Last year I applied to Mit, Columbia gsapp and Rice as an international student. I submitted my apps 2 hours prior to deadline for Mit and rice and like 1 day before the deadline of gsapp. My results were all announced at the same days as other applicants since I have checked from gradcafe. So it didn't matter for me but maybe it differs according to applicants. You should check each of their website to see their admissions are rolling basis. best of luck

Submit a bit early in case you realize at the last minute forgot something, but the more time can put into portfolio & application = the better.

When I was teaching we got >70% of all applications within the last week before the deadline.  It doesn't make any difference to your chances - the admissions committee doesn't look at any of them until a couple weeks after the deadline, after the administrative people have been all through them to make sure they're complete and that references, transcripts, etc. have been received.  

The exception to this is if you happen to apply to a less competitive program with "rolling admissions": those programs offer spots to all qualified candidates with complete applications, until their class for the following year fills up, and then they stop admitting for that year.