I had the same problem as you:
Log into the Mi account which has acquired unlocking permission on your device. Make sure it's the same with the account you sent your request. It's recommended to connect to the Internet for Mi account to sync.
http://en.miui.com/thread-246705-1-1.html
Auto Sync is disabled when screen is off (Smart Network)
I use BetterBatteryStats (root requiered): you can read everything about "Deep Sleep"
http://forum.xda-developers.com/showthread.php?t=1179809
Smart Network: http://forum.xda-developers.com/xposed/modules/mod-t2908637
Xposed for MIUI...
I bought a Remdi Note 3 Pro this week and I'm very happy with purchase. Great ROM (I'm coming from Stock/CM) and a impressive standby battery life! -> 99% deepsleep -0,3%/h - thank you and the german translation team for your hard work!