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.

Bounding and predicting the generalization gap of overparameterized neural networks remains a central open problem in theoretical machine learning. Neural network optimization trajectories have been proposed to possess fractal structure, leading to bounds and generalization measures based on notions of fractal dimension on these trajectories. Prominently, both the Hausdorff dimension and the persistent homology dimension have been proposed to correlate with generalization gap, thus serving as a measure of generalization. This work performs an extended evaluation of these topological generalization measures. We demonstrate that fractal dimension fails to predict generalization of models trained from poor initializations. We further identify that the $\ell^2$ norm of the final parameter iterate, one of the simplest complexity measures in learning theory, correlates more strongly with the generalization gap than these notions of fractal dimension. Finally, our study reveals the intriguing manifestation of model-wise double descent in persistent homology-based generalization measures. This work lays the ground for a deeper investigation of the causal relationships between fractal geometry, topological data analysis, and neural network optimization.

Disobeying the classical wisdom of statistical learning theory, modern deep neural networks generalize well even though they typically contain millions of parameters. Recently, it has been shown that the trajectories of iterative optimization algorithms can possess fractal structures, and their generalization error can be formally linked to the complexity of such fractals. This complexity is measured by the fractal's intrinsic dimension, a quantity usually much smaller than the number of parameters in the network. Even though this perspective provides an explanation for why overparametrized networks would not overfit, computing the intrinsic dimension (e.g., for monitoring generalization during training) is a notoriously difficult task, where existing methods typically fail even in moderate ambient dimensions. In this study, we consider this problem from the lens of topological data analysis (TDA) and develop a generic computational tool that is built on rigorous mathematical foundations. By making a novel connection between learning theory and TDA, we first illustrate that the generalization error can be equivalently bounded in terms of a notion called the 'persistent homology dimension' (PHD), where, compared with prior work, our approach does not require any additional geometrical or statistical assumptions on the training dynamics. Then, by utilizing recently established theoretical results and TDA tools, we develop an efficient algorithm to estimate PHD in the scale of modern deep neural networks and further provide visualization tools to help understand generalization in deep learning. Our experiments show that the proposed approach can efficiently compute a network's intrinsic dimension in a variety of settings, which is predictive of the generalization error.