Dynamical versus Bayesian Phase Transitions in a Toy Model of Superposition

TL;DR

This paper explores phase transitions in a simplified model of superposition using Singular Learning Theory, revealing geometric critical points that influence Bayesian and SGD learning behaviors and suggesting a sequential learning mechanism.

Contribution

It introduces a closed-form formula for the theoretical loss in the Toy Model of Superposition and identifies geometric critical points affecting phase transitions in Bayesian and SGD training.

Findings

Regular $k$-gons are critical points in the model.

The local learning coefficient determines phase transitions.

SGD and Bayesian learning follow a sequential journey through parameter space.

Abstract

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.