Dynamical versus Bayesian Phase Transitions in a Toy Model of Superposition
Primary Area: learning theory
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Keywords: phase transition, toy model of superposition, Bayesian statistics, singular learning theory
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TL;DR: Phase transitions in a toy model of superposition over training are well-described by singular learning theory
Abstract: We investigate phase transitions in a Toy Model of Superposition (TMS) \citep{elhage2022superposition} 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.
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Submission Number: 6519
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