DeepLoop: Depth Scaling for Looped Transformers

Shuzhen Li; Yifan Zhang; Jiacheng Guo; Quanquan Gu; Mengdi Wang

cs.LGJul 15, 2026

DeepLoop: Depth Scaling for Looped Transformers

Authors: Shuzhen Li, Yifan Zhang, Jiacheng Guo, Quanquan Gu, Mengdi Wang

Organizations: Princeton University · University of California, Los Angeles

Abstract

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 $κ_R$ . 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, \textbf{DeepLoop}, keeps the Post-LN DeepNorm architecture and sets $α=(2N)^{1/2}$ and $β=(8N)^{-1/2}$ 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 and improves validation loss and downstream accuracy once recurrent depth is activated. These results show that stable recurrent depth requires residual scaling rules that account for parameter visits, not only nominal layer count.

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On the Residual Scaling of Looped Transformers: Stability and Transferability

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 = \lambda/(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.

Shaowen Wang, Bingrui Li, Ge Zhang +3

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Apr 22, 2026cs.LG

How Much Is One Recurrence Worth? Iso-Depth Scaling Laws for Looped Language Models

We measure how much one recurrence is worth to a looped (depth-recurrent) transformer, in equivalent unique parameters. From an iso-depth pretraining sweep across recurrence counts

r \in \{1, 2, 4, 8\}

spanning

{\sim}50\times

in training compute, we fit a joint scaling law

L = E + A\,(N_\text{once} + r^{\varphi} N_\text{rec})^{-α} + B\,D^{-β}

and measure a recurrence-equivalence exponent

\varphi = 0.46

. Intuitively,

\varphi

tells us whether looping a block

r

times is equivalent in validation loss to

r

unique blocks of a non-looped model (full equivalence,

\varphi{=}1

) or to a single block run repeatedly with no capacity gain (

\varphi{=}0

). Our

\varphi = 0.46

sits in between, so replacing unique blocks with shared recurrences increases validation loss at matched training compute. For example, at

r{=}4

a 410M looped model performs on par with a 580M non-looped model, but incurs the training cost of a 1B non-looped one. We demonstrate the utility of

\varphi

as a diagnostic tool on two case studies: commonly used truncated backpropagation lowers

\varphi

0.38

, indicating that the loop mechanism is poorly trained under truncation, even though validation loss decreases. Conversely, hyperconnections raise

\varphi

0.65

, a genuine capacity gain. Our method separates true loop improvements from training-side gains, a distinction raw validation loss cannot make.

Kristian Schwethelm, Daniel Rueckert, Georgios Kaissis

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May 9, 2026cs.LG

Sparse Layers are Critical to Scaling Looped Language Models

Looped language models repeat a set of transformer layers through depth, reducing memory costs and providing natural early-exit points at loop boundaries. However, looped models do not scale as favorably as standard transformers with unique layers. We compare standard and Mixture-of-Experts (MoE) transformers, with and without looping, and find two main results. First, we find Looped-MoE models scale better than the standard baseline while dense looped models do not. We trace this to routing divergence between loops: in Looped-MoE models, different experts are activated on each pass through the same shared layers, recovering expressivity without additional parameters. Our second finding is that looped models have better compute-quality trade-offs with early exits than standard models. Because each loop ends with the same layers that produce the final output, loop boundaries are superior exit points, as confirmed by earlier output convergence at these points. In sum, we provide a clear direction for scaling looped models: a Looped-MoE model with early exits can not only beat standard transformers at scale, but also enable significant memory and inference savings with minimal degradation in quality.

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