I have my iPad using shared WiFi from my Mac. It's not plugged in, nothing's updating, nothing's open. But unless I have it on airplane mode I see it draining all the bandwidth from my Mac. Is there some app or settings view that shows what processes are using bandwidth?
On an unjailbroken device you can't see this information.
On iOS 7, you can only see cellular data usage ( Settings > Cellular network > Usage )
Maybe you could just try to kill one by one the apps that are running in background.