How Model Growth, Recursion, and Boundary Operators Influence Scaling Exponents

Zixi Chen; Akshay Vegesna; Samip Dahal; Andrew Gordon Wilson

cs.LGSep 16, 2026

How Model Growth, Recursion, and Boundary Operators Influence Scaling Exponents

Authors: Zixi Chen, Akshay Vegesna, Samip Dahal, Andrew Gordon Wilson

Organizations: New York University · Q Labs

Abstract

Scaling laws predict how loss decreases with increases in computation. We show, contrary to conventional wisdom, that architectural interventions can modify scaling exponents in pre-training, leading to power-law improvements in performance as computation increases. As an anchoring point, we consider the architectural formulation of looped transformers. Although not typically used in this way, looping, also known as recursive depth, provides a mechanism for model growth, by increasing the number of loops during training. Model growth, with and without shared weights, provides the biggest changes to the scaling exponents. In particular, a 7.4B model growth architecture matches GPT-3 13B on CORE with roughly $20\times$ less compute, and has compute efficiency gains that increase with scale. Moreover, simply using a boundary operator in a vanilla transformer, which normalizes and injects an earlier block, also provides an exponent increase, although to a lesser extent. In the data-constrained, multi-epoch setting, standard looping has a useful regularizing effect, where we find it is compute-optimal to increase the number of loops with scale. These results can be understood through the lens of computational depth: for a given computational budget, we wish to increase the usable depth of the transformer, which can lead to efficiency gains that increase with scale.

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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.

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and

β=(8N)^{-1/2}

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N

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