On the Limitations of Fractal Dimension as a Measure of Generalization

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. In this talk, I will present an empirical evaluation of these persistent homology-based generalization measures, with an in-depth statistical analysis. This 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; and reveal the intriguing manifestation of model-wise double descent in these topological generalization measures. This is joint work with Charlie B. Tan, Qiquan Wang, Michael M. Bronstein and Anthea Monod.