On the limitations of fractal dimension as a measure of generalization
OA Location
Author(s)
Tan, CB
García-Redondo, I
Wang, Q
Bronstein, MM
Monod, A
Type
Conference Paper
Abstract
Bounding and predicting the generalization gap of overparameterized neural networks remains a central open problem in theoretical machine learning. There is a recent and growing body of literature that proposes the framework of fractals to model optimization trajectories of neural networks, motivating generalization bounds and measures based on the fractal dimension of the trajectory. Notably, the persistent homology dimension has been proposed to correlate with the generalization gap. This paper performs an empirical evaluation of these persistent homology-based generalization measures, with an in-depth statistical analysis. Our study reveals confounding effects in the observed correlation between generalization and topological measures due to the variation of hyperparameters. We also observe that fractal dimension fails to predict generalization of models trained from poor initializations. We lastly reveal the intriguing manifestation of model-wise double descent in these topological generalization measures. Our work forms a basis for a deeper investigation of the causal relationships between fractal geometry, topological data analysis, and neural network optimization.
Date Issued
2025-02-01
Date Acceptance
2024-12-01
Citation
Advances in Neural Information Processing Systems, 2025, 37, pp.60309-60334
ISBN
9798331314385
ISSN
1049-5258
Publisher
Curran Associates, Inc.
Start Page
60309
End Page
60334
Journal / Book Title
Advances in Neural Information Processing Systems
Volume
37
Copyright Statement
© 2024 NEURIP.
Source
38th Conference on Neural Information Processing Systems (NeurIPS 2024)
Publication Status
Published
Start Date
2024-12-10
Finish Date
2024-12-15
Coverage Spatial
Vancouver, Canada