Found and corrected a mistake on someone's else paper -- praxis?
I have not been in such situation. However, I do not see what is the problem for you to write a paper complementing or correcting the previous paper? Isn't this how science works?
In my opinion, it is very toxic culture in academia to consider such thing as inappropriate. Those previous auathors are humans. Assuming good faith, that was what they knew and what was to the best of their knowledge, and to the best of the reviewers' knowledge, at the time of their publication. You got something, great, you have the right to get the credit for it. I feel it is unfair that you inform the authors offline. You should write it, and publish it too.
Also, we should not forget that you might be wrong too! I do not mean to offend anyone. But if we remember that we are all humans and have limitations in our knowledge about exploring this world, we would take these issues in a more relaxed way.
Edited: OP mentioned that the paper has been published in conference (not just on ArXiv as was implied by phrasing).
This can be tricky.
The main thing is that you get due credit, and that the scientific community is aware of the mistake.
If the mistake is in a key theorem that is the basis for the entire paper, then the authors should retract the paper. As unpleasant as this may be, it is the only ethical and fair thing to do. If the authors aren't doing this of their own volition, then you should (I would argue that you are in fact ethically obliged to do so!):
- Make 100% sure that you are right in your claim. It's not entirely clear whether the theorem is "dead in the water" i.e. the claim is demonstrably false (via a counterexample that you've constructed), or that the claim might be true, but their proof is wrong, and you couldn't have come up with a proof either. If you have a counterexample, then you've really managed to kill the paper. Otherwise it's still an open problem (not a bad thing, just a different scenario).
- Consulting with your advisor/senior member of the community, contact the conference program chairs to inform them of the mistake. You should let them know where the paper is wrong; you can also mention that you have a correction that you're keeping to yourself, but you are not obliged to provide it. A paper with a wrong proof should never appear in any conference.
This paper should not be published. If it was published in a high-profile ML conference (ICML/NeurIPS/COLT etc.), then the organizers can be trusted to take it from there.
You must understand though that if you go and report this to the organizers without the authors being on board, then this may result in some unpleasant interactions with them in the future (to be fair, the authors brought this upon themselves by not owning up, but still - people have egos and pride).
Next up, The ideal scenario is that you email the authors, the authors agree to working with you, adopt your proof technique and add you as a coauthor (assuming you’ve corrected an important theorem). This is assuming that the authors are being reasonable and don’t have a fragile ego.
To make that happen you’ll need to phrase the email carefully “I think that I can show that Theorem 1 holds under some minor assumptions, can you take a look at my proof? I’d love to collaborate with you on a future version of the paper” or something like that.
If that doesn’t happen (radio silence after you email them or worse: them saying they had already thought of this idea or some other nonsense), you need to get your credit somehow (e.g. by getting a supervisor/mentor to intervene). You could concurrently start writing your own version and upload to ArXiv (so it’s publicly timestamped), referencing the original.
Most cases I’ve seen followed a merger of authors but there are some distasteful instances, be prepared and be pleasant and you’ll be fine!
If you think that you have found the proof of a paper, or you can (with evidence) notify the community of the wrong/improper/incomplete proof, you can itself publish it as a research article.
For example, look at the following comment (to be) published in IEEE Trans. on Wireless Communication: Comments on “Coverage Analysis of Multiuser Visible Light Communication Networks”
Linke can be found here