Two Parameters, Two Gradient Scales

A network has two parameters. $\theta_A$ sits in a busy part of the network and its gradient has held steady at $g_A = 2$ for many steps. $\theta_B$ is the weight on a rarely-seen feature, and its gradient has held steady at $g_B = 0.02$.

Both are trained with RMSProp:

$s_t = \rho\, s_{t-1} + (1-\rho)\, g_t^2, \qquad \theta_{t+1} = \theta_t - \frac{\alpha}{\sqrt{s_t} + \epsilon}\, g_t$

with $\alpha = 0.01$, $\rho = 0.9$, and $\epsilon = 10^{-8}$ (ignorable). Because each gradient has been constant for so long, each parameter's $s_t$ has settled to a constant.

Plain gradient descent would move $\theta_A$ by $0.02$ per step and $\theta_B$ by $0.0002$ per step — a hundredfold gap. Once $s$ has settled, how big is each parameter's RMSProp update?