Loss of material trainability through an unusual transition

Material training is a method to endow materials with specific responses through external driving. We study the complexity of attainable responses, as expressed in the number of sites that are simultaneously controlled. With increased complexity, convergence to the desired response becomes very slow. The training error decays as a power law with an exponent that varies continuously and vanishes at a critical threshold, marking the limit of trainable responses. We study how the transition affects the vibrational properties. Approaching the critical threshold, low-frequency modes proliferate, approaching zero frequency. This implies that training causes material degradation and that training fails due to competing spurious low-frequency modes. We propose that the excess low-frequency spectrum is due to atypical local structures with bonds that nearly align. Our work explains how the presence of an exotic critical point affects the convergence of training, and could be relevant for understanding learning in physical systems.