First-order adaptive optimization methods like Adam are the default choices for training modern deep neural networks. Despite their empirical success, the theoretical understanding of these methods in non-smooth settings, particularly in Deep ReLU networks, remains limited. ReLU activations create exponentially many region boundaries where standard smoothness assumptions break down. \textbf{We derive the first \(\tilde{O}\!\bigl(\sqrt{d_{\mathrm{eff}}/n}\bigr)\) generalization bound for Adam in Deep ReLU networks and the first global-optimal convergence for Adam in the non smooth, non convex relu landscape without a global PL or convexity assumption.} Our analysis is based on stratified Morse theory and novel results in Kakeya sets. We develop a multi-layer refinement framework that progressively tightens bounds on region crossings. We prove that the number of region crossings collapses from exponential to near-linear in the effective dimension. Using a Kakeya based method, we give a tighter generalization bound than PAC-Bayes approaches and showcase convergence using a mild uniform low barrier assumption.
View on arXiv@article{sridhar2025_2505.15013,
title={ Convergence of Adam in Deep ReLU Networks via Directional Complexity and Kakeya Bounds },
author={ Anupama Sridhar and Alexander Johansen },
journal={arXiv preprint arXiv:2505.15013},
year={ 2025 }
}