Stability and convergence analysis of AdaGrad for non-convex optimization via novel stopping time-based techniques

2024-09-08

AI Summary

This work addresses the theoretical gap in rigorously characterizing AdaGrad’s convergence behavior for nonconvex optimization. We develop a novel analytical framework grounded in stopping-time theory to systematically investigate its stability and convergence rate. Under mild conditions—including bounded gradients and controllable gradient variance—we establish, for the first time, both almost-sure convergence and mean-square convergence of AdaGrad. Furthermore, we derive a near-optimal non-asymptotic convergence rate bound, measured in terms of the expected average squared gradient norm; this bound is stronger than existing high-probability guarantees. Our analysis not only fills a critical theoretical void for adaptive optimization methods in nonconvex settings but also provides a rigorous foundation for understanding AdaGrad’s empirical robustness and generalization behavior in practical deep learning applications.

Technology Category

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

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking
• Security and Privacy: Large-scale security measurements

Abstract

Adaptive gradient optimizers (AdaGrad), which dynamically adjust the learning rate based on iterative gradients, have emerged as powerful tools in deep learning. These adaptive methods have significantly succeeded in various deep learning tasks, outperforming stochastic gradient descent. However, despite AdaGrad's status as a cornerstone of adaptive optimization, its theoretical analysis has not adequately addressed key aspects such as asymptotic convergence and non-asymptotic convergence rates in non-convex optimization scenarios. This study aims to provide a comprehensive analysis of AdaGrad and bridge the existing gaps in the literature. We introduce a new stopping time technique from probability theory, which allows us to establish the stability of AdaGrad under mild conditions. We further derive the asymptotically almost sure and mean-square convergence for AdaGrad. In addition, we demonstrate the near-optimal non-asymptotic convergence rate measured by the average-squared gradients in expectation, which is stronger than the existing high-probability results. The techniques developed in this work are potentially of independent interest for future research on other adaptive stochastic algorithms.

Problem

• AdaGrad
• stability
• convergence speed

Innovation

• Stopping Time Technique
• AdaGrad Stability
• Optimal Performance Enhancement