I would say regret takes a present participle (among other things*), not an infinitive.
'I regret not having piano lessons when I was young'.
'He's always regretted taking cocaine.'
'For the rest of her life she'll regret going to Australia.'
* Other uses:
'You'll regret it if you're not home by midnight.'
'I called you stupid. I regret that.'
'He'll regret the day he called me a liar.'
Student or Learner