Adaptive gradient methods are workhorses in deep learning. However, the convergence guarantees of adaptive gradient methods for nonconvex optimization have not been sufficiently studied. In this paper, we provide a sharp analysis of a recently proposed adaptive gradient method namely partially adaptive momentum estimation method (Padam) (Chen and Gu, 2018), which admits many existing adaptive gradient methods such as AdaGrad, RMSProp and AMSGrad as special cases. Our analysis shows that, for smooth nonconvex functions, Padam converges to a first-order stationary point at the rate of $O\big((\sum_{i=1}^d\|\mathbf{g}_{1:T,i}\|_2)^{1/2}/T^{3/4} + d/T\big)$, where $T$ is the number of iterations, $d$ is the dimension, $\mathbf{g}_1,\ldots,\mathbf{g}_T$ are the stochastic gradients, and $\mathbf{g}_{1:T,i} = [g_{1,i},g_{2,i},\ldots,g_{T,i}]^\top$. Our theoretical result also suggests that in order to achieve faster convergence rate, it is necessary to use Padam instead of AMSGrad. This is well-aligned with the empirical results of deep learning reported in Chen and Gu (2018).

本文提供了关于一类自适应梯度方法（包括AMSGrad，RMSProp和AdaGRad）在光滑非凸函数优化方面的收敛性分析，证明了期望下自适应梯度方法能够收敛到一阶稳定点，同时还证明了AMSGrad，RMSProp和AdaGrad的收敛速率，这些结论有助于更好地理解自适应梯度方法在优化非凸目标时的机制。

非凸优化的自适应梯度方法收敛性研究