  Large language models exhibit exceptional generalization capabilities,
primarily attributed to the utilization of diversely sourced data. However,
conventional practices in integrating this diverse data heavily rely on
heuristic schemes, lacking theoretical guidance. This research tackles these
limitations by investigating strategies based on low-cost proxies for data
mixtures, with the aim of streamlining data curation to enhance training
efficiency. Specifically, we propose a unified scaling law, termed
$\textbf{BiMix}$, which accurately models the bivariate scaling behaviors of
both data quantity and mixing proportions. We conduct systematic experiments
and provide empirical evidence for the predictive power and fundamental
principles of $\textbf{BiMix}$. Notably, our findings reveal that
entropy-driven training-free data mixtures can achieve comparable or even
better performance than more resource-intensive methods. We hope that our
quantitative insights can shed light on further judicious research and
development in cost-effective language modeling.
