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.

本研究解决了AdaGrad在非凸优化中渐近和非渐近收敛率理论分析不足的问题。通过引入来自概率理论的新停时技术，建立了AdaGrad在温和条件下的稳定性，并推导出几乎必然和均方的渐近收敛形式，展示了平均平方梯度的接近最优非渐近收敛率。这为未来其他自适应随机算法的研究提供了潜在的独立技术。 

AdaGrad在非凸优化中的渐近与非渐近收敛分析：基于新停时分析的方法