Where does “at any rate” come from?

“At any rate” literally means that “the rate doesn’t matter”.

If I were to say “my computer can process commands at any rate that I can enter commands”, then I’m saying that my computer is not limited in any way, I can send commands as fast as I want.

However, you’re asking for a specific usage. “at any rate” when it is used to go back to an older discussion topic. This isn’t a literal meaning, but rather a figurative (idiomatic?) meaning.

Let me first rephrase my example:

• My computer can process commands at any rate that I can enter commands.
• My computer can process commands, regardless of the rate with which I send them.
• Sometimes, I enter commands much faster than other times, but the computer can process them anyway.
• Sometimes, I enter commands much faster than other times, but the computer can process them at any rate.

What I am telling you is that we should not be discussing the rate at which I can send commands, because whatever the rate is, I am telling you my computer will definitely be able to handle it. We could spend time trying to determine the rate at which I send commands, but it would be wasted time because it does not matter, the computer will handle it anyway.

When you say “regardless”, you are trying to stop talking about a specific topic because it does not influence the other topic you’re trying to talk about.

I don’t know if the gun he was holding was an automatic rifle or a pistol. Regardless (of which type of gun he was carrying), he was carrying a gun in public!

I’ll also give you an example where “regardless” or “at any rate” (they are interchangeably in this specific contect) is used to revert to an older topic.

A : Hey, I bumped into C last night. She said you’ve been stalking her.
B : Oh. She gets like that when she’s drunk.
A : I don’t think that’s relevant to this discussion. She said you’ve been calling her and leaving her creepy messages.
B : Let me just ask you this: was she smoking a cigarette? Because she only smokes after she’s been drinking for a while, so that must mean that she was drunk at the time!
A : Regardless / At any rate , I heard the voicemail messages you left her, and they are really creepy.

B was introducing a new discussion topic (how to establish whether C was drunk or not). A is shutting that topic down, because it does not matter. He already has proof that B left creepy messages, and it is therefore irrelevant to argue whether C was truthful or not.