According to the American Heritage Dictionary:

Regret -
To feel sorry, disappointed or distressed about.
To MOURN.
Distress over a desire unfulfilled or an ACTION performed or NOT PERFORMED.
An expression of GRIEF or disappointment.

Regrets are nothing to be ashamed of. And having regrets does not mean denying one's past experiences nor whom one is. Quite the contrary. It is Honesty. And only through honesty can one truly embrace one's Real self.
Regret, as per the dictionary explanation above, mentions the words mourn and grief. So if someone we loved died (especially prematurely) and we loved them and miss them -- that is regret.
Or if one wishes one had taken action to save a loved one's life but didn't,
that is regret.
Mourning a loved one's passing and experiencing grief over an action not performed, is only natural and normal.