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The paper investigates the effective parameter scaling of depth-recurrent language models by fitting a joint scaling law that accounts for both unique and recurrent parameters. They introduce a recurrence-equivalence exponent, $\varphi$, to quantify the effective capacity gain from looping a block multiple times. Through an iso-depth sweep, they find $\varphi = 0.46$, suggesting diminishing returns from recurrence, and demonstrate its utility by showing how truncated backpropagation lowers $\varphi$ while hyperconnections raise it.
Looping a language model block four times only gives you the effective capacity of 1.4 additional unique blocks, but costs as much to train as 2.4.
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 $L = E + A\,(N_\text{once} + r^{\varphi} N_\text{rec})^{-\alpha} + B\,D^{-\beta}$ 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.
