Stability and Generalization in Looped Transformers

arXiv:2604.15259v3 Announce Type: replace Abstract: Looped transformers promise test-time compute scaling by spending more iterations on harder problems, but it remains unclear which architectural choices let them extrapolate to harder problems at test time rather than memorize training-specific solutions. We introduce a fixed-point based framework for analyzing looped architectures along three axes of stability -- local attractivity, input-dependence, and geometry -- and use it to characterize when fixed-point iteration yields meaningful predictions. Theoretically, we prove that looped networks without recall have countable fixed points and cannot achieve strong input-dependence at any spectral regime, while recall with outer normalization reliably produces a regime in which fixed points are simultaneously locally-attractive, locally smooth in the input, and supported by stable backpropagation. Empirically, we train small looped transformers on chess, sudoku, and prefix-sums and find that downstream performance tracks the framework's predictions across tasks and architectural configurations. We additionally introduce internal recall, a novel recall placement variant allowing an identity residual path, and exploit its contrasting fixed-point behavior to test our framework's predictions in a controlled setting.

Abstract:Looped transformers promise test-time compute scaling by spending more iterations on harder problems, but it remains unclear which architectural choices let them extrapolate to harder problems at test time rather than memorize training-specific solutions. We introduce a fixed-point based framework for analyzing looped architectures along three axes of stability -- local attractivity, input-dependence, and geometry -- and use it to characterize when fixed-point iteration yields meaningful predictions. Theoretically, we prove that looped networks without recall have countable fixed points and cannot achieve strong input-dependence at any spectral regime, while recall with outer normalization reliably produces a regime in which fixed points are simultaneously locally-attractive, locally smooth in the input, and supported by stable backpropagation. Empirically, we train small looped transformers on chess, sudoku, and prefix-sums and find that downstream performance tracks the framework's predictions across tasks and architectural configurations. We additionally introduce internal recall, a novel recall placement variant allowing an identity residual path, and exploit its contrasting fixed-point behavior to test our framework's predictions in a controlled setting.

10 main pages, 30 total. Accepted to NeurIPS 2026 Main Track

Machine Learning (cs.LG); Artificial Intelligence (cs.AI)

arXiv:2604.15259 [cs.LG]

(or arXiv:2604.15259v3 [cs.LG] for this version)

https://doi.org/10.48550/arXiv.2604.15259 Focus to learn more arXiv-issued DOI via DataCite

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Stability and Generalization in Looped Transformers

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