On the $O(\frac{\sqrt{d}}{T^{1/4}})$ Convergence Rate of RMSProp and Its Momentum Extension Measured by $\ell_1$ Norm: Better Dependence on the Dimension

Although adaptive gradient methods have been extensively used in deep learning, their convergence rates have not been thoroughly studied, particularly with respect to their dependence on the dimension. This paper considers the classical RMSProp and its momentum extension and establishes the convergence rate of $\frac{1}{T}\sum_{k=1}^TE\left[\|\nabla f(x^k)\|_1\right]\leq O(\frac{\sqrt{d}}{T^{1/4}})$ measured by $\ell_1$ norm without the bounded gradient assumption, where $d$ is the dimension of the optimization variable and $T$ is the iteration number. Since $\|x\|_2\ll\|x\|_1\leq\sqrt{d}\|x\|_2$ for problems with extremely large $d$, our convergence rate can be considered to be analogous to the $\frac{1}{T}\sum_{k=1}^TE\left[\|\nabla f(x^k)\|_2\right]\leq O(\frac{1}{T^{1/4}})$ one of SGD measured by $\ell_1$ norm.