Make sure you don't fall foul of TSA rules and regulations. The good old days of coming to the US on a 'visitor visa' and completing a pilot certificate are long gone.
Flight schools have to spend a bunch of time making sure they are in compliance, a private instructor (such as myself) does not hold the authorization to issue visas for flight training.
If you have a foreign address on your pilot certificate the TSA will cross reference it against your immigration history and if they find that you trained 'under the radar' then you may find yourself in hot water. This may take years for them to do, but this informatuin came from a TSA agent doing an audit on a flight school I used to be associated with.