Fourier fractal dimension to predict the generalization of deep neural networks

• URL: http://arxiv.org/abs/2606.08308v1
• Date: Sat, 06 Jun 2026 19:40:54 GMT
• Title: Fourier fractal dimension to predict the generalization of deep neural networks
• Abstract summary: We propose a novel generalization measure based on the Fourier fractal dimension of the network's weight variations.<n>By analyzing the characteristic function of the Lévy-driven differential equations in the frequency domain, we extract a metric that robustly captures the geometric complexity of the learning process.<n>Our method achieves state-of-the-art Kendall rank correlation, outperforming a wide array of existing norm-based, margin-based, and PAC-Bayesian measures.
• Score: 0.0
• License: http://creativecommons.org/licenses/by/4.0/
• Abstract: 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.

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