Before revision:  Then, we show that PAGE obtains the optimal convergence results O(n+\sqrt{n}{\epsilon^2}) (} finite-sum }{\epsilon^2}) (online) matching our lower bounds for both nonconvex finite-sum } and online regimes, which establish that PAGE is an optimal method .
After revision:  Then, we show that PAGE obtains the optimal convergence results O(n+\sqrt{n}{\epsilon^2}) (} finite-sum }{\epsilon^2}) (online) matching our lower bounds for both nonconvex finite-sum } and online problems .