Think Shallow, Solve Deep: Controlling Recurrent Dynamics for Reliable Test-Time Depth
arXiv:2608.18222v1 Announce Type: new Abstract: Recurrent-depth reasoners aim to solve harder problems by iterating their update longer at test time, but additional iterations can improve, preserve, or degrade an answer. We show that a measurable property of the trained operator, its finite-time dynamical regime (estimated as settling, marginal, or drifting), indicates which of these occurs. We give a sufficient condition for depth-safety: once an operator's per-step displacement is small relative to the decoder margin, the decoded answer cannot change under further iterations. Empirically, on algorithmic tasks trained from $800$ unaugmented examples per difficulty tier, settling operators do not degrade with added depth, and on some tasks convert it into higher accuracy on harder unseen instances (Sudoku, $0.19$ to $0.34$ past the training horizon). A single terminal fixed-point objective moves the regime and the depth behavior together: removing it induces drift and removes the gains, and adding it to a generic recurrence yields depth-safe extrapolation on carry propagation. We give four operational criteria for useful test-time depth, use them to catalogue failure modes, and, as a consistency check, apply the same measurements to Huginn-3.5B, which falls in the non-settling family.
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[Submitted on 18 Aug 2026]
Title:Think Shallow, Solve Deep: Controlling Recurrent Dynamics for Reliable Test-Time Depth
View a PDF of the paper titled Think Shallow, Solve Deep: Controlling Recurrent Dynamics for Reliable Test-Time Depth, by Ivan Viakhirev and Kirill Borodin and Amirah Almutairi and Serguei Barannikov and Maxim Abramov and Grach Mkrtchian
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Abstract:Recurrent-depth reasoners aim to solve harder problems by iterating their update longer at test time, but additional iterations can improve, preserve, or degrade an answer. We show that a measurable property of the trained operator, its finite-time dynamical regime (estimated as settling, marginal, or drifting), indicates which of these occurs. We give a sufficient condition for depth-safety: once an operator's per-step displacement is small relative to the decoder margin, the decoded answer cannot change under further iterations. Empirically, on algorithmic tasks trained from $800$ unaugmented examples per difficulty tier, settling operators do not degrade with added depth, and on some tasks convert it into higher accuracy on harder unseen instances (Sudoku, $0.19$ to $0.34$ past the training horizon). A single terminal fixed-point objective moves the regime and the depth behavior together: removing it induces drift and removes the gains, and adding it to a generic recurrence yields depth-safe extrapolation on carry propagation. We give four operational criteria for useful test-time depth, use them to catalogue failure modes, and, as a consistency check, apply the same measurements to Huginn-3.5B, which falls in the non-settling family.
Comments: Submitted to the Thirty-Ninth AAAI Conference on Artificial Intelligence (AAAI-27)
Subjects:
Machine Learning (cs.LG); Computation and Language (cs.CL)
Cite as: arXiv:2608.18222 [cs.LG]
(or arXiv:2608.18222v1 [cs.LG] for this version)
https://doi.org/10.48550/arXiv.2608.18222
arXiv-issued DOI via DataCite (pending registration)
Submission history
From: Kirill Borodin [view email] [v1] Tue, 18 Aug 2026 18:05:53 UTC (383 KB)
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