One second is long enough that it can put a user off from using your app though. Take notifications on phones for example. I know several people who would benefit from a habitual use of phone notifications, but they never stick to using them because the process of opening (or switching over to) the notification app and navigating its UI to leave a notification takes too long. Instead they write a physical sticky note, because it has a faster "startup time".
All depends on the type of interaction.
A high usage one, absolutely improve the time of it.
Loading the profile page? Isn't done often so not really worth it unless it's a known and vocal issue.
https://xkcd.com/1205/ gives a good estimate.