Its hard to speed up the Russians advance into East Prussia but by very aggressive you mean willing to take great risks there are options
Zhilinsky could have pushed ahead with the full complement of 1st and 2nd Armies instead of whittling away about 11 divisions in useless rear duties. Even better if he would have, once it was obvious the main thrust was west, stripped the northern fortresses of their mobile artillery and sent it ahead
With moves like that, the Russians are likely to actually win Tannenberg. Whether the Germans fall back behind the Vistula or allow themselves to be destroyed is up to the local commanders
If the Germans retire behind the Vistula, the Russians secure their northern flank, spare themselves hundreds of thousands of casualties and prevent any serious damage from the German breakthrough at Gorlice-Tarnow.
If the Germans get trapped in East Prussia, the Russians force a rapid end to the Schefflien Plan as the Germans are going to have to find two armies instead of two corps to deal with the East