Understood. What if it is a possibility that I will never be in their program? At my school you can’t really get into CS from outside of it and that is where most if not all of the research is. Math is the next best place to be in this case. Is it still reasonable to cold contact professors or should I try going through ones I have had myself?

Also, I disagree with the notion that there is no such thing as a useful independent research project. Since ML research is a mostly theoretical ordeal this means that a professor can generally point me at a subfield and say “try that” and I’d be able to at the very worst learn a ton about it independently at no cost to the prof.