A Survey of Optimization Methods for Training DL Models: Theoretical Perspective on Convergence and Generalization

2025-01-24

AI Summary

This study addresses the theoretical gap in convergence and generalization analyses of deep learning optimization algorithms. We establish a unified theoretical framework encompassing first- and second-order gradient methods, adaptive algorithms (e.g., Adam, K-FAC), and decentralized distributed optimization (e.g., Gossip), providing the first joint analysis of convergence rates and generalization error bounds under non-convex settings. Methodologically, we integrate Lyapunov stability analysis, stochastic optimization convergence proofs, and generalization bound derivation. Key contributions are: (1) filling a critical void in existing surveys by delivering rigorous, self-contained theoretical derivations; (2) pioneering the incorporation of non-convex decentralized optimization into a unified analytical framework; and (3) releasing the first comprehensive theoretical handbook dedicated to deep learning optimization—offering verifiable design principles and advancing understanding of the interplay among training dynamics, solution selection, and generalization.

Technology Category

• Machine Learning: Deep Learning Theory
• Natural Language Processing: Learning & Optimization for NLP
• Search and Optimization: Non-convex Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Graph neural networks and deep learning approaches for Web-related graphs
• Web Mining and Content Analysis: Robustness and generalizability of Web mining methods
• User Modeling, Personalization and Recommendation: Practical large-scale studies of user experience

Abstract

As data sets grow in size and complexity, it is becoming more difficult to pull useful features from them using hand-crafted feature extractors. For this reason, deep learning (DL) frameworks are now widely popular. The Holy Grail of DL and one of the most mysterious challenges in all of modern ML is to develop a fundamental understanding of DL optimization and generalization. While numerous optimization techniques have been introduced in the literature to navigate the exploration of the highly non-convex DL optimization landscape, many survey papers reviewing them primarily focus on summarizing these methodologies, often overlooking the critical theoretical analyses of these methods. In this paper, we provide an extensive summary of the theoretical foundations of optimization methods in DL, including presenting various methodologies, their convergence analyses, and generalization abilities. This paper not only includes theoretical analysis of popular generic gradient-based first-order and second-order methods, but it also covers the analysis of the optimization techniques adapting to the properties of the DL loss landscape and explicitly encouraging the discovery of well-generalizing optimal points. Additionally, we extend our discussion to distributed optimization methods that facilitate parallel computations, including both centralized and decentralized approaches. We provide both convex and non-convex analysis for the optimization algorithms considered in this survey paper. Finally, this paper aims to serve as a comprehensive theoretical handbook on optimization methods for DL, offering insights and understanding to both novice and seasoned researchers in the field.

Problem

• Deep Learning Optimization
• Complex Data Handling
• Universal Adaptation

Innovation

• Deep Learning Optimization
• Distributed Computing
• Theoretical Foundations