DeepLoop: Depth Scaling for Looped Transformers

Looped Transformers scale sequential computation by applying a compact stack of physical blocks for multiple rounds, increasing unrolled depth without increasing stored parameters. This reuse changes the residual-scaling problem: in an untied Transformer, each residual branch receives and applies its own parameter update, whereas in a looped Transformer one shared update aggregates gradients from repeated visits and is read back by those same visits in the next linearized forward pass. We formalize this tied-depth effect through a first-order perturbation bound controlled by a visit-alignment coefficient ${\kappa }_R$. The bound recovers the DeepNorm exponent when visits decorrelate, but in the conservative aligned regime it requires the exponent to increase from $1/4$ to $1/2$ as loop count grows at fixed physical depth. The resulting method, DeepLoop, keeps the Post-LN DeepNorm architecture with a normalized branch input and sets $\alpha =(2N{)}^{1/2}$ and $\beta =(8N{)}^{-1/2}$ for unrolled depth $N$. On GPT-style looped language models at GPT-2 small and GPT-2 medium scale, DeepLoop is neutral when no physical block is revisited, improves validation loss at every activated loop count, and improves average downstream accuracy in all but one of the evaluated settings. These results show that stable recurrent depth requires residual scaling rules that account for parameter visits, not only nominal layer count.