I’m afraid that I’m not sure precisely what your measure is, and I think this is because you have given zero precise examples: even of its subcomponents. For example, here are two optimization problems:
1) You have to output 10 million bits. The goal is to output them so that no two consecutive bits are different.
2) You have to output 10 million bits. The goal is to output them so that when interpreted as an MP3 file, they would make a nice sounding song.
Now, the solution space for (1) consists of two possibilities (all 1s, all 0s) out of 2^10000000, for a total of 9,999,999 bits. The solution space for (2) is millions of times wider, leading to fewer bits. However, intuitively, (2) is a much harder problem and things that optimized (2) are actually doing more of the work of intelligence, after all (1) can be achieved in a few lines of code and very little time or space, while (2) takes much more of these resources.
(2) is a pretty complex problem, but can you give some specifics for (1)? Is it exactly 9,999,999 bits? If so, is this the ‘optimization power’? Is this a function of the size of the solution space and the size of the problem space only? If there was another program attempting to produce a sequence of 100 million bits coding some complex solution to a large travelling salesman problem, such that only two bitstrings suffice, would this have the same amount of optimization power?, or is it a function of the solution space itself and not just its size?
Without even a single simple example, it is impossible to narrow down your answer enough to properly critique it. So far I see it as no more precise than Legg and Hutter’s definition.
Eliezer,
I’m afraid that I’m not sure precisely what your measure is, and I think this is because you have given zero precise examples: even of its subcomponents. For example, here are two optimization problems:
1) You have to output 10 million bits. The goal is to output them so that no two consecutive bits are different.
2) You have to output 10 million bits. The goal is to output them so that when interpreted as an MP3 file, they would make a nice sounding song.
Now, the solution space for (1) consists of two possibilities (all 1s, all 0s) out of 2^10000000, for a total of 9,999,999 bits. The solution space for (2) is millions of times wider, leading to fewer bits. However, intuitively, (2) is a much harder problem and things that optimized (2) are actually doing more of the work of intelligence, after all (1) can be achieved in a few lines of code and very little time or space, while (2) takes much more of these resources.
(2) is a pretty complex problem, but can you give some specifics for (1)? Is it exactly 9,999,999 bits? If so, is this the ‘optimization power’? Is this a function of the size of the solution space and the size of the problem space only? If there was another program attempting to produce a sequence of 100 million bits coding some complex solution to a large travelling salesman problem, such that only two bitstrings suffice, would this have the same amount of optimization power?, or is it a function of the solution space itself and not just its size?
Without even a single simple example, it is impossible to narrow down your answer enough to properly critique it. So far I see it as no more precise than Legg and Hutter’s definition.