LayerNorm as Implicit Gain Control in Looped Transformers
In pre-LayerNorm looped transformers, LayerNorm inside the recurrent block acts as an implicit gain controller: by coupling the block's local Lipschitz constant inversely to the activation scale, it renders the recurrence Jacobian non-normal -- asymptotically contractive at every verified fixed point even where its operator norm exceeds 1 -- so the true stability budget is the spectral margin, not an operator-norm bound. That margin depletes as the carry $ρ\to 1$, and a minority of initializations never converge to a fixed point at all, so the diagonal carry constraint $ρ(\bar{A}) < 1$ is necessary but not sufficient for convergence of the full recurrence. Training experiments across six ...
| Authors | Matthias M. M. Buehlmaier |
|---|---|
| Primary category | cs.LG |
| Published | 12 July 2026 |
| Last updated at source | 12 July 2026 |
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