Is there a fundamental difference between building a set on N logistic regressions in 1 vs all fashion, as compared to training a single mutlinomial logistic regression? Put another way, are there any optimization techniques that treat the 1 to N classes logistic regression problem in a way that's markedly different from N independent regressions?
Intuitively is seems like the answer ought to be yes, since there should be a lot of information sharing between various problems if two classes are similar. But since I'm not entirely well versed on how common 1 to N solvers actually work, I can't tell if I'm right or these problems are treated in ways that fundamentally the same.
I think that we can see that there could be a difference between the two models, but I'm not entirely sure. Googling the matter revealed a few arcane discussing about the subject, but I was unable to find an authoritative discussion of the matter.