THEORY-074: A phase transition of the Bayesian posterior and a transition in the training trajectory are different events, and only the first is well defined

Source

Chen et al. (2023), LIT-544 — ARXIV-2310.06301. The distinction between the two kinds of transition, the 5 → 6 prediction and its observation in the Toy Model of Superposition, and the Bayesian Antecedent Hypothesis are all this paper’s.

The account

Two things in this literature are called phase transitions and they are indexed by different variables.

The Bayesian one. The local free energy of a region W_α dominated by a critical point is

F_n(W_α) = n L_n(w*_α) + λ_α log n − (m_α − 1) log log n + O_p(1)

so, approximately, F_n ≈ min_α [n L_n(w*_α) + λ_α log n + c_α]. At small n the λ log n term dominates and the posterior sits on a simple region even at high loss; as n grows the n L_n term takes over and it moves to a lower-loss, more complex region. A transition is where the ordering swaps. This is Occam’s razor as an automatic consequence of the free energy, indexed by sample size, and it is a theorem.

The dynamical one. A plateau in the training curve followed by a sudden drop, indexed by step. This is what everybody actually observes and, as the source puts it, its formal status has remained elusive.

They are not the same event. “There is no necessary relation between these two kinds of transitions.” A Bayesian transition need not have a dynamical counterpart — the regions may be far apart or separated by barriers SGD does not cross. The paper’s own headline Bayesian transition is an example.

The evidence, in one model

The Toy Model of Superposition at r = 2 hidden dimensions, where regular k-gons are proved to be critical points and their local learning coefficients can be derived rather than estimated:

• The free energy formula predicts the 5 → 6 transition at n_cr ≈ 600; MCMC over the posterior shows it at 600 ≤ n ≤ 700. A number, predicted and then observed.
• The same k-gon critical points explain the plateaus of SGD training curves — which the paper argues is no coincidence, since SLT ties posterior phases to singularities of the KL divergence and nonlinear dynamics ties trajectory behaviour to singularities of a potential.
• And the 5 → 6 transition has not been observed as a dynamical transition. The strongest Bayesian result in the paper is the one with no training-time counterpart.

The Bayesian Antecedent Hypothesis

The conjecture that dynamical transitions encountered in neural network training do have Bayesian antecedents. Its obstruction is stated: a Bayesian transition α → β requires λ to increase, so a dynamical transition that reduces loss without increasing degeneracy has nowhere to come from. The paper’s analysis finds antecedents for the dynamical transitions it observed, with two cases where the analysis is inconclusive.

Why it is Proposed, and what is not in doubt

The distinction is not in doubt — it is definitional, and the 5 → 6 case demonstrates the two can come apart. What is Proposed is the bridge: whether the transitions people care about are governed by the theory that is well-defined. That is the BAH, it is labelled a hypothesis by its authors, and it is supported in a model with two hidden dimensions and a critical-point classification the paper declines to call exhaustive.

Why the record needed this

Every grokking mechanism it holds is dynamical. THEORY-072 (a walk down the weight norm), THEORY-071 (norm moving from a memorising circuit to a more efficient one) and THEORY-070 (leaving the lazy regime) are all claims about a trajectory, all Proposed, and none is general. THEORY-073’s stages are dynamical too. Everything THEORY-075 supplies is Bayesian.

Without this document the record would hold two literatures that use the same words for different objects, and its own summary of them would blur the two — which is the failure it filed ADR-031 and the two-collapse-accounts precedent to avoid.

It also sharpens SOTA-200. A capability appearing suddenly at some scale is a claim about n; a capability appearing suddenly during training is a claim about t. The practice’s third check now says to move the regime and see whether the discontinuity moves; n is one of those regimes, and this is the only work the record holds that predicts where the crossing should be.