias-free (leaky) ReLU networks, for the first time outside the lazy learning regime. While deep bias-free ReLU networks are more expressive than their two-layer counterparts, they still share a number of similarities with deep linear networks. These similarities enable us to leverage insights from linear networks to understand certain ReLU newtorks. Overall, our results show that some properties previously established for bias-free ReLU networks arise due to equivalence to linear networks.

Primary Area: learning theory

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Submission Number: 4748