Organizations: Xi’an Jiaotong University · Shanghai Jiao Tong University · Shanghai Artificial Intelligence Laboratory
Abstract
Understanding why deep neural networks (DNNs) fail to generalize to unseen samples remains a long-standing challenge. Existing studies mainly examine changes in externally observable factors such as data, representations, or outputs, yet offer limited insight into how a model's internal decision mechanism evolves from training to test. To address this gap, we introduce Decision Pattern Shift (DPS), a new perspective that defines generalization through the stability of internal decision patterns and quantifies failure as their deviation from those learned during training. Specifically, we represent each sample's decision pattern as a GradCAM-based channel-contribution vector, which captures how feature channels collectively support a prediction, and we propose the DPS metric to measure its discrepancy from the class-average pattern. Empirical analyses across multiple datasets and architectures show that, (i) decision patterns form a highly structured, class-consistent space with strong intra-class cohesion and low inter-class confusion, enabling direct analysis of a model's decision logic; (ii) the DPS magnitude correlates linearly with the generalization gap (nearly all Pearson r > 0.8), revealing generalization as a systematic drift in the model's internal decision mechanism; (iii) the DPS spectrum organizes diverse generalization degradation scenarios (covering ideal generalization, in-distribution degradation, domain shift, out-of-distribution, and shortcut learning) into a continuous trajectory, providing a unified explanation of their failure modes. These findings open up new possibilities for early generalization-risk detection, failure-mode diagnosis, and channel-level defect localization.
We derive a differential equation that governs the evolution of the generalization gap when a model is trained by gradient descent-based methods. This differential equation is driven by two key quantities, a contraction factor that brings together trajectories corresponding to slightly different datasets, and a perturbation factor that accounts for them training on different datasets. The coupled decay of contraction and perturbation guarantees a controlled accumulation of generalization gap during training. We analyze this differential equation to show that the generalization gap is given by a quadratic form that consists of an ``effective Gram matrix'' that depends upon the training trajectory and a certain residual of the predictor at initialization. Our framework is applicable to general deep networks and smooth loss functions. In numerical experiments on different neural network architectures, datasets and sample sizes, we show that this quadratic form accurately captures the actual generalization gap. We also show how to instantiate our framework in a number of examples via analytical calculations. For example, for high-dimensional linear regression, our framework matches existing calculations of generalization gap in the literature exactly in under-parameterized, over-parameterized and critical regimes.
Understanding generalization remains a central challenge in machine learning because it requires jointly considering data, architecture, and training dynamics. In this paper, we develop a theoretical framework that characterizes how these factors jointly shape generalization performance throughout training. More precisely, we study a broad class of neural networks trained under the ℓ2 loss by gradient descent (GD) with weight decay, and prove the convergence of GD to a neighbourhood of the global minimizers of the empirical loss. By partitioning the space based on the input data, we then decompose the population error into data error, optimization error, and prediction variation error, and bound them separately. In particular, for the prediction variation error, which measures the oscillations of the learned function, we propose (local) approximate homogeneity and derive explicit cellwise and layerwise bounds for its evolution along the training trajectory. These bounds yield two important implications: a necessary condition of improved generalization explains differences in layerwise generalization behavior; a sufficient condition describes delayed generalization and provides a theoretical characterization of grokking.
Yuqing Wang, Ioannis G. Kevrekidis, Mikhail Belkin
Understanding the generalization performance of over-parameterized neural networks has become a central topic in deep learning theory. While recent advances, particularly works under the Neural Tangent Kernel (NTK) regime, have shed light on the behavior of shallow architectures, the statistical generalization properties of deep neural networks (DNNs), especially in regression tasks, remain far less understood. In this paper, we make significant progress toward closing this gap by providing a comprehensive generalization analysis of DNNs trained using gradient-based methods. First, we establish, for the first time, a crucial connection between the learning dynamics of a DNN with smooth activation functions trained via gradient-based methods and those of kernel methods, showing that gradient-based methods on over-parameterized DNNs can fully inherit the favorable learning dynamics of their kernel counterparts. Building on this connection and the well-established optimality of kernel methods, we derive the first known minimax-optimal rates for the excess population risk of both gradient descent (GD) and stochastic gradient descent (SGD), under the assumption that network width scales polynomially with the sample size. Our results demonstrate that, with sufficient width, DNNs trained by GD or SGD can achieve generalization performance comparable to kernel-based methods.