Exercise 1:
The empty set is the only one. For any nonempty set X, you could pick as a counterexample:
Exercise 2:
The agent will choose an option which scores better than the threshold .
It's a generalization of satisficers, these latter are thresholders such as is nonempty.
Exercise 3:
Exercise 4:
I have discovered a truly marvelous-but-infinite solution of this, which this finite comment is too narrow to contain.
Exercise 5:
The generalisable optimisers are the following:
i.e. argmin will choose the minimal points.
i.e. satisfice will choose an option which dominates some fixed anchor point . Note that since R is only equipped with a preorder, it means it might be a more selective optimiser (if not total, it's harder to get an option better then the anchor). More interestingly, if there are indifferent options with the anchor (some x / and ), it could choose it rather than the anchor even if there is no gain to do so. This degree of freedom could be eventually not desirable.
Exercise 6:
Interesting problem.
First of all, is there a way to generalize the trick?
The first idea would be to try to find some equivalent to the destested element . For context-dependant optimiser such as better-than-average, there isn't.
A more general way to try to generalize the trick would be the following question:
For some fixed , and , could we find some other such as and
i.e. is there a general way to replace values outside of without modify the result of the constrained optimisation?
Answer: no. Counter-example: The optimiser for some infinite set X and finite nonempty sets and R.
So it seems there is no general trick here. But why bother? We should refocus on what we mean by constrained optimisation in general, and it has nothing to do with looking for some u'. What we mean is value outside are totally irrelevant.
How? For any of the current example we have here, what we actually want is not , but : we only apply the optimiser on the legal set.
Problem: in the current formalism, an optimiser has type , so I don't see obvious way to define the "same" optimiser on a different X. and the others here are implicitly parametrized, so it's not that a problem, but we have to specify this. This suggests to look for categories (e.g. for argmax...).
Even with finite sets, it doesn't work because the idea to look for "closest to " is not what we looking for.
Let a class of students, scoring within a range , . Let the (uniform) better-than-average optimizer, standing for the professor picking any student who scores better than the mean.
Let (the professor despises Charlie and ignores him totally).
If u(Alice) = 5 and u(Bob) = 4, their average is 4.5 so only Alice should be picked by the constrained optimisation.
Howewer, with your proposal, you run into trouble with u' defined as u'(Alice) = u(Alice), u'(Bob) = u(Bob), and u'(Charlie) = 0.
The average value for this u' is , and both Alice and Bob scores better than 3: . The size of the intersection with is then maximal, so your proposal suggests to pick this set as the result. But the actual result should be , because the score of Charlie is irrelevant to constrained optimisation.
Natural language is lossy because the communication channel is narrow, hence the need for lower-dimensional representation (see ML embeddings) of what we're trying to convey. Lossy representations is also what Abstractions are about.
But in practice, you expect Natural Abstractions (if discovered) cannot be expressed in natural language?
There is one catch: in principle, there could be multiple codes/descriptions which decode to the same message. The obvious thing to do is then to add up the implied probabilities of each description which produces the same message. That indeed works great. However, it turns out that just taking the minimum description length - i.e. the length of the shortest code/description which produces the message - is a good-enough approximation of that sum in one particularly powerful class of codes: universal Turing machines.
Is this about K-complexity is silly; use cross-entropy instead?
Do we really have such good interpretations for such examples? It seems to me that we have big problems in the real world because we don't.
We do have very high-level interpretations, but not enough to have solid guarantees. After all, we have a very high-level trivial interpretation of our ML models: they learn! The challenge is not just to have clues, but clues that are relevant enough to address safety concerns in relation to impact scale (which is the unprecedented feature of the AI field).
Pretty cool!
Just to add, although I think you already know: we don't need to have a reflexive understanding of your DT to put it into practice, because messy brains rather than provable algo etc....
And I always feel it's kinda unfair to dismiss as orthogonal motivations "valuing friendliness or a sense of honor" because they might be evolutionarily selected heuristics to (sort of) implement such acausal DT concerns!
Great! Isn't it generalizable to any argmin/argmax issues? Especialy thinking about the argmax decision theories framework, which is a well-known difficulty for safety concerns.
Similarly, in EA/action-oriented discussions, there is a reccurent pattern like:
Eager-to-act padawan: If world model/moral theory X is best likely to be true (due to evidence y z...), we need to act accordingly with the controversial Z! Seems best EU action!
Experienced jedi: Wait for a minute. You have to be careful with this way of thinking, because there are unkwown unknown, unilateralist curse and so on. A better decision-making procedure is to listen several models, severals moral theories, and to look for strategies acceptable by most of them.
It's fixed ;)