Authors: Mario Tuci, Caner Korkmaz, Umut Şimşekli, Tolga Birdal
Organizations: INRIA, CNRS, Département d’Informatique de l’Ecole Normale Supérieure / PSL, France · Department of Computing, Imperial College London, United Kingdom
Training modern neural networks often relies on large learning rates, operating at the edge of stability, where the optimization dynamics exhibit oscillatory and chaotic behavior. Empirically, this regime often yields improved generalization performance, yet the underlying mechanism remains poorly understood. In this work, we represent stochastic optimizers as random dynamical systems, which often converge to a fractal attractor set (rather than a point) with a smaller intrinsic dimension. Building on this connection and inspired by Lyapunov dimension theory, we introduce a novel notion of dimension, coined the `sharpness dimension', and prove a generalization bound based on this dimension. Our results show that generalization in the chaotic regime depends on the complete Hessian spectrum and the structure of its partial determinants, highlighting a complexity that cannot be captured by the trace or spectral norm considered in prior work. Experiments across various MLPs and transformers validate our theory while also providing new insights into the recently observed phenomenon of grokking.
One of the fundamental challenges in the deep learning community is to theoretically understand how well a deep neural network generalizes to unseen data. However, current approaches often yield generalization bounds that are either too loose to be informative of the true generalization error or only valid to the compressed nets. In this study, we present a simple yet non-vacuous generalization bound from the optimization perspective. We achieve this goal by leveraging that the hypothesis set accessed by stochastic gradient algorithms is essentially fractal-like and thus can derive a tighter bound over the algorithm-dependent Rademacher complexity. The main argument rests on modeling the discrete-time recursion process via a continuous-time stochastic differential equation driven by fractional Brownian motion. Numerical studies demonstrate that our approach is able to yield plausible generalization guarantees for modern neural networks such as ResNet and Vision Transformer, even when they are trained on a large-scale dataset (e.g. ImageNet-1K).
Predicting the generalization performance of deep neural networks without relying on hold-out validation data is a fundamental challenge in machine learning. While Stochastic Gradient Descent (SGD) drives the optimization of these highly parameterized models, its heavy-tailed, non-Gaussian dynamics induce complex, scale-invariant trajectories in the parameter space. In this paper, we propose a novel generalization measure based on the Fourier fractal dimension of the network's weight variations. By analyzing the characteristic function of the Lévy-driven stochastic differential equations in the frequency domain, we extract a metric that robustly captures the geometric complexity of the learning process. Furthermore, we introduce a customized Fourier-based optimizer designed to actively regularize this fractal dimension during training. Extensive empirical evaluations on the CIFAR-10, SVHN, and MNIST datasets demonstrate that our proposed Fourier generalization measure exhibits a strong correlation with the actual generalization gap. Our method achieves state-of-the-art Kendall rank correlation coefficients, outperforming a wide array of existing norm-based, margin-based, and PAC-Bayesian measures. Ultimately, this work highlights the potential of frequency-domain fractal analysis as both a powerful predictor for model generalizability and a principled foundation for developing more stable optimization algorithms.
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.