English (United Kingdom)

https://curated-unify.zendy.io/wp-json/zendy-region/v1/featured_content/oa?rat=en

https://curated-unify.zendy.io/wp-json/zendy-region/v1/highlighted_journal/

Global: Zendy Plus annual

Presents the access of premium content as premium feature

Premium Content

Presents the keyphrase highlighting as premium feature

Keyphrase Highlighting

Presents the summarisation as premium feature

Summarisation

Insights

Presents the pdf analysis as premium feature

PDF Analysis

Presents the zaia usage as premium feature

ZAIA

Global: Zendy Tools annual

Global: Zendy Free Plan

Global: Zendy Tools monthly

Global: Zendy Plus monthly

Although adaptive gradient methods have been extensively used in deeplearning, their convergence rates proved in the literature are all slower thanthat of SGD, particularly with respect to their dependence on the dimension.This paper considers the classical RMSProp and its momentum extension andestablishes the convergence rate of $\frac{1}{T}\sum_{k=1}^T E\left[\|\nablaf(x^k)\|_1\right]\leq O(\frac{\sqrt{d}C}{T^{1/4}})$ measured by $\ell_1$ normwithout the bounded gradient assumption, where $d$ is the dimension of theoptimization variable, $T$ is the iteration number, and $C$ is a constantidentical to that appeared in the optimal convergence rate of SGD. Ourconvergence rate matches the lower bound with respect to all the coefficientsexcept the dimension $d$. Since $\|x\|_2\ll\|x\|_1\leq\sqrt{d}\|x\|_2$ forproblems with extremely large $d$, our convergence rate can be considered to beanalogous to the $\frac{1}{T}\sum_{k=1}^T E\left[\|\nabla f(x^k)\|_2\right]\leqO(\frac{C}{T^{1/4}})$ rate of SGD in the ideal case of $\|\nablaf(x)\|_1=\varTheta(\sqrt{d}\|\nabla f(x)\|_2)$.

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