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

• URL: http://arxiv.org/abs/2604.21106v2
• Date: Mon, 27 Apr 2026 13:50:06 GMT
• Title: How Much Is One Recurrence Worth? Iso-Depth Scaling Laws for Looped Language Models
• Abstract summary: We measure how much one extra recurrence is worth to a looped (depth-recurrent) language model.<n>Our method applies to any looped LM and separates true loop improvements from token-budget gains.
• Score: 33.509836193149795
• License: http://creativecommons.org/licenses/by/4.0/
• Abstract: We measure how much one extra recurrence is worth to a looped (depth-recurrent) language model, in equivalent unique parameters. From an iso-depth sweep of 116 pretraining runs 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 recover a new 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 each additional recurrence predictably 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 measurement tool on two probes. Truncated backpropagation lowers $\varphi$ to $0.38$, indicating that the loop mechanism is poorly trained under truncation, even though validation loss decreases. Conversely, hyperconnections raise $\varphi$ to $0.65$, a genuine capacity gain. Our method applies to any looped LM and separates true loop improvements from token-budget gains.

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