Okay, it looks like we are roughly on the same page :)
(Yes, my name is Robert, I go by Rob)
I think it is definitely possible that the ideas involved in the construction (e.g. making a “supertrader”) may lead to good practical algorithms. But it seems like this issue is orthogonal to the dominance issue. In other words, if you had shown me the constructions first and then the dominance results, I would not have been any more (or less) optimistic about the constructions after seeing those results than before.
It seems to me like there are two totally distinct ingredients here. First, we have a framework for specifying models and choosing which to use (like VC theory), including an idea about model averaging/fusion. Second, we have a trick involving enumerating a countably infinite set. The second part doesn’t seem relevant to the finite time case: if I’m running LIA and you tell me that you’ve changed it so it only enumerates traders up to T3↑↑↑3 and then stops adding more, this will ruin all of the LI criterion guarantees, but it will not change any of the output I’ll see in my lifetime, and the method for combining traders will remain as useful (or not) as it was before.
It’s interesting to compare this to structural risk minimization, where we also have a potential infinity (nested classes of increasing VC dimension), but we are given a good ordering for them (increasing VC dimension). One could do just as well asymptotically by choosing any other order for the classes: they are countable, so you will eventually hit the best one. But in practice the VC dimension order is crucial. The arbitrary enumeration in LIA is unsatisfying to me in the same way an arbitrary ordering of the VC classes would be. (Edit: you could even say SRM succeeds because it lets you avoid counting up to infinity, by giving you an ordering under which the minimum guaranteed risk class for your sample occurs early enough that you can actually find it.)
Okay, it looks like we are roughly on the same page :)
(Yes, my name is Robert, I go by Rob)
I think it is definitely possible that the ideas involved in the construction (e.g. making a “supertrader”) may lead to good practical algorithms. But it seems like this issue is orthogonal to the dominance issue. In other words, if you had shown me the constructions first and then the dominance results, I would not have been any more (or less) optimistic about the constructions after seeing those results than before.
It seems to me like there are two totally distinct ingredients here. First, we have a framework for specifying models and choosing which to use (like VC theory), including an idea about model averaging/fusion. Second, we have a trick involving enumerating a countably infinite set. The second part doesn’t seem relevant to the finite time case: if I’m running LIA and you tell me that you’ve changed it so it only enumerates traders up to T3↑↑↑3 and then stops adding more, this will ruin all of the LI criterion guarantees, but it will not change any of the output I’ll see in my lifetime, and the method for combining traders will remain as useful (or not) as it was before.
It’s interesting to compare this to structural risk minimization, where we also have a potential infinity (nested classes of increasing VC dimension), but we are given a good ordering for them (increasing VC dimension). One could do just as well asymptotically by choosing any other order for the classes: they are countable, so you will eventually hit the best one. But in practice the VC dimension order is crucial. The arbitrary enumeration in LIA is unsatisfying to me in the same way an arbitrary ordering of the VC classes would be. (Edit: you could even say SRM succeeds because it lets you avoid counting up to infinity, by giving you an ordering under which the minimum guaranteed risk class for your sample occurs early enough that you can actually find it.)