Convergence of Adam in Deep ReLU Networks via Directional Complexity and Kakeya Bounds

• URL: http://arxiv.org/abs/2505.15013v1
• Date: Wed, 21 May 2025 01:34:16 GMT
• Title: Convergence of Adam in Deep ReLU Networks via Directional Complexity and Kakeya Bounds
• Authors: Anupama Sridhar, Alexander Johansen,
• Abstract summary: First-order adaptive optimization methods like Adam are the default choices for training modern deep neural networks.<n>We develop a multi-layer refinement framework that progressively tightens bounds on region crossings.<n>We prove that the number of region crossings collapses from exponential to near-linear in the effective dimension.
• Score: 49.1574468325115
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: 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.

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