Dynamical versus Bayesian Phase Transitions in a Toy Model of
  Superposition
Zhongtian Chen Edmund Lau Jake Mendel Susan Wei Daniel Murfet
We investigate phase transitions in a Toy Model of Superposition (TMS) using
Singular Learning Theory (SLT). We derive a closed formula for the theoretical
loss and, in the case of two hidden dimensions, discover that regular $k$-gons
are critical points. We present supporting theory indicating that the local
learning coefficient (a geometric invariant) of these $k$-gons determines phase
transitions in the Bayesian posterior as a function of training sample size. We
then show empirically that the same $k$-gon critical points also determine the
behavior of SGD training. The picture that emerges adds evidence to the
conjecture that the SGD learning trajectory is subject to a sequential learning
mechanism. Specifically, we find that the learning process in TMS, be it
through SGD or Bayesian learning, can be characterized by a journey through
parameter space from regions of high loss and low complexity to regions of low
loss and high complexity.