In this paper, we study the convergence of the Adaptive Moment Estimation
(Adam) algorithm under unconstrained non-convex smooth stochastic
optimizations. Despite the widespread usage in machine learning areas, its
theoretical properties remain limited. Prior researches primarily investigated
Adam's convergence from an expectation view, often necessitating strong
assumptions like uniformly stochastic bounded gradients or problem-dependent
knowledge in prior. As a result, the applicability of these findings in
practical real-world scenarios has been constrained. To overcome these
limitations, we provide a deep analysis and show that Adam could converge to
the stationary point in high probability with a rate of $\mathcal{O}\left({\rm
poly}(\log T)/\sqrt{T}\right)$ under coordinate-wise "affine" variance noise,
not requiring any bounded gradient assumption and any problem-dependent
knowledge in prior to tune hyper-parameters. Additionally, it is revealed that
Adam confines its gradients' magnitudes within an order of
$\mathcal{O}\left({\rm poly}(\log T)\right)$. Finally, we also investigate a
simplified version of Adam without one of the corrective terms and obtain a
convergence rate that is adaptive to the noise level.

研究了自适应矩估计算法（Adam）在无约束非凸平滑随机优化中的收敛性，证明了 Adam 能够在很高的概率下以 $O (poly (log T)/√T)$ 的速率收敛到稳定点，不需要任何有界梯度假设和问题相关的先验知识来调整超参数，同时还研究了一个简化版本的 Adam 算法以适应噪声水平。