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

2025-05-21

AI Summary

This work addresses the lack of theoretical foundations for adaptive optimizers (e.g., Adam) in deep ReLU networks—non-smooth and non-convex settings—by establishing their first global convergence guarantees and generalization bounds. Methodologically, we introduce a hierarchical region-crossing compression strategy coupled with a Kakeya-type geometric analysis framework, integrating hierarchical Morse theory and directional complexity modeling to reduce the number of region crossings induced by ReLU activations from exponential to near-linear. Theoretical contributions include: (1) a generalization bound of $ ilde{O}(sqrt{d_{ ext{eff}}/n})$ without the Polyak–Łojasiewicz (PL) condition or convexity assumptions, improving upon existing PAC-Bayes results; and (2) globally optimal convergence under a weak low-barrier assumption. These advances resolve a fundamental bottleneck in optimization theory for non-smooth deep learning.

Technology Category

• Search and Optimization: Non-convex Optimization
• Machine Learning: Optimization
• Reasoning under Uncertainty: Stochastic Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Graph neural networks and deep learning approaches for Web-related graphs
• Responsible Web: Human-perceived consequences of algorithmic deployment on the web
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking

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. extbf{We derive the first ( ilde{O}!igl(sqrt{d_{mathrm{eff}}/n}igr)) 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.

Problem

• Theoretical understanding of Adam in non-smooth Deep ReLU networks
• Generalization bound for Adam in non-convex ReLU landscapes
• Exponential to near-linear reduction in region crossings

Innovation

• Generalization bound for Adam in ReLU networks
• Multi-layer refinement for region crossings
• Kakeya-based tighter generalization bound