Unless you’re saying that an optimization goal must produce a well-ordering of possible environment states (which isn’t true for any definition of optimization I’ve ever heard of in an AI context)
You mean an ordering? The reals aren’t well-ordered.
If there’s no ordering, there’s circular preferences.
In any case, that’s not what I was talking about.
For example, “optimize the number of electoral votes I gain in the upcoming US presidential election”.
Compare the expected number of electoral votes with and without the optimizer. The difference gives you how powerful the optimizer is, and it will almost never be zero.
That’s equivalent to asserting that the axiom of choice is untrue. That’s not derivable from the other axioms, and in fact the axiom of choice is often used in mathematics. (This is entirely irrelevant to the [long-dead] discussion, however.)
You mean an ordering? The reals aren’t well-ordered.
Shoot, you’re right. I believe I meant a strict ordering; it’s been a while since I last studied set theory.
I’m confused as to what you mean by an optimizer now, though. It sounds like you mean something along the lines of a utility-based agent, but expected utility in this context is an attribute of a hypothesis relative to a model, not of the hypothesis relative to the world, and we’re just as free to define models as we are to define optimization objectives. Previously I’d been thinking in terms of a more general agent, which needn’t use a concept of utility and whose performance relative to an objective is found in retrospect.
Previously I’d been thinking in terms of a more general agent, which needn’t use a concept of utility and whose performance relative to an objective is found in retrospect.
It doesn’t need to use utility explicitly. It’s just whatever objective it tends to gravitate towards.
I’m not entirely sure what you’re saying in the rest of the comment.
The reason I’m talking about “expected value” is that an optimizer must be able to work in a variety of environments. This is equivalent to talking about a probability distribution of environments.
You mean an ordering? The reals aren’t well-ordered.
If there’s no ordering, there’s circular preferences.
In any case, that’s not what I was talking about.
Compare the expected number of electoral votes with and without the optimizer. The difference gives you how powerful the optimizer is, and it will almost never be zero.
That’s equivalent to asserting that the axiom of choice is untrue. That’s not derivable from the other axioms, and in fact the axiom of choice is often used in mathematics. (This is entirely irrelevant to the [long-dead] discussion, however.)
Shoot, you’re right. I believe I meant a strict ordering; it’s been a while since I last studied set theory.
I’m confused as to what you mean by an optimizer now, though. It sounds like you mean something along the lines of a utility-based agent, but expected utility in this context is an attribute of a hypothesis relative to a model, not of the hypothesis relative to the world, and we’re just as free to define models as we are to define optimization objectives. Previously I’d been thinking in terms of a more general agent, which needn’t use a concept of utility and whose performance relative to an objective is found in retrospect.
It doesn’t need to use utility explicitly. It’s just whatever objective it tends to gravitate towards.
I’m not entirely sure what you’re saying in the rest of the comment.
The reason I’m talking about “expected value” is that an optimizer must be able to work in a variety of environments. This is equivalent to talking about a probability distribution of environments.
I mean a well-ordering, though I’ll admit that was a bit unclear in context. Possible environment states are a set, not points on the real line.