Learning to Extrapolate in Depth with Looped Transformers

Abstract:Looped transformers reuse a single block across iterations, raising a natural question: how far can a block trained at one depth be safely extrapolated to greater depths? We study this question for all-pairs graph connectivity, where harder graphs require longer-range information propagation. We show that under exchangeability and symmetric positive initialization, population training at depth K converges to an update rule governed by a single effective parameter that balances graph propagation against fitting structure unresolved at training depth. This trade-off creates a finite extrapolation window: too few test-time loops provide insufficient propagation, while too many eventually create false connections. We derive explicit endpoints for this window and bound the graph hardness solvable beyond training as a function of training depth and hardness. In particular, a test depth chosen solely from the training distribution can guarantee recovery on substantially harder graphs, with a certified depth gain that grows exponentially as training depth outpaces training hardness. Experiments validate the predicted windows and show how compute budget and initialization affect the optimal training depth.

Machine Learning (cs.LG)

arXiv:2610.14456 [cs.LG]

(or arXiv:2610.14456v1 [cs.LG] for this version)

https://doi.org/10.48550/arXiv.2610.14456 Focus to learn more arXiv-issued DOI via DataCite (pending registration)

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