Will increasing your internet speed reduce lag on xbox live?

Not necessarily, because most of the time lag is coming from the host's internet connection, and he/she is the one responsible for hosting the game. But it never hurts to increase your internet speed so like unless it's costing you go for it.

EDIT: If you are truly don't want any lag, get about a 1mbps+ upload and you should usually be the host in about every game, that should be plenty to host up to 12 players with little to no lag. If you don't wish to host, then try increasing your download, and don't go into foreign lobbies, over sea hosters are terrible with a lot of lag.