Modern deep neural networks usually have large parameter scales and nonlinear hierarchical structures, and they have achieved strong performance in computer vision. However, the source of their generalization performance remains difficult to explain using traditional statistical learning theory. Among the factors that may affect visual generalization, data scale, model complexity, and input modalities are fundamental and controllable variables. This study empirically analyzes how these three factors influence model generalization performance. Specifically, in a preliminary experiment, we construct a one-dimensional nonlinear function and vary the number of training samples and the polynomial degree to observe the effects of data scale and model complexity on model performance. In the main experiments, we compare model performance on CIFAR-10 and CIFAR-100 under different training data scales, model architectures, and input modalities. The experimental results show that increasing the training data scale consistently improves generalization performance, whereas changes in model complexity do not provide stable gains. In addition, removing color information degrades model performance, while explicit prior features such as gradients, edges, and wavelets have inconsistent effects across different model architectures. Overall, this study provides an empirical analysis of the relationships among data scale, model complexity, input modalities, and visual generalization performance. Code and experimental logs are available at: https://github.com/YidiZhouluo/DeepLearning-Empirical-Studies/tree/main/Exp_01.
Sharpness and complexity are two central factors in the generalization analysis of deep neural networks. Existing quantitative evaluations of generalization measures have largely focused on individual scalar measures, leaving the joint explanatory power of sharpness and complexity largely unexplored. This work studies how far sharpness and complexity can jointly explain generalization. We use linear regression and introduce a Pareto-based analysis to quantitatively evaluate the joint explanatory power of these two factors. Beyond the existing parameter-level definitions, we further propose realizations of sharpness and complexity that are closer to function space and less dependent on raw parameter representations. We find that function-oriented definitions of these two quantities expand the explanatory scope of the two-factor view beyond what is achieved by existing parameter-level metrics. Overall, our results support the sharpness-complexity perspective as an informative lens for understanding generalization across diverse settings. At the same time, the remaining failures indicate that whether this two-factor view can serve as a complete theory of generalization remains open.
Understanding how performance scales jointly with model size and data is a central problem in modern machine learning. Existing theoretical works on scaling laws typically describe generalization as a function of data or compute, often in fixed-feature or infinite-width regimes and for online SGD. Here, we instead study how generalization scales with the number of trainable parameters and the number of samples in a feature-learning model. We analyze ℓ2-regularized empirical test error minimization in a quadratic two-layer network in a finite-sample setting with structured data. This setting allows for an explicit characterization of the generalization error as a function of the number of samples, model width, and regularization. Our results reveal a phase diagram with distinct scaling regimes as the number of parameters varies. In particular, the generalization error follows data-dependent power laws controlled by the spectral structure of the target. We further characterize the transitions between regimes, including the onset of interpolation, and their impact on generalization.
We address the fundamental question of why deep neural networks generalize by establishing a pointwise generalization theory for fully connected networks. This framework resolves long-standing barriers to characterizing the rich nonlinear feature-learning regime and builds a new statistical foundation for representation learning. For each trained model, we characterize the hypothesis via a pointwise Riemannian Dimension, derived from the eigenvalues of the learned feature representations across layers. This establishes a principled framework for deriving hypothesis-dependent, representation-aware generalization bounds. These bounds offer a systematic upgrade over approaches based on model size, products of norms, and infinite-width linearizations, yielding guarantees that are orders of magnitude tighter in both theory and experiment. Analytically, we identify the structural properties and mathematical principles that explain the tractability of deep networks. Empirically, the pointwise Riemannian Dimension exhibits substantial feature compression, decreases with increased over-parameterization, and captures the implicit bias of optimizers. Taken together, our results indicate that deep networks are mathematically tractable in practical regimes and that their generalization is sharply explained by pointwise, feature-spectrum-aware complexity.