Is there known any complexity class containing online counterparts of optimization problems?
One tricky aspect of defining complexity classes for online problems is that there's in principle no limit on what kinds of computations I can do once I've read the input. In other words, online problems are hard even if I have (for example) an NP oracle processing the input once it arrives. 

It's conceivable that with a more limited processor, even simpler prediction tasks become harder to perform, but in general the difficulty of designing online algorithms comes from the ability of the adversary to change the input after you've built a prediction model.