Real-world machine learning systems are often trained using a mix of data sources with varying cost and quality. Understanding how the size and composition of a training dataset affect model performance is critical for advancing our understanding of generalization, as well as designing more effective data collection policies. We show
that there is a simple scaling law that predicts the loss incurred by a model even under varying dataset composition. Our work expands recent observations of scaling laws for log-linear generalization error in the i.i.d setting and uses this to cast model performance prediction as a learning problem. Using the theory of optimal experimental design, we derive a simple rational function approximation to generalization error that can be fitted using a few model training runs. Our approach can achieve highly accurate ($r^2\approx .9$) predictions of model performance under substantial extrapolation in two different standard supervised learning tasks and is accurate ($r^2 \approx .83$) on more challenging machine translation and question answering tasks where many baselines achieve worse-than-random performance.