  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.
