(This talk was given at a public online event on Sunday July 12th. Scott Garrabrant is responsible for the talk, Justis Mills edited the transcript.

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Scott Garrabrant: I'm going to be working in the logical induction paradigm, which means that I'm going to have this P_n thing, which assigns probabilities to logical sentences.

Basically all you need to know about it is that the probabilities that it assigns to logical sentences will be good. In particular, they'll be good on sentences that are parameterised by n, so for large n, P_nwill have good beliefs about sentences that have n as a parameter.

This will allow us to build algorithms that can use beliefs about their own outputs as part of their algorithm, because the output of a deterministic algorithm is a logical sentence.

Today I’ll present some algorithms that use self-prediction.

Here's the first one.

A_n predicts whether or not it's going to output left. If the probability to output left is less than one half, then it outputs left. Otherwise, it outputs right. It predicts what it would do, and then it does the opposite.

So for n large, it converges to randomly choosing between left and right, because if it's overdoing left then it would do right instead, and vice versa.

We can also make a biased version of this.

Here's an algorithm that, if it predicts that it outputs left with probability less than P then it outputs left, and otherwise outputs right.

The only way this algorithm can work is outputting left with probability P. In fact the previous example was a special case of this with P = ½.

We can use this general self-prediction method to basically create pseudo-randomness for algorithms. Instead of saying “flip a coin,” I can say “try to predict what you would do, then do the opposite.”

Third, here's an algorithm that's trying to do some optimization. It tries to predict whether it has a higher chance of winning if it picks left than if it picks right. If it predicts that it has a higher chance of winning with left, it picks left, and otherwise it picks right.

We're going to specifically think about this algorithm (and the following algorithms) in an example case where what it means to win is the following: if you choose left, you always win, and if you choose right, you win with probability ½. Intuitively, we’d like the algorithm in this case to choose left, since left is a guaranteed win and right isn’t.

However, it turns out this algorithm might not choose left — because if it always chooses right, it might not have well calibrated probabilities about what would happen if it chose left instead.

Here's an alternate algorithm. (We leave out W_n and assume it is defined as before.)

If this one predicts that it definitely won't go left — more precisely, if it predicts that the probability that it goes left is less than ε — then it goes left. If it predicts it definitely doesn't go right, then it goes right. And otherwise it does the thing that gives it a higher probability of winning.

To make this I took the last algorithm and I stapled onto it a process that says, "Use our randomization technique to go left every once in a while and right every once in a while." And because of this, it'll end up getting well calibrated probabilities about what would happen if it went left, even if it starts out only going right.

This algorithm, in the same game that we expressed before, will converge to going left with probability 1 - ε. The only time that it will go right is when the clause about exploration happens, when it very strongly believes that it will go left.

Now we get to the one that I'm excited about. This algorithm is very similar to the last one, but with a subtle difference.

This one still has the same exploration behaviour. But now, rather than predicting what makes winning more likely, it says: “Conditioned on my having won, what do I predict that I did?” And this gives it a distribution over actions.

This algorithm predicts what it did conditional on having won, and then copies that distribution. It just says, “output whatever I predict that I output conditioned on my having won”.

This is weird, because it feels more continuous. It feels like you’d move more gradually to just choosing the same thing all the time. If I win with probability ½, and then I apply this transform, it feels like you don't just jump all the way to always.

But it turns out that you do reach the same endpoint, because the only fixed point of this process is going to do the same as the last algorithm’s. So this algorithm turns out to be functionally the same as the previous one.

Questions

Ben Pace: Isn't the error detection requirement in algorithm 5 awful? It seems a shame that I can only have accurate beliefs if I add a weird edge case that I will pretend that I will sometimes do, even though I probably never will.

Scott Garrabrant: Yeah, it’s pretty awful. But it’s not really pretending! It will in fact do them with probability ε.

Ben Pace: But isn’t it odd to add that as an explicit rule that forces you into having accurate beliefs?

Scott Garrabrant: I agree that it's awful.

Stuart Armstrong: Do you want to say more about the problems that we're trying to avoid by pasting on all these extra bits?

Scott Garrabrant: I feel like I'm not really trying to avoid problems. I was mostly trying to give examples of ways that you can use self-prediction in algorithms. There was really only one paste-in, which is pasted on this exploration. And that feels like the best thing we have right now to deal with problems like “spurious counterfactuals”. There's just this problem where if you never try something, then you don't get good beliefs about what it's like to try it. And so then you can miss out on a lot of opportunities. Unfortunately, the strategy of always trying all options doesn't work very well either. It’s a strategy to avoid being locked out of a better option, but it has weaknesses.

Stuart Armstrong: Right, that’s all fine. I want to point out that Jan Leike also had a different way of dealing with a problem like that, that is more a probability thing than a logic thing. I think there are solutions to that issue in probability or convergence literature, particularly Thompson sampling.

Sam Eisenstat: What does the algorithm tree actually do? Doesn't that converge to going left or going right infinitely many times?

Scott Garrabrant: It depends on the logical inductor. We can't prove that it always converges to going left because you could have the logical inductor always predict that the probability of winning given that you go right is approximately one half, and the probability of winning given that you go left is approximately zero. And it never goes left, it always goes right. Because it never goes left, it never gets good feedback on what would happen if it went left.

David Manheim: Have you read Gary Drescher's book, Good and Real?

Scott Garrabrant: No, I haven't.

David Manheim: It seems like what this is dealing with is the same as what he tries to figure out when he talks about a robot that doesn't cross the street even if there are no cars coming; because since he has never crossed the street he doesn't know whether or not he gets hit if he does try to cross. So why doesn't he ever do it? The answer is, well, he has no reason not to. So you're stuck. Which means that, if we want to avoid this, as long as it's a logical counterfactual, he has to have some finite probability of crossing the street, in order to have a computable probability to figure out whether or not he should. It seems like this is addressing that with the added benefit of actually dealing with logical uncertainty.

Scott Garrabrant: Yeah. I don't think I have more of a response than that.

Daniel Filan: If I think about machine learning, you can do exploration to find good strategies. Then there are questions like “how fast do you converge?” It seems like it might be the case that algorithm 5 converges slower than algorithm 4, because you're doing this weird, opposite conditional thing. I was wondering if you have thoughts about convergence speeds, because to me they seem fundamental and important.

Scott Garrabrant: Yeah. I'm guessing that there's not going to be a meaningful difference. I mean, it's going to converge slower just because it's more complicated, and so it'll take longer for the logical inductor to have good beliefs about it. But the first clause isn't going to happen very much for small ε, and so they're basically just going to do the same thing. I'm not sure though.

One point is that, while here we're doing ε-exploration, there are ways to do better, such that the rate at which you explore is density zero, such that you explore less and less for large n, which is maybe a similar thing. I don't think there's going to be a meaningful difference in terms of convergence except for coming from the complexity of the algorithm, causing it to take longer to have good beliefs about it.