Official Review · NEURIPS 2024

- The authors provided a novel topological-complexity-based uniform generalization error bound, constructed on the $\alpha$-weighted lifetime sums or positive magnitude. This bound shows better correlation with the generalization error compared to existing bounds. - The authors proposed an implementation scheme based on dissimilarity measures between neural networks, enabling the quantification of generalization across different model architectures without the need for domain or problem-specific analysis. - The authors confirmed that their topological complexity term exhibits better correlation with the actual generalization gap in large-scale neural networks, such as ViT and GNN. - More Realistic Generalization Error Bound - This paper addresses a significant limitation of previous topological complexity-based generalization error bounds, which only hold under discrete parameter updates based on SGD or Adam, or when the number of iterations is taken to infinity. By overcoming this limitation, the bound provided in this paper offers a more practical and realistic understanding of generalization performance. - Additionally, the paper presents a method to compute this complexity for large-scale neural networks such as ViTs and GNNs, enhancing its practical applicability. - Innovative Combination of Two Interesting Topological Metrics for Generalization Error Bound - The authors leverage the fact that the $\alpha$-weighted lifetime sums based on minimum spanning trees, a quantity from persistent homology, are related to a pseudo metric, and apply this to generalization error analysis. - They also focus on a topological quantity called Magnitude and its connection to metric spaces, successfully providing another generalization error bound. - These achievements are the result of effectively combining techniques from topological data analysis, which goes beyond mere application and demonstrates significant originality. - Correlation with Generalization Performance Confirmed through Numerical Experiments on Large-Scale Neural Networks - The proposed topological complexity is computationally feasible even for large-scale neural networks. This is crucial for understanding the generalization performance of models like LLMs in the future. - Concerns Regarding Assumptions - This bound only holds under bounded losses. Thus, as the upper bound B increases, the bound becomes vacuous and diverges under unbounded losses. - Additionally, the relationship between the upper bound B of the loss and the topological complexity is unclear, making the overall interpretation of the bound difficult. For instance, under large B, the correlation of the topological complexity might be negated, resulting in a bound that does not correlate well overall. Providing an intuitive discussion on this aspect would help clarify the significance of the bound. - I am not well-versed in the assumptions based on pseudo metrics, so I cannot assess the validity of the (1,