How Many Samples Are Enough for Learning Across Domains?

We study hierarchical domain generalization as a problem of extrapolation from finite observed regions to an entire instance space, replacing i.i.d. sampling with arbitrary domain hierarchies. We show that the central obstruction is not only the complexity of the hypothesis class, but the train/test domain partition through which evidence is revealed. In particular, no matter how small the class or how large the training size, some partition makes generalization fail for some target. These results suggest that modern generalization theory must treat domain structure as a first-class object.

The small-sample learning problem remains a fundamental challenge in machine learning because limited training data lead to unstable model estimation and generalization. Structural Risk Minimization (SRM) has long been regarded as a principled solution under the classical i.i.d. assumption. However, domain generalization (DG) violates this assumption, leaving the theoretical role of SRM in DG largely unexplored. To bridge this gap, we establish the first theoretical guarantees for SRM in DG under mild assumptions. Specifically, based on the concept of stability, we derive learning consistency and generalization error bounds and prove that these bounds become tight when the hypotheses satisfy the stability condition. Building upon this, under a specific hypothesis space assumption, we establish stability, learning, and generalization bounds for SRM. We further discuss the applicability of these bounds to deep learning. This work establishes theoretical foundations for SRM under distribution shifts and sheds light on the design of robust DG algorithms in small-sample scenarios.

Recent research has established empirical scaling laws to predict model performance on multi-domain data mixtures. However, a theoretical understanding of these model loss behaviors remains absent. In this work, we propose a unified framework to explain the underlying mechanics of data mixing. Our approach extends theoretical perspectives originally developed for standard neural scaling laws (e.g., Kaplan and Chinchilla) to the multi-domain setting. Based on the distributional assumption that domains overlap on fundamental skills while diverging on specialized skills, we identify two key factors that govern the domain losses of models trained on different data mixtures: \textit{Capacity Competition}, where the allocation of finite model capacity couples domain losses globally, and \textit{Noise Reduction}, where optimal weights shift toward harder-to-learn domains to minimize overall noise. Empirical evaluations show that our framework outperforms existing baselines by fitting the loss landscape with a lower Mean Relative Error and identifying higher-performing training mixtures. Most importantly, our model successfully extrapolates across scales, predicting highly effective mixtures for large, unseen scales using parameters fitted on smaller ones. In addition, our model achieves these results using significantly fewer parameters compared to previous empirical laws. Our code is available at https://github.com/meiqwq/Explaining-Data-Mixing-Scaling-Laws.