learning theory still doesn't use description lengths, and would be perfectly happy with rules that have long descriptions as long as we delineate a small set of those rules
Any delineation of a small set of rules leads immediately to a short description length for the rules. You just need to encode the index of the rule in the set, costing log(N) bits for a set of size N.
Note that MDL is not the same as algorithmic information theory (definition-up-to-a-constant comes up in AIT, not MDL), though they're of course related.
You just need to encode the index of the rule in the set, costing log(N) bits for a set of size N.
I see everyone's assuming that some things, by their nature, always go to infinity (e.g. number of samples) while others stay constant (e.g. rule set). This is a nice convention, but not always realistic - and it certainly wasn't mentioned in the original formulation of the problem of learning, where everything is finite. If you really want things to grow, why don't you then allow the set itself to be specified by increasingly convoluted algorithms as N goe...
I declare this Open Thread open for discussion of Less Wrong topics that have not appeared in recent posts.