I don’t think that stopping at N does some kind of “disproportionate” damage to the LI. For example, Theorem 4.2.1 in the LI paper requires one trader per each ϵ>0 and sequence of theorems. If this trader is in the selection, then the probabilities of the theorems will converge to 1 within ϵ. Similarly, in my paper you need the gambler ΓSMk to ensure your forecaster converges to incomplete model M within 1/k.
You can do SRM for any sequence of nested classes of finite VC dimension. For example, if you have a countable set of hypotheses {hn}, you can take classes to be Hn:={hm}m<n. This is just as arbitrary as in LI. The thing is, the error bound that SRM satisfies depends on the actual class in which the reference hypothesis lies. So, compared to a hypothesis in a very high class, SRM can converge very slowly (require very large sample size). This is completely analogous to the inequality I gave before, where ξ(k) appears in the denominator of the bound, so gamblers that are “far away” in the “prior” can win a lot of bets before the forecaster catches up. SRM is useful iff you a priori expect “low” hypotheses to be good approximations. For example, suppose you want to fit a polynomial to some data but you don’t know what degree to use. SRM gives you a rule that determines the degree automatically, from the data itself. However, the reason this rule has good performance is because we expect most functions in the real world to be relatively “smooth” and therefore well-approximable by a low degree polynomial.
Ordering by description complexity is perfectly computable in itself, it just means that we fix a UTM, represent traders by programs (on which we impose a time bound, otherwise it really is uncomputable), and weight each trader by 2^{-program length}. It would be interesting to find some good property this thing has. If we don’t impose a time bound (so we are uncomputable), then the Bayesian analogue is Solomonoff induction, which has the nice property that it only “weakly” depends on the choice of UTM. Will different UTMs gives “similar” LIs? Off the top of my head, I have no idea! When we add the time bound it gets more messy since time is affect by translation between UTMs, so I’m not sure how to formulate an “invariance” property even in the Bayesian case. Looking through Schmidhuber’s paper on the speed prior, I see ey do have some kind of invariance (section 4) but I’m too lazy to understand the details right now.
Replying to Rob.
I don’t think that stopping at N does some kind of “disproportionate” damage to the LI. For example, Theorem 4.2.1 in the LI paper requires one trader per each ϵ>0 and sequence of theorems. If this trader is in the selection, then the probabilities of the theorems will converge to 1 within ϵ. Similarly, in my paper you need the gambler ΓSMk to ensure your forecaster converges to incomplete model M within 1/k.
You can do SRM for any sequence of nested classes of finite VC dimension. For example, if you have a countable set of hypotheses {hn}, you can take classes to be Hn:={hm}m<n. This is just as arbitrary as in LI. The thing is, the error bound that SRM satisfies depends on the actual class in which the reference hypothesis lies. So, compared to a hypothesis in a very high class, SRM can converge very slowly (require very large sample size). This is completely analogous to the inequality I gave before, where ξ(k) appears in the denominator of the bound, so gamblers that are “far away” in the “prior” can win a lot of bets before the forecaster catches up. SRM is useful iff you a priori expect “low” hypotheses to be good approximations. For example, suppose you want to fit a polynomial to some data but you don’t know what degree to use. SRM gives you a rule that determines the degree automatically, from the data itself. However, the reason this rule has good performance is because we expect most functions in the real world to be relatively “smooth” and therefore well-approximable by a low degree polynomial.
Ordering by description complexity is perfectly computable in itself, it just means that we fix a UTM, represent traders by programs (on which we impose a time bound, otherwise it really is uncomputable), and weight each trader by 2^{-program length}. It would be interesting to find some good property this thing has. If we don’t impose a time bound (so we are uncomputable), then the Bayesian analogue is Solomonoff induction, which has the nice property that it only “weakly” depends on the choice of UTM. Will different UTMs gives “similar” LIs? Off the top of my head, I have no idea! When we add the time bound it gets more messy since time is affect by translation between UTMs, so I’m not sure how to formulate an “invariance” property even in the Bayesian case. Looking through Schmidhuber’s paper on the speed prior, I see ey do have some kind of invariance (section 4) but I’m too lazy to understand the details right now.