I've read ADAM paper (https://arxiv.org/abs/1412.6980) and found out a suspicious part.
In Lemma 10.3 of Appendix, \begin{align*} \sqrt{\|g_{1:T,i}\|_2^2 - g_{T,i}^2} & \le \|g_{1:T,i}\|_2 - \frac{g_{T,i}^2}{2\|g_{1:T,i}\|_2}\\ & \le \|g_{1:T,i}\|_2 - \frac{g_{T,i}^2}{2\sqrt{TG_\infty^2}} \end{align*}
They set the bound of 2-norm, $\|g_{1:T,i}\|_2$ of gradient using $G_\infty$, not $G_2$.
Infinity norm is less than or equal to 2-norm, so I think the proof might be wrong.
Can someone tell me what went wrong? Thanks.