Abstract: Despite their prevalence in deep-learning communities, over-parameterized models convey high demands of computational costs for proper training. This work studies the fine-grained, modular-level learning dynamics of over-parameterized models to attain a more efficient and fruitful training strategy. Empirical evidence reveals that when scaling down into network modules, such as heads in self-attention models, we can observe varying learning patterns implicitly associated with each module's trainability. To describe such modular-level learning capabilities, we introduce a novel concept dubbed modular neural tangent kernel (mNTK), and we demonstrate that the quality of a module's learning is tightly associated with its mNTK's principal eigenvalue $\lambda_{\max}$. A large $\lambda_{\max}$ indicates that the module learns features with better convergence, while those miniature ones may impact generalization negatively. Inspired by the discovery, we propose a novel training strategy termed Modular Adaptive Training (MAT) to update those modules with their $\lambda_{\max}$ exceeding a dynamic threshold selectively, concentrating the model on learning common features and ignoring those inconsistent ones. Unlike most existing training schemes with a complete BP cycle across all network modules, MAT can significantly save computations by its partially-updating strategy and can further improve performance. Experiments show that MAT nearly halves the computational cost of model training and outperforms the accuracy of baselines.