Time Transfer: On Optimal Learning Rate and Batch Size In The Infinite Data Limit

One of the main challenges in optimal scaling of large language models (LLMs) is the prohibitive cost of hyperparameter tuning, particularly learning rate $\eta$ and batch size $B$. While techniques like $\mu$P (Yang et al., 2022) provide scaling rules for optimal $\eta$ transfer in the infinite model size limit, the optimal scaling behavior in the infinite data size limit ($T \to \infty$) remains unknown. We fill in this gap by observing for the first time an interplay of three optimal $\eta$ scaling regimes: $\eta \propto \sqrt{T}$, $\eta \propto 1$, and $\eta \propto 1/\sqrt{T}$ with transitions controlled by $B$ and its relation to the time-evolving critical batch size $B_\mathrm{crit} \propto T$. Furthermore, we show that the optimal batch size is positively correlated with $B_\mathrm{crit}$: keeping it fixed becomes suboptimal over time even if learning rate is scaled optimally. Surprisingly, our results demonstrate that the observed optimal $\eta$ and $B$ dynamics are preserved with $\mu$P model scaling, challenging the conventional view of $B_\mathrm{crit}$ dependence solely on loss value. Complementing optimality, we examine the sensitivity of loss to changes in learning rate, where we find the sensitivity to decrease with $T \to \infty$ and to remain constant with $\mu$P model scaling. We hope our results make the first step towards a unified picture of the joint optimal data and model scaling.