Asymptotic and Non-Asymptotic Convergence Analysis of AdaGrad for Non-Convex Optimization via Novel Stopping Time-based Analysis

Adaptive optimizers have emerged as powerful tools in deep learning, dynamically adjusting the learning rate based on iterative gradients. These adaptive methods have significantly succeeded in various deep learning tasks, outperforming stochastic gradient descent (SGD). However, although AdaGrad is a cornerstone adaptive optimizer, its theoretical analysis is inadequate in addressing asymptotic convergence and non-asymptotic convergence rates on non-convex optimization. This study aims to provide a comprehensive analysis and complete picture of AdaGrad. We first introduce a novel stopping time technique from probabilistic theory to establish stability for the norm version of AdaGrad under milder conditions. We further derive two forms of asymptotic convergence: almost sure and mean-square. Furthermore, we demonstrate the near-optimal non-asymptotic convergence rate measured by the average-squared gradients in expectation, which is rarely explored and stronger than the existing high-probability results, under the mild assumptions. The techniques developed in this work are potentially independent of interest for future research on other adaptive stochastic algorithms.