How large can those constants be? How complex may the environment be maximally for an ideal, but still realistic, agi agent (thus not a solomonof or AIXI agent) to be still succesful? Does somebody know how to calculate (and formalise) this?
I'm not sure if this makes much sense. An "ideal" agent is not going to be a "realistic" agent. The bigger your computer and the better your software more complexity your agent will be able to deal with.
The only way I could see that it would make sense would be if you could come up with an algorithm and prove that it made the best possible usage of time and space in terms of achieving its goals. Then the constants you are talking about would be set by this algorithm and the size of the biggest computer you could get.
Not even an educated guess?
But I think some things can be said:
Suppose perception of the environment is just a bit at a time: ...010100010010010111010101010...
In the random case: for any sequence of length l the number of possible patterns is 2^l. Completely hopeless, unless prediction precision need decreases also exponentially with l. But that is not realistic. You then know nothing, but you want nothing also.
Yes, this defines the limiting case for Solomonoff Induction...
in the logarithmic case: the number of possible patterns of length l increases logarithmically with l: #p < constant * log(l). If the constant is not to high this environment can be learned easily. There is no need for vagueness
Not true. Just because the sequence is very compressible in a Kolmogorov sense doesn't imply that it's easy to learn. For example you could have some sequence where the computation time of the n-th bit take n^1000 computation cycles. There is only one pattern and it's highly compressible as it has a pretty short algorithm however there is no way you'll ever learn what the pattern is.
I suppose the point I'm trying to make is that complexity of the environment is not all. It's is also important to know how many of the complexity can be ignored.
Yes. The real measure of how difficult an environment is is not the complexity of the environment, but rather the complexity of the simplest solution to the problem that you need to solve in that environment.
Shane
P.S. one of these days I'm going to get around to replying to your other emails to me!! sorry about the delay!
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