Just for reference, the lowest limit for competition aerobatics, depending on the competence level of the pilot, is between 700 and 1500 feet. Drop below that and your penalized. Drop significantly (>100') below that and you're disqualified.
For training we try to use a lower limit of 3500 feet, airspace and weather permitting. Mainly for safety, obviously, but also to limit noise on the ground.
Display Aerobatics requires a DA, as mentioned, which needs to be renewed regularly - I think every year before the display season starts. And even those with DAs will very carefully plan their sequences so that that lower limit is only reached in a safe, controllable manner.
Brian Lecomber has a column in Flyer Magazine and regularly writes about this sort of stuff.