Some people insist that “regime” should be used only in reference to governments, and that people who say they are following a dietary regime should instead use “regimen”; but “regime” has been a synonym of “regimen” for over a century, and is widely accepted in that sense.
However “regiment” is an error in this sense. The only way you could follow a strict regiment would be to march behind a highly disciplined military unit. Your diet or exercise routine is not a “regiment.”
List of errors