Source-linked AI summary
Dynamical phase selection controls compute scaling in looped transformers
Gunn Kim
TL;DR
The paper asks what determines inference cost in looped transformers when training and task performance are held fixed. It analyzes trained recurrences as dynamical systems and finds that initialization selects distinct bifurcation phases, which determine compute-time scaling: fold phases obey universal critical laws, whereas Neimark–Sacker phases do not.
Problem
Inference cost is not fixed by training objectives, accuracy, or linear stability conditions alone; it depends on the fixed-point structure and bifurcation governing relaxation.
Method
The authors train weight-tied recurrent transformers, classify solutions by continued Jacobian spectra and fixed-point fate, and test fold dynamics with a one-dimensional normal-form reduction.
Results
Networks with identical architecture, objective, and accuracy realize initialization-dependent fold, Neimark–Sacker, rigid, and monostable phases; fold networks satisfy parameter-free scaling, while Neimark–Sacker networks lack an analogous compute-time law.
Takeaways & Limitations
Test-time compute is a dynamical property of the solution selected by training, not of architecture alone.
Takeaways & Limitations
The observed bifurcation classification applies to the models explored and is not an exclusion theorem; the fold-derived law is analyzed only for the fold phase.
Abstract
from arXiv · showhide
A looped transformer performs inference by iterating a weight-tied map, making its computation a dynamical process whose cost is set by the resulting inference dynamics. Here we show that networks with identical architecture and objective, trained to identical accuracy, nevertheless realize distinct dynamical phases depending strongly on initialization, and that the bifurcation defining each phase determines how test-time compute scales. The phases are distinguished by their bifurcation mechanisms, including a saddle-node fold and a Neimark-Sacker-type transition to bounded nonstationary motion. In the fold phase, a one-dimensional normal-form reduction predicts both the relaxation-time and spectral-gap amplitudes from local derivatives of the trained map, yielding the parameter-free relation $τ(\varepsilon)[1-λ_{\max}(-\varepsilon)]\toπ$. Composed with a regular distribution of problem difficulty, the same critical slowing down produces the workload-level tail $P(τ>N)\sim N^{-2}$. In the Neimark--Sacker phase, the fold scaling law disappears rather than merely changing its prefactor. Thus, test-time compute is not determined by architecture alone. It is governed by the dynamical phase of the solution found by training.