On the Residual Scaling of Looped Transformers: Stability and Transferability

• URL: http://arxiv.org/abs/2606.18524v1
• Date: Tue, 16 Jun 2026 22:39:13 GMT
• Title: On the Residual Scaling of Looped Transformers: Stability and Transferability
• Abstract summary: We show that $1/N$ scaling improves trainability and yields better loss than $1/!sqrtN$ scaling across loop counts.<n>Experiments on looped Transformers confirm that $1/N$ scaling improves trainability and yields better loss than $1/!sqrtN$ scaling across loop counts.
• Score: 31.27468588849646
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: Looped (weight-tied) Transformers apply a shared residual block $N$ times ($h \leftarrow h + \varepsilon\,f(h)$, same $f$ at each step), increasing effective depth without adding parameters. Prior depth-scaling analyses prescribe $\varepsilon = 1/\!\sqrt{L}$ for depth-$L$ residual networks. We show that this is insufficient for looped architectures: weight sharing makes residual updates correlated across iterations, requiring the stronger scaling $\varepsilon = 1/N$. For multi-layer blocks ($L$ unique layers looped $N$ times), we derive a factored parameterization $\varepsilon = λ/(N\!\sqrt{L})$ that separates the two sources of growth: $1/N$ controls the within-layer loop correlation, and $1/\!\sqrt{L}$ controls the across-layer variance. A key consequence is that the optimal learning rate depends only on the number of unique layers $L$, not on the loop count $N$, enabling direct hyperparameter transfer from small to large $N$ without retuning. Experiments on looped Transformers confirm that $1/N$ scaling improves trainability and yields better loss than $1/\!\sqrt{N}$ scaling across loop counts.

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