The president said he had considerable regret in accepting the resignation of "one of our nation's finest soldiers."

VOA: special.2010.06.26

In the next example you can use "I'm sorry" not to express regret for your personal fault.

下一个例句中，“I'm sorry”的使用并不是用来表达对个人过失的抱歉。

I'm sorry 课堂 - SpeakingMax英语口语达人

The young man feels regret, which only grows as he falls in love with her daughter.

VOA: special.2009.05.04