and the number of possible models for T rounds is exponential in T
??? Here n is the number of other people betting. It’s a constant.
Within a single application of online learning, n is a constant, but that doesn’t mean we can’t look at the consequences of it having particular values, even values that vary with other parameters. But, you seem to be agreeing with the main points that if you use all possible models (or “super-people”) the regret bound is meaningless, and that in order to reduce the number of models so it is not meaningless, while also keeping a good model that is worth performing almost as well as, you need structural assumptions.
even if the “true hypothesis” isn’t in the family of models we consider
I agree you don’t need the model that is right every round, but you do need the model to be right in a lot of rounds. You don’t need a perfect model, but you need a model that is as correct as you want your end results to be.
maybe even adversarial data
I think truly adversarial data gives a result that is within the regret bounds, as guaranteed, but still uselessly inaccurate because the data is adversarial against the collection of models (unless the collection is so large you aren’t really bounding regret).
Within a single application of online learning, n is a constant, but that doesn’t mean we can’t look at the consequences of it having particular values, even values that vary with other parameters. But, you seem to be agreeing with the main points that if you use all possible models (or “super-people”) the regret bound is meaningless, and that in order to reduce the number of models so it is not meaningless, while also keeping a good model that is worth performing almost as well as, you need structural assumptions.
I agree you don’t need the model that is right every round, but you do need the model to be right in a lot of rounds. You don’t need a perfect model, but you need a model that is as correct as you want your end results to be.
I think truly adversarial data gives a result that is within the regret bounds, as guaranteed, but still uselessly inaccurate because the data is adversarial against the collection of models (unless the collection is so large you aren’t really bounding regret).