LIT-544: Dynamical versus Bayesian Phase Transitions in a Toy Model of Superposition

Chen, Lau, Mendel, Wei and Murfet (2023) — ARXIV-2310.06301. Read as NOTE-288.

Key takeaways

• The distinction. A Bayesian phase transition is a change in which region of parameter space the posterior concentrates on, as a function of sample size n, and it is mathematically well defined in SLT. A dynamical transition is a change in the SGD trajectory as a function of training step, and its “formal status has remained elusive”. The paper’s title is its thesis.
• The mechanism, in one formula. The local free energy is F_n(W_α) = n L_n(w*_α) + λ_α log n − (m_α − 1) log log n + O_p(1), approximated as F_n ≈ min_α [n L_n(w*_α) + λ_α log n + c_α]. At small n the λ log n term dominates, so the posterior prefers low complexity even at high loss; at large n the n L_n term dominates and it prefers low loss at higher complexity. The transitions are where the ordering swaps. That is Occam’s razor as an internal, automatic, n-dependent mechanism.
• A quantitative prediction, confirmed. For the Toy Model of Superposition with r = 2 hidden dimensions, regular k-gons are proved to be critical points, and their local learning coefficients are derived. The free energy formula predicts the 5 → 6 transition at n_cr ≈ 600; MCMC over the posterior shows it at 600 ≤ n ≤ 700.
• And the honest negative. “There is no necessary relation between these two kinds of transitions.” The 5 → 6 Bayesian transition has not been observed as a dynamical transition — the regions may be separated by energy barriers SGD does not cross.
• The Bayesian Antecedent Hypothesis. The conjecture that dynamical transitions encountered in training do have Bayesian antecedents. Its main obstruction is that a Bayesian transition requires λ to increase. The paper’s own analysis finds antecedents for its observed dynamical transitions with two inconclusive cases.

Standing in the anthology

Active, and the most load-bearing single idea in this cluster for a record that holds three Proposed grokking mechanisms and a practice about emergent capabilities. Everything in THEORY-072, THEORY-071 and THEORY-070 is dynamical. Everything SLT proves is Bayesian. This paper is where the field says out loud that those are different events and supplies the one worked example in which both are computed.

It is a toy model, deliberately. Two hidden dimensions, high sparsity limit, and the classification of critical points is not claimed to be exhaustive. What that buys is the only setting in which the theoretical λ is available, the prediction is numerical, and the comparison is therefore a test rather than an analogy.