There's no evidence to suggest he did
He possibly feels a small bit of remorse for killing Mace Windu in Revenge of the Sith, that's about it and it doesn't last long:
Anakin Skywalker knelt in the rain.
He was looking at a hand. The hand had brown skin. The hand held a lightsaber. The hand had a charred oval of tissue where it should have been attached to an arm.
"What have I done?" Was it his voice? It must have been. Because it was his question.
"What have I done?"
Revenge of the Sith Chapter 17: "The Face of the Dark"
After this point, when Anakin completely falls to the Dark Side and is subsumed by Darth Vader, there's no evidence that he feels any remorse whatsoever for his actions. The intention of George Lucas, which so far has been continued by the Lucasfilm Story Group under Disney, is that Luke is the only thing that can really penetrate Vader's Darkness; this is something Dave Filoni discussed in a 2016 interview with IGN, on the subject of Vader's confrontation with Ahsoka on Star Wars Rebels (emphasis mine):
We both agreed that the one thing that Ahsoka can't represent in any way is any path of redemption for Vader, or the hope that that's there because Luke is the only one that's going to be able to make that happen. And that's the story that we see. It was our belief that we could have this moment, but it's not even a moment of hesitation for Vader. Because he says then, "You will die," and he means it.
