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Armok_GoB comments on The mathematics of reduced impact: help needed - Less Wrong
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This confuses me. Doesn't the "randomness" of quantum mechanics drown out and smooth over such effects, especially given multiple worlds where there's no hidden pseudorandom number generator that can be perpetuated in unknown ways?
I don't think so. Butterfly effects in classical universes should translate into butterfly effects over many worlds.
If we use trace distance to measure the distance between distributions outside of the box (and trace out the inside of the box) we don't seem to get a butterfly effect. But these things are a little hard to reason about so I'm not super confident (my comment above was referring to probabilities of measurements rather than entire states of affairs, as suggested in the OP, where the randomness more clearly washes out).
So today we were working on the Concreteness / Being Specific kata.
I can't visualize how "trace distance" makes this not happen.
I believe the Oracle approach may yet be recovered, even in light of this new flaw you have presented.
There are techniques to prevent sneezing and if AI researchers were educated in them then such a scenario could be avoided.
(Downvote? S/he is joking and in light of how most of these debates go it's actually pretty funny.)
I've provided two responses, which I will try to make more clear. (Trace distance is just a precise way of measuring distance between distributions; I was trying to commit to an actual mathematical claim which is either true or false, in the spirit of precision.):
My sneezing may be causally connected to the occurrence of a hurricane. However, given that I sneezed, the total probability of a hurricane occurring wasn't changed. It was still equal to the background probability of a hurricane occurring, because many other contributing factors--which have a comparable contribution to the probability of a hurricane in florida--are determined randomly. Maybe for reference it is helpful to think of the occurrence of a hurricane as an XOR of a million events, at least one of which is random. If you change one of those events it "affects" whether a hurricane occurs, but you have to exert a very special influence to make the probability of a hurricane be anything other than 50%. Even if the universe were deterministic, if we define these things with respect to a bounded agent's beliefs then we can appeal to complexity-theoretic results like Yao's XOR lemma and get identical results. If you disagree, you can specify how your mathematical model of hurricane occurrence differs substantially.
This just isn't true. In the counterfactual presented the state of the universe where there is no sneeze will result - by the very operation of phsyics - in a hurricane while the one with a sneeze will not. (Quantum Mechanics considerations change the deterministic certainty to something along the lines of "significantly more weight in resulting Everett Branches without than resulting Everett Branches with" - the principle is unchanged.)
Although this exact state of the univrse not likely to occur - and having sufficient knowledge to make the prediction in advance is even more unlikely - it is certainly a coherent example of something that could occur. As such it fulfills the role of illustrating what can happen when a small intervention results in significant influence.
You seem to be (implicitly) proposing a way of mapping uncertainty about whether there may be a hurricane and then forcing them upon the universe. This 'background probability' doesn't exist anywhere except in ignorance of what will actually occur and the same applies to 'are determined randomly'. Although things with many contributing factors can be hard to predict things just aren't 'determined randomly' - at least not according to physics we have access to. (The aforementioned caveat regarding QM and "will result in Everett Branches with weights of..." applies again.)
This is helpful for explaining where your thinking has gone astray but a red herring when it comes to think about the actual counterfactual. It is true that if the occurrence of a hurricane is an XOR of a million events then if you have zero evidence about any one of those million events then a change in another one of the events will not tell you anything about the occurrence of a hurricane. But that isn't the how the (counterf)actual universe is.
I don't quite understand your argument. Lets set aside issues about logical uncertainty, and just talk about quantum randomness for now, to make things clearer? It seems to make my case weaker. (We could also talk about the exact way in which this scheme "forces uncertainty onto the universe," by defining penalty in terms of the AI's beliefs P, at the time of deciding what disciple to produce, about future states of affairs. It seems to be precise and to have the desired functionality, though it obviously has huge problems in terms of our ability to access P and the stability of the resulting system.)
Why isn't this how the universe is? Is it the XOR model of hurricane occurrence which you are objecting to? I can do a little fourier analysis to weaken the assumption: my argument goes through as long as the occurrence of a hurricane is sufficiently sensitive to many different inputs.
Is it the supposed randomness of the inputs which you are objecting to? It is easy to see that if you have a very tiny amount of independent uncertainty about a large number of those events, then a change in another one of those events will not tell you much about the occurrence of a hurricane. (If we are dealing with logical uncertainty we need to appeal to the XOR lemma, otherwise we can just look at the distributions and do easy calculations.)
There is a unique special case in which learning about one event is informative: the case where you have nearly perfect information about nearly all of the inputs, i.e., where all of those other events do not depend on quantum randomness . As far as I can tell, this is an outlandish scenario when looking at any realistic chaotic system--there are normally astronomical numbers of independent quantum events.
Is it the difference between randomness and quantum events that you are objecting to? I suggested tracing out over the internals of the box, which intuitively means that quantum events which leave residues in the box (or dump waste heat into the box) are averaged over. Would the claim seem truer if we traced over more stuff, say everything far away from Earth, so that more quantum processes looked like randomness from the perspective of our distance measure? It doesn't look to me like it matters. (I don't see how you can make claims about quantumness and randomness being different without getting into this sort of technical detail. I agree that if we talk about complete states of affairs, then quantum mechanics is deterministic, but this is neither coherent nor what you seem to be talking about.)
I'm not going to argue further about the main point. Eliezer has failed to convince you and I know my own explanations are not nearly as clear as he can be so I don't think we would get anywhere. I'll just correct one point, which I'll concede minor in as much as it doesn't change the conclusion anyway, since the XOR business is of only tangential relevance to the question at hand.
The case where learning about one of the XORed variables is informative is not nearly perfect information about nearly all of the inputs. As a matter of plain mathematics you need any information at all about each and every one of the other variables. (And then the level of informativeness is obviously dependent on degree of knowledge, particularly the degree of knowledge with respect to those events that you know least about.)
It drops off exponentially with the number of variables about which you don't have nearly perfect information. "Not much" seems like an extremely fair description of 2^(-billion), and distinguishing between that and 0 seems pedantic unless the proposal treated 0 somehow specially.
Not arguing seems fine. It is a strange and unusually straightforward seeming thing to disagree about, and I am genuinely perplexed as to what is going on, but I don't think it matters too much or even touches on Eliezer's actual objections.
Yes. And when translated into the original counterfactual this equates to how determining how difficult it is for a superintelligence in a box to predict that the sneeze will cause a hurricane. I rather suspect that Eliezer is aware that this is a difficult task. He is probably also aware that even a perfect Bayesian would have difficulty (of the exponential kind) when it comes to predicting a hurricane from a sneeze. In fact when it comes to proof of concept counterfactuals the whole point (and a lot of the fun) is to choose extreme examples that make the point stand out in stark detail.
For those that are not comfortable dealing with counterfactuals that harness logical extremes allow me to propose a somewhat more plausible scenario - one which ensures the Oracle will have a significant chance of predicting a drastic butterfly effect to emerge from:
INPUT: Does Turing Machine 2356234534 halt?
POSSIBLE OUTPUTS: YES; NO; <NO RESPONSE>
ORACLE'S STREAM OF THOUGHT:
I, for one, would love to see continued dialogue between you and Eliezer on this topic — on that returns to Eliezer's original objections.
It's even better/worse, since we're operating on multiple worlds quantum mechanics, and many of those random events happens after the AI has stopped having an influence... If you have the AI output a bit, and then XOR it with a random bit, what bit the AI outputs has literally zero impact no matter how you count: you end up with one universe in which 1 was outputed and one in wich 0 was outputed.
... I guess this is based on the assumption that there's no difference between "universe A sees 1 and universe B sees 0" and "universe A sees 0 and universe B sees 1"... but blobs of amplitude having indexical identities like that seems like an incredibly silly notion to me.
Seems like "minimize impact" is being applied at the wrong granularity, if a large deliberate impact is required to cancel out a large incidental one. If we break open the "utility-function maximizing agent" black box, and apply the minimum-impact rule to subgoals instead of actions, it might work better. (This does, however, require an internal architecture that supports a coherent notion of "subgoal", and maintains it in spite of suboptimality through self modifications - both large cans of worms.)
What "minimum impact rule"? How is "impact" computed so that applying it to "subgoals" changes anything?