## Posts

Mapping Out Alignment 2020-08-15T01:02:31.489Z
AlexMennen's Shortform 2019-12-08T04:51:09.960Z
When wishful thinking works 2018-09-01T23:43:00.858Z
Safely and usefully spectating on AIs optimizing over toy worlds 2018-07-31T18:30:38.392Z
A comment on the IDA-AlphaGoZero metaphor; capabilities versus alignment 2018-07-11T01:03:03.305Z
Logical uncertainty and mathematical uncertainty 2018-07-01T00:33:53.000Z
Logical uncertainty and Mathematical uncertainty 2018-06-26T01:08:20.559Z
More on the Linear Utility Hypothesis and the Leverage Prior 2018-02-26T23:53:35.605Z
Value learning subproblem: learning goals of simple agents 2017-12-18T02:05:15.000Z
Against the Linear Utility Hypothesis and the Leverage Penalty 2017-12-14T18:38:53.103Z
Being legible to other agents by committing to using weaker reasoning systems 2017-12-03T07:49:13.000Z
Metamathematics and probability 2017-09-22T04:04:55.000Z
Metamathematics and Probability 2017-09-22T03:07:57.156Z
Density Zero Exploration 2017-08-17T00:43:51.000Z
Logical Induction with incomputable sequences 2017-08-17T00:39:01.000Z
Existential risk from AI without an intelligence explosion 2017-05-25T16:44:04.809Z
Modal Combat for games other than the prisoner's dilemma 2017-02-26T08:10:05.000Z
Superintelligence via whole brain emulation 2016-08-17T04:11:28.914Z
An approach to the Agent Simulates Predictor problem 2016-04-09T06:28:04.000Z
Two kinds of population ethics, and Current-Population Utilitarianism 2014-06-17T22:26:31.924Z
Selfish reasons to reject the repugnant conclusion in practice 2014-06-05T06:41:41.682Z
Prisoner's dilemma tournament results 2013-07-09T20:50:43.753Z
Prisoner's Dilemma (with visible source code) Tournament 2013-06-07T08:30:20.271Z
VNM agents and lotteries involving an infinite number of possible outcomes 2013-02-21T21:58:30.280Z
Harsanyi's Social Aggregation Theorem and what it means for CEV 2013-01-05T21:38:42.693Z
Interpersonal and intrapersonal utility comparisons 2013-01-04T03:05:36.635Z
A utility-maximizing varient of AIXI 2012-12-17T03:48:05.173Z
Meetup : Berkeley meetup: choice blindness 2012-09-25T05:31:44.630Z
Meetup : Berkeley meetup 2012-09-19T03:21:52.634Z
Meetup : Berkeley meetup: board game night 2012-09-03T17:27:47.017Z
Friendship and happiness generation 2011-11-25T20:52:18.009Z
Silicon Valley AI/Machine Learning study group 2011-09-23T17:18:02.239Z
Fiction: Letter from the End 2011-08-09T23:04:59.534Z
Assumption of positive rationality 2011-04-28T16:41:50.161Z
Creationism's effect on the progress of our understanding of evolution 2011-03-28T20:36:26.967Z
Social ethics vs decision theory 2011-02-20T04:14:54.389Z
Resolving the unexpected hanging paradox 2011-01-25T19:32:50.111Z
Could/would an FAI recreate people who are information-theoretically dead by modern standards? 2011-01-22T21:11:19.278Z
Does Evidential Decision Theory really fail Solomon's Problem? 2011-01-11T04:53:36.930Z
A Paradox in Timeless Decision Theory 2010-10-25T03:09:06.462Z
Human inability to assign numerical probabilities 2010-09-30T04:42:26.633Z

Comment by alexmennen on Superintelligence via whole brain emulation · 2020-09-21T19:58:17.190Z · LW · GW

All that is indeed possible, but not guaranteed. The reason I was speculating that better brain imaging wouldn't be especially useful for machine learning in the absence of better neuron models is that I'd assume that the optimization pressure that went into the architecture of brains was fairly heavily tailored to the specific behavior of the neurons that those brains are made of, and wouldn't be especially useful relative to other neural network design techniques that humans come up with when used with artificial neurons that behave quite differently. But sure, I shouldn't be too confident of this. In particular, the idea of training ML systems to imitate brain activation patterns, rather than copying brain architecture directly, is a possible way around this that I hadn't considered.

Comment by alexmennen on Superintelligence via whole brain emulation · 2020-09-19T23:44:27.313Z · LW · GW

No. Scanning everything and then waiting until we have a good enough neuron model might work fine; it's just that the scan wouldn't give you a brain emulation until your neuron model is good enough.

Comment by alexmennen on An overview of 11 proposals for building safe advanced AI · 2020-06-19T05:10:16.939Z · LW · GW

For individual ML models, sure, but not for classes of similar models. E.g. GPT-3 presumably was more expensive to train than GPT-2 as part of the cost to getting better results. For each of the proposals in the OP, training costs constrain how complex a model you can train, which in turn would affect performance.

Comment by alexmennen on Intuitive Lagrangian Mechanics · 2020-06-15T07:28:46.674Z · LW · GW

I'm confused about the motivation for in terms of time dilation in general relativity. I was under the impression that general relativity doesn't even have a notion of gravitational potential, so I'm not sure what this would mean. And in Newtonian physics, potential energy is only defined up to an added constant. For to represent any sort of ratio (including proper time/coordinate time), V would have to be well-defined, not just up to an arbitrary added constant.

I also had trouble figuring out the relationship between the Euler-Lagrange equation and extremizing S. The Euler-Lagrange equation looks to me like just a kind of funny way of stating Newton's second law of motion, and I don't see why it should be equivalent to extremizing action. Perhaps this would be obvious if I knew some calculus of variations?

Comment by alexmennen on Relaxed adversarial training for inner alignment · 2020-06-09T08:22:08.359Z · LW · GW

I'm concerned about Goodhart's law on the acceptability predicate causing severe problems when the acceptability predicate is used in training. Suppose we take some training procedure that would otherwise result in an unaligned AI, and modify the training procedure by also including the acceptability predicate in the loss function during training. This results the end product that has been trained to appear to satisfy the intended version of the acceptability predicate. One way that could happen is if it actually does satisfy what was intended by the acceptability predicate, which is great. But otherwise, we have made the bad behavior of the final product more difficult to detect, essentially by training the AI to be deceptively aligned.

Comment by alexmennen on An overview of 11 proposals for building safe advanced AI · 2020-06-08T06:17:38.239Z · LW · GW

Is there a difference between training competitiveness and performance competitiveness? My impression is that, for all of these proposals, however much resources you've already put into training, putting more resources into training will continue to improve performance. If this is the case, then whether a factor influencing competitiveness is framed as affecting the cost of training or as affecting the performance of the final product, either way it's just affecting the efficiency with which putting resources towards training leads to good performance. Separating competitiveness into training and performance competitiveness would make sense if there's a fixed amount of training that must be done to achieve any reasonable performance at all, but past that, more training is not effective at producing better performance. My impression is that this isn't usually what happens.

Comment by alexmennen on Scott Garrabrant's problem on recovering Brouwer as a corollary of Lawvere · 2020-05-04T18:33:53.650Z · LW · GW
Let α be the least countable ordinal such that there is no polynomial-time computable recursive well-ordering of length α.

Proof: Let be any computable well-ordering of . Let be the number of steps it takes to compute whether or not . Let (notice I'm using the standard ordering on , so this is the maximum of a finite set, and is thus well-defined). is computable in time. Let be a bijective pairing function such that both the pairing function and its inverse are computable in polynomial time. Now let be the well-ordering of given by , if is not for any , and if neither nor is of the form for any . Then is computable in polynomial time, and the order type of is plus the order type of , which is just the same as the order type of if that order type is at least .

The fact that you said you think is makes me suspect you were thinking of the least countable ordinal such that there is no recursive well-ordering of length that can be proven to be a recursive well-ordering in a natural theory of arithmetic such that, for every computable function, there's a program computing that function that the given theory can prove is total iff there's a program computing that function in polynomial time.

Comment by alexmennen on An Orthodox Case Against Utility Functions · 2020-04-20T01:41:36.830Z · LW · GW
This makes Savage a better comparison point, since the Savage axioms are more similar to the VNM framework while also trying to construct probability and utility together with one representation theorem.

Sure, I guess I just always talk about VNM instead of Savage because I never bothered to learn how Savage's version works. Perhaps I should.

As a representation theorem, this makes VNM weaker and JB stronger: VNM requires stronger assumptions (it requires that the preference structure include information about all these probability-distribution comparisons), where JB only requires preference comparison of events which the agent sees as real possibilities.

This might be true if we were idealized agents who do Bayesian updating perfectly without any computational limitations, but as it is, it seems to me that the assumption that there is a fixed prior is unreasonably demanding. People sometimes update probabilities based purely on further thought, rather than empirical evidence, and a framework in which there is a fixed prior which gets conditioned on events, and banishes discussion of any other probability distributions, would seem to have some trouble handling this.

Doesn't pointless topology allow for some distinctions which aren't meaningful in pointful topology, though?

Sure, for instance, there are many distinct locales that have no points (only one of which is the empty locale), whereas there is only one ordinary topological space with no points.

Isn't the approach you mention pretty close to JB? You're not modeling the VNM/Savage thing of arbitrary gambles; you're just assigning values (and probabilities) to events, like in JB.

Assuming you're referring to "So a similar thing here would be to treat a utility function as a function from some lattice of subsets of (the Borel subsets, for instance) to the lattice of events", no. In JB, the set of events is the domain of the utility function, and in what I said, it is the codomain.

Comment by alexmennen on An Orthodox Case Against Utility Functions · 2020-04-20T01:18:38.288Z · LW · GW
In the Savage framework, an outcome already encodes everything you care about.

Yes, but if you don't know which outcome is the true one, so you're considering a probability distribution over outcomes instead of a single outcome, then it still makes sense to speak of the probability that the true outcome has some feature. This is what I meant.

So the computation which seems to be suggested by Savage is to think of these maximally-specified outcomes, assigning them probability and utility, and then combining those to get expected utility. This seems to be very demanding: it requires imagining these very detailed scenarios.

You do not need to be able to imagine every possible outcome individually in order to think of functions on or probability distributions over the set of outcomes, any more than I need to be able to imagine each individual real number in order to understand the function or the standard normal distribution.

It seems that you're going by an analogy like Jeffrey-Bolker : VNM :: events : outcomes, which is partially right, but leaves out an important sense in which the correct analogy is Jeffrey-Bolker : VNM :: events : probability distributions, since although utility is defined on outcomes, the function that is actually evaluated is expected utility, which is defined on probability distributions (this being a distinction that does not exist in Jeffrey-Bolker, but does exist in my conception of real-world human decision making).

Comment by alexmennen on An Orthodox Case Against Utility Functions · 2020-04-12T03:06:53.639Z · LW · GW

I agree that the considerations you mentioned in your example are not changes in values, and didn't mean to imply that that sort of thing is a change in values. Instead, I just meant that such shifts in expectations are changes in probability distributions, rather than changes in events, since I think of such things in terms of how likely each of the possible outcomes are, rather than just which outcomes are possible and which are ruled out.

Comment by alexmennen on An Orthodox Case Against Utility Functions · 2020-04-10T04:55:56.801Z · LW · GW

It seems to me that the Jeffrey-Bolker framework is a poor match for what's going on in peoples' heads when they make value judgements, compared to the VNM framework. If I think about how good the consequences of an action are, I try to think about what I expect to happen if I take that action (ie the outcome), and I think about how likely that outcome is to have various properties that I care about, since I don't know exactly what the outcome will be with certainty. This isn't to say that I literally consider probability distributions in my mind, since I typically use qualitative descriptions of probability rather than numbers in [0,1], and when I do use numbers, they are very rough, but this does seem like a sort of fuzzy, computationally limited version of a probability distribution. Similarly, my estimations of how good various outcomes are are often qualitative, rather than numerical, and again this seems like a fuzzy, computationally limited version of utility function. In order to determine the utility of the event "I take action A", I need to consider how good and how likely various consequences are, and take the expectation of the 'how good' with respect to the 'how likely'. The Jeffrey-Bolker framework seems to be asking me to pretend none of that ever happened.

Comment by alexmennen on An Orthodox Case Against Utility Functions · 2020-04-09T04:09:37.498Z · LW · GW
Say I have a computer that will simulate an arbitrary Turing machine T, and will award me one utilon when that machine halts, and do nothing for me until that happens. With some clever cryptocurrency scheme, this is a scenario I could actually build today.

No, you can't do that today. You could produce a contraption that will deposit 1 BTC into a certain bitcoin wallet if and when some computer program halts, but this won't do the wallet's owner much good if they die before the program halts. If you reflect on what it means to award someone a utilon, rather than a bitcoin, I maintain that it isn't obvious that this is even possible in theory.

Why in the world would one expect a utility function over an uncountable domain to be computable?

There is a notion of computability in the continuous setting.

As far as I can see, the motivation for requiring a utility function to be computable is that this would make optimization for said utility function to be a great deal easier.

This seems like a strawman to me. A better motivation would be that agents that actually exist are computable, and a utility function is determined by judgements rendered by the agent, which is incapable of thinking uncomputable thoughts.

Comment by alexmennen on An Orthodox Case Against Utility Functions · 2020-04-09T03:09:18.120Z · LW · GW

I think we're going to have to back up a bit. Call the space of outcomes and the space of Turing machines . It sounds like you're talking about two functions, and . I was thinking of as the utility function we were talking about, but it seems you were thinking of .

You suggested should be computable but should not be. It seems to me that should certainly be computable (with the caveat that it might be a partial function, rather than a total function), as computation is the only thing Turing machines do, and that if non-halting is included in a space of outcomes (so that is total), it should be represented as some sort of limit of partial information, rather than represented explicitly, so that is continuous.

In any case, a slight generalization of Rice's theorem tells us that any computable function from Turing machines to reals that depends only of the machine's semantics must be constant, so I suppose I'm forced to agree that, if we want a utility function that is defined on all Turing machines and depends only on their semantics, then at least one of or should be uncomputable. But I guess I have to ask why we would want to assign utilities to Turing machines.

Comment by alexmennen on An Orthodox Case Against Utility Functions · 2020-04-08T23:48:48.572Z · LW · GW

It's not clear to me what this means in the context of a utility function.

Comment by alexmennen on An Orthodox Case Against Utility Functions · 2020-04-08T05:08:40.698Z · LW · GW

I'm not sure what it would mean for a real-valued function to be enumerable. You could call a function enumerable if there's a program that takes as input and enumerates the rationals that are less than , but I don't think this is what you want, since presumably if a Turing machine halting can generate a positive amount of utility that doesn't depend on the number of steps taken before halting, then it could generate a negative amount of utility by halting as well.

I think accepting the type of reasoning you give suggests that limit-computability is enough (ie there's a program that takes and produces a sequence of rationals that converges to , with no guarantees on the rate of convergence). Though I don't agree that it's obvious we should accept such utility functions as valid.

Comment by alexmennen on An Orthodox Case Against Utility Functions · 2020-04-08T02:27:47.306Z · LW · GW
we need not assume there are "worlds" at all. ... In mathematics, it brings to mind pointless topology.

I don't think the motivation for this is quite the same as the motivation for pointless topology, which is designed to mimic classical topology in a way that Jeffrey-Bolker-style decision theory does not mimic VNM-style decision theory. In pointless topology, a continuous function of locales is a function from the lattice of open sets of to the lattice of open sets of . So a similar thing here would be to treat a utility function as a function from some lattice of subsets of (the Borel subsets, for instance) to the lattice of events.

My understanding of the Jeffrey-Bolker framework is that its primary difference from the VNM framework is not its pointlessness, but the fact that it comes with a prior probability distribution over outcomes, which can only be updated by conditioning on events (i.e. updating on evidence that has probability 1 in some worlds and probability 0 in the rest). VNM does not start out with a prior, and allows any probability distribution over outcomes to be compared to any other, and Jeffrey-Bolker only allows comparison of probability distributions obtained by conditioning the prior on an event. Of course, this interpretation requires a fair amount of reading between the lines, since the Jeffrey-Bolker axioms make no explicit mention of any probability distribution, but I don't see any other reasonable way to interpret them, since if asked which of two events is better, I will often be unable to answer without further information, since the events may contain worlds of widely varying utility. Associating an event with a fixed prior conditioned on the event gives me this additional information needed to answer the question, and I don't see how any others could work. Starting with a prior that gets conditioned on events that correspond to the agent's actions seems to build in evidential decision theory as an assumption, which makes me suspicious of it.

In the Jeffrey-Bolker treatment, a world is just a maximally specific event: an event which describes everything completely. But there is no requirement that maximally-specific events exist.

This can be resolved by defining worlds to be minimal non-zero elements of the completion of the Boolean algebra of events, rather than a minimal non-zero event. This is what you seemed to be implicitly doing later with the infinite bitstrings example, where the events were clopen subsets of Cantor space (i.e. sets of infinite bitstrings such that membership in the set only depends on finitely many bits), and this Boolean algebra has no minimal non-zero elements (maximally-specific events), but the minimal non-zero elements of its completion correspond to infinite bitstrings, as desired.

Comment by alexmennen on Might humans not be the most intelligent animals? · 2019-12-24T23:26:43.753Z · LW · GW

I guess what I was trying to say is (although I think I've partially figured out what you meant; see next paragraph), cultural evolution is a process that acquires adaptations slowly-ish and transmits previously-acquired adaptations to new organisms quickly, while biological evolution is a process that acquires adaptations very slowly and transmits previously-acquired adaptations to new organisms quickly. You seem to be comparing the rate at which cultural evolution acquires adaptations to the rate at which biological evolution transmits previously-acquired adaptations to new organisms, and concluding that cultural evolution is slower.

Re-reading the part of your post where you talked about AI takeoff speeds, you argue (which I hadn't understood before) that the rise of humans was fast on an evolutionary timescale, and slow on a cultural timescale, so that if it was due to an evolutionary change, it must involve a small change that had a large effect on capabilities, so that a large change will occur very suddenly if we mimic evolution quickly, while if it was due to a cultural change, it was probably a large change, so mimicking culture quickly won't produce a large effect on capabilities unless it is extremely quick.

This clarifies things, but I don't agree with the claim. I think slow changes in the intelligence of a species is compatible with fast changes in its capabilities even if the changes are mainly in raw innovative ability rather than cultural learning. Innovations can increase ability to innovate, causing a positive feedback loop. A species could have high enough cultural learning ability for innovations to be transmitted over many generations without having the innovative ability to ever get the innovations that will kick off this loop. Then, when they start slowing gaining innovative ability, the innovations accumulated into cultural knowledge gradually increase, until they reach the feedback loop and the rate of innovation becomes more determined by changes in pre-existing innovations than by changes in raw innovative ability. There doesn't even have to be any evolutionary changes in the period in which innovation rate starts to get dramatic.

If you don't buy this story, then it's not clear why the changes being in cultural learning ability rather than in raw innovative ability would remove the need for a discontinuity. After all, our cultural learning ability went from not giving us much advantage over other animals to "accumulating decisive technological dominance in an evolutionary eyeblink" in an evolutionary eyeblink (quotation marks added for ease of parsing). Does this mean our ability to learn from culture must have greatly increased from a small change? You argue in the post that there's no clear candidate for what such a discontinuity in cultural learning ability could look like, but this seems just as true to me for raw innovative ability.

Perhaps you could argue that it doesn't matter if there's a sharp discontinuity in cultural learning ability because you can't learn from a culture faster than the culture learns things to teach you. In this case, yes, perhaps I would say that AI-driven culture could make advancements that look discontinuous on a human scale. Though I'm not entirely sure what that would look like, and I admit it does sound kind of soft-takeoffy.

Comment by alexmennen on Might humans not be the most intelligent animals? · 2019-12-24T20:46:50.182Z · LW · GW

The abilities we obtained from architectural changes to our brains also came from a slow, accumulated process, taking even longer than cultural evolution does.

Comment by alexmennen on Might humans not be the most intelligent animals? · 2019-12-24T18:58:10.860Z · LW · GW

There's more than one thing that you could mean by raw innovative capacity separate from cultural processing ability. First, you could mean someone's ability to innovate on their own without any direct help from others on the task at hand, but where they're allowed to use skills that they previously acquired from their culture. Second, you could mean someone's counterfactual ability to innovate on their own if they hadn't learned from culture. You seem to be conflating these somewhat, though mostly focusing on the second?

The second is underspecified, as you'd need to decide what counterfactual upbringing you're assuming. If you compare the cognitive performance of a human raised by bears to the cognitive performance of a bear in the same circumstances, this is unfair to the human, since the bear is raised in circumstances that it is adapted for and the human is not, just like comparing the cognitive performance of a bear raised by humans to that of a human in the same circumstances would be unfair to the bear. Though a human raised by non-humans would still make a more interesting comparison to non-human animals than Genie would, since Genie's environment is even less conducive to human development (I bet most animals wouldn't cognitively develop very well if they were kept immobilized in a locked room until maturity either).

I think this makes the second notion less interesting than the first, as there's a somewhat arbitrary dependence on the counterfactual environment. I guess the first notion is more relevant when trying to reason specifically on genetics as opposed to other factors that influence traits, but the second seems more relevant in other contexts, since it usually doesn't matter to what extent someone's abilities were determined by genetics or environmental factors.

I didn't really follow your argument for the relevance of this question to AI development. Why should raw innovation ability be more susceptible to discontinuous jumps than cultural processing ability? Until I understand the supposed relevance to AI better, it's hard for me to say which of the two notions is more relevant for this purpose.

I'd be very surprised if any existing non-human animals are ahead of humans by the first notion, and there's a clear reason in this case why performance would correlate strongly with social learning ability: social learning will have helped people in the past develop skills that they keep in the present. Even for the second notion, though it's a bit hard to say without pinning down the counterfactual more closely, I'd still expect humans to outperform all other animals in some reasonable compromise environment that helps both develop but doesn't involve them being taught things that the non-humans can't follow. I think there are still reasons to expect social learning ability and raw innovative capability to be correlated even in this sense, because higher general intelligence will help for both; original discovery and understanding things that are taught to you by others both require some of the same cognitive tools.

Comment by alexmennen on AlexMennen's Shortform · 2019-12-08T04:51:10.077Z · LW · GW

Theorem: Fuzzy beliefs (as in https://www.alignmentforum.org/posts/Ajcq9xWi2fmgn8RBJ/the-credit-assignment-problem#X6fFvAHkxCPmQYB6v ) form a continuous DCPO. (At least I'm pretty sure this is true. I've only given proof sketches so far)

The relevant definitions:

A fuzzy belief over a set is a concave function such that (where is the space of probability distributions on ). Fuzzy beliefs are partially ordered by . The inequalities reverse because we want to think of "more specific"/"less fuzzy" beliefs as "greater", and these are the functions with lower values; the most specific/least fuzzy beliefs are ordinary probability distributions, which are represented as the concave hull of the function assigning 1 to that probability distribution and 0 to all others; these should be the maximal fuzzy beliefs. Note that, because of the order-reversal, the supremum of a set of functions refers to their pointwise infimum.

A DCPO (directed-complete partial order) is a partial order in which every directed subset has a supremum.

In a DCPO, define to mean that for every directed set with , such that . A DCPO is continuous if for every , .

Lemma: Fuzzy beliefs are a DCPO.

Proof sketch: Given a directed set , is convex, and . Each of the sets in that intersection are non-empty, hence so are finite intersections of them since is directed, and hence so is the whole intersection since is compact.

Lemma: iff is contained in the interior of and for every such that , .

Proof sketch: If , then , so by compactness of and directedness of , there should be such that . Similarly, for each such that , there should be such that . By compactness, there should be some finite subset of such that any upper bound for all of them is at least .

Lemma: .

Proof: clear?

Comment by alexmennen on TurnTrout's shortform feed · 2019-11-26T23:37:25.091Z · LW · GW

The part about derivatives might have seemed a little odd. After all, you might think, is a discrete set, so what does it mean to take derivatives of functions on it. One answer to this is to just differentiate symbolically using polynomial differentiation rules. But I think a better answer is to remember that we're using a different metric than usual, and isn't discrete at all! Indeed, for any number , , so no points are isolated, and we can define differentiation of functions on in exactly the usual way with limits.

Comment by alexmennen on TurnTrout's shortform feed · 2019-11-26T23:28:49.532Z · LW · GW

The theorem: where is relatively prime to an odd prime and , is a square mod iff is a square mod and is even.

The real meat of the theorem is the case (i.e. a square mod that isn't a multiple of is also a square mod . Deriving the general case from there should be fairly straightforward, so let's focus on this special case.

Why is it true? This question has a surprising answer: Newton's method for finding roots of functions. Specifically, we want to find a root of , except in instead of .

To adapt Newton's method to work in this situation, we'll need the p-adic absolute value on : for relatively prime to . This has lots of properties that you should expect of an "absolute value": it's positive ( with only when ), multiplicative (), symmetric (), and satisfies a triangle inequality (; in fact, we get more in this case: ). Because of positivity, symmetry, and the triangle inequality, the p-adic absolute value induces a metric (in fact, ultrametric, because of the strong version of the triangle inequality) . To visualize this distance function, draw giant circles, and sort integers into circles based on their value mod . Then draw smaller circles inside each of those giant circles, and sort the integers in the big circle into the smaller circles based on their value mod . Then draw even smaller circles inside each of those, and sort based on value mod , and so on. The distance between two numbers corresponds to the size of the smallest circle encompassing both of them. Note that, in this metric, converges to .

Now on to Newton's method: if is a square mod , let be one of its square roots mod . ; that is, is somewhat close to being a root of with respect to the p-adic absolute value. , so ; that is, is steep near . This is good, because starting close to a root and the slope of the function being steep enough are things that helps Newton's method converge; in general, it might bounce around chaotically instead. Specifically, It turns out that, in this case, is exactly the right sense of being close enough to a root with steep enough slope for Newton's method to work.

Now, Newton's method says that, from , you should go to . is invertible mod , so we can do this. Now here's the kicker: , so . That is, is closer to being a root of than is. Now we can just iterate this process until we reach with , and we've found our square root of mod .

Exercise: Do the same thing with cube roots. Then with roots of arbitrary polynomials.

Comment by alexmennen on AlphaStar: Impressive for RL progress, not for AGI progress · 2019-11-02T19:51:47.196Z · LW · GW
The impressive part is getting reinforcement learning to work at all in such a vast state space

It seems to me that that is AGI progress? The real world has an even vaster state space, after all. Getting things to work in vast state spaces is a necessary pre-condition to AGI.

Comment by alexmennen on What are we assuming about utility functions? · 2019-10-03T16:54:44.047Z · LW · GW

Ok, I see what you mean about independence of irrelevant alternatives only being a real coherence condition when the probabilities are objective (or otherwise known to be equal because they come from the same source, even if there isn't an objective way of saying what their common probability is).

But I disagree that this makes VNM only applicable to settings in which all sources of uncertainty have objectively correct probabilities. As I said in my previous comment, you only need there to exist some source of objective probabilities, and you can then use preferences over lotteries involving objective probabilities and preferences over related lotteries involving other sources of uncertainty to determine what probability the agent must assign for those other sources of uncertainty.

Re: the difference between VNM and Bayesian expected utility maximization, I take it from the word "Bayesian" that the way you're supposed to choose between actions does involve first coming up with probabilities of each outcome resulting from each action, and from "expected utility maximization", that these probabilities are to be used in exactly the way the VNM theorem says they should be. Since the VNM theorem does not make any assumptions about where the probabilities came from, these still sound essentially the same, except with Bayesian expected utility maximization being framed to emphasize that you have to get the probabilities somehow first.

Comment by alexmennen on What are we assuming about utility functions? · 2019-10-03T06:10:34.388Z · LW · GW

I think you're underestimating VNM here.

only two of those four are relevant to coherence. The main problem is that the axioms relevant to coherence (acyclicity and completeness) do not say anything at all about probability

It seems to me that the independence axiom is a coherence condition, unless I misunderstand what you mean by coherence?

correctly point out problems with VNM

I'm curious what problems you have in mind, since I don't think VNM has problems that don't apply to similar coherence theorems.

VNM utility stipulates that agents have preferences over "lotteries" with known, objective probabilities of each outcome. The probabilities are assumed to be objectively known from the start. The Bayesian coherence theorems do not assume probabilities from the start; they derive probabilities from the coherence criteria, and those probabilities are specific to the agent.

One can construct lotteries with probabilities that are pretty well understood (e.g. flipping coins that we have accumulated a lot of evidence are fair), and you can restrict attention to lotteries only involving uncertainty coming from such sources. One may then get probabilities for other, less well-understood sources of uncertainty by comparing preferences involving such uncertainty to preferences involving easy-to-quantify uncertainty (e.g. if A is preferred to B, and you're indifferent between 60%A+40%B and "A if X, B if not-X", then you assign probability 60% to X. Perhaps not quite as philosophically satisfying as deriving probabilities from scratch, but this doesn't seem like a fatal flaw in VNM to me.

I do not expect agent-like systems in the wild to be pushed toward VNM expected utility maximization. I expect them to be pushed toward Bayesian expected utility maximization.

I understood those as being synonyms. What's the difference?

Comment by alexmennen on [AN #66]: Decomposing robustness into capability robustness and alignment robustness · 2019-10-02T00:58:31.657Z · LW · GW
I do, however, believe that the single step cooperate-defect game which they use to come up with their factors seems like a very simple model for what will be a very complex system of interactions. For example, AI development will take place over time, and it is likely that the same companies will continue to interact with one another. Iterated games have very different dynamics, and I hope that future work will explore how this would affect their current recommendations, and whether it would yield new approaches to incentivizing cooperation.

It may be difficult for companies to get accurate information about how careful their competitors are being about AI safety. An iterated game in which players never learn what the other players did on previous rounds is the same as a one-shot game. This points to a sixth factor that increases chance of cooperation on safety: high transparency, so that companies may verify their competitors' cooperation on safety. This is closely related to high trust.

Comment by alexmennen on A Critique of Functional Decision Theory · 2019-09-16T03:40:17.491Z · LW · GW

I object to the framing of the bomb scenario on the grounds that low probabilities of high stakes are a source of cognitive bias that trip people up for reasons having nothing to do with FDT. Consider the following decision problem: "There is a button. If you press the button, you will be given $100. Also, pressing the button has a very small (one in a trillion trillion) chance of causing you to burn to death." Most people would not touch that button. Using the same payoffs and probabilies in a scenario to challenge FDT thus exploits cognitive bias to make FDT look bad. A better scenario would be to replace the bomb with something that will fine you$1000 (and, if you want, also increase the chance of of error).

But then, it seems to me, that FDT has lost much of its initial motivation: the case for one-boxing in Newcomb’s problem didn’t seem to stem from whether the Predictor was running a simulation of me, or just using some other way to predict what I’d do.

I think the crucial difference here is how easily you can cause the predictor to be wrong. In the case where the predictor simulates you, if you two-box, then the predictor expects you to two-box. In the case where the predictor uses your nationality to predict your behavior, Scots usually one-box, and you're Scottish, if you two-box, then the predictor will still expect you to one-box because you're Scottish.

But now suppose that the pathway by which S causes there to be money in the opaque box or not is that another agent looks at S...

I didn't think that was supposed to matter at all? I haven't actually read the FDT paper, and have mostly just been operating under the assumption that FDT is basically the same as UDT, but UDT didn't build in any dependency on external agents, and I hadn't heard about any such dependency being introduced in FDT; it would surprise me if it did.

Comment by alexmennen on A Critique of Functional Decision Theory · 2019-09-16T02:15:11.538Z · LW · GW
I don't know if I'm a simulation or a real person.

A possible response to this argument is that the predictor may be able to accurately predict the agent without explicitly simulating them. A possible counter-response to this is to posit that any sufficiently accurate model of a conscious agent is necessarily conscious itself, whether the model takes the form of an explicit simulation or not.

Comment by alexmennen on Troll Bridge · 2019-09-15T16:34:59.969Z · LW · GW

I think the counterfactuals used by the agent are the correct counterfactuals for someone else to use while reasoning about the agent from the outside, but not the correct counterfactuals for the agent to use while deciding what to do. After all, knowing the agent's source code, if you see it start to cross the bridge, it is correct to infer that it's reasoning is inconsistent, and you should expect to see the troll blow up the bridge. But while deciding what to do, the agent should be able to reason about purely causal effects of its counterfactual behavior, screening out other logical implications.

Also, counterfactuals which predict that the bridge blows up seem to be saying that the agent can control whether PA is consistent or inconsistent.

Disagree that that's what's happening. The link between the consistency of the reasoning system and the behavior of the agent is because the consistency of the reasoning system controls the agent's behavior, rather than the other way around. Since the agent is selecting outcomes based on their consequences, it does make sense to speak of the agent choosing actions to some extent, but I think speaking of logical implications of the agent's actions on the consistency of formal systems as "controlling" the consistency of the formal system seems like an inappropriate attribution of agency to me.

Comment by alexmennen on A Primer on Matrix Calculus, Part 2: Jacobians and other fun · 2019-08-19T16:44:00.348Z · LW · GW
I suppose why that's not why we're minimizing determinant, but rather frobenius norm.

Yes, although another reason is that the determinant is only defined if the input and output spaces have the same dimension, which they typically don't.

Comment by alexmennen on A Primer on Matrix Calculus, Part 3: The Chain Rule · 2019-08-18T19:46:36.750Z · LW · GW
First, a vector can be seen as a list of numbers, and a matrix can be seen as an ordered list of vectors. An ordered list of matrices is... a tensor of order 3. Well not exactly. Apparently some people are actually disappointed with the term tensor because a tensor means something very specific in mathematics already and isn't just an ordered list of matrices. But whatever, that's the term we're using for this blog post at least.

It's true that tensors are something more specific than multidimensional arrays of numbers, but Jacobians of functions between tensor spaces (that being what you're using the multidimensional arrays for here) are, in fact, tensors.

Comment by alexmennen on A Primer on Matrix Calculus, Part 2: Jacobians and other fun · 2019-08-18T04:58:29.491Z · LW · GW
What this means is for the Jacobian is that the determinant tells us how much space is being squished or expanded in the neighborhood around a point. If the output space is being expanded a lot at some input point, then this means that the neural network is a bit unstable at that region, since minor alterations in the input could cause huge distortions in the output. By contrast, if the determinant is small, then some small change to the input will hardly make a difference to the output.

This isn't quite true; the determinant being small is consistent with small changes in input making arbitrarily large changes in output, just so long as small changes in input in a different direction make sufficiently small changes in output.

The frobenius norm is nothing complicated, and is really just a way of describing that we square all of the elements in the matrix, take the sum, and then take the square root of this sum.

An alternative definition of the frobenius norm better highlights its connection to the motivation of regularizing the Jacobian frobenius in terms of limiting the extent to which small changes in input can cause large changes in output: The frobenius norm of a matrix J is the root-mean-square of |J(x)| over all unit vectors x.

Comment by alexmennen on If physics is many-worlds, does ethics matter? · 2019-07-10T17:37:58.100Z · LW · GW

"Controlling which Everett branch you end up in" is the wrong way to think about decisions, even if many-worlds is true. Brains don't appear to rely much on quantum randomness, so if you make a certain decision, that probably means that the overwhelming majority of identical copies of you make the same decision. You aren't controlling which copy you are; you're controlling what all of the copies do. And even if quantum randomness does end of mattering in decisions, so that a non-trivial proportion of copies of you make different decisions from each other, then you would still presumably want a high proportion of them to make good decisions; you can do your part to bring that about by making good decisions yourself.

Comment by alexmennen on If physics is many-worlds, does ethics matter? · 2019-07-10T17:24:44.198Z · LW · GW
Consider reading a real physicist's take on the issue

This seems phrased to suggest that her view is "the real physicist view" on the multiverse. You could also read what Max Tegmark or David Deutsch, for instance, have to say about multiverse hypotheses and get a "real physicist's" view from them.

Also, she doesn't actually say much in that blog post. She points out that when she says that multiverse hypotheses are unscientific, she doesn't mean that they're false, so this doesn't seem especially useful to someone who wants to know whether there actually is a multiverse, or is interested in the consequences thereof. She says "there is no reason to think we live in such multiverses to begin with", but proponents of multiverse hypotheses have given reasons to support their views, which she doesn't address.

Comment by alexmennen on Von Neumann’s critique of automata theory and logic in computer science · 2019-05-29T01:17:38.188Z · LW · GW

#1 (at the end) sounds like complexity theory.

Some of what von Neumann says makes it sound like he's interested in a mathematical foundation for analog computing, which I think has been done by now.

Comment by alexmennen on And My Axiom! Insights from 'Computability and Logic' · 2019-01-17T02:11:03.435Z · LW · GW
On several occasions, the authors emphasize how the intuitive nature of "effective computability" renders futile any attempt to formalize the thesis. However, I'm rather interested in formalizing intuitive concepts and therefore wondered why this hasn't been attempted.

Formalizing the intuitive notion of effective computability was exactly what Turing was trying to do when he introduced Turing machines, and Turing's thesis claims that his attempt was successful. If you come up with a new formalization of effective computability and prove it equivalent to Turing computability, then in order to use this as a proof of Turing's thesis, you would need to argue that your new formalization is correct. But such an argument would inevitably be informal, since it links a formal concept to an informal concept, and there already have been informal arguments for Turing's thesis, so I don't think there is anything really fundamental to be gained from this.

Comment by alexmennen on And My Axiom! Insights from 'Computability and Logic' · 2019-01-17T01:57:56.471Z · LW · GW

Consider the halting set; ... is not enumerable / computable.
...
Here, we should be careful with how we interpret "information". After all, coNP-complete problems are trivially Cook reducible to their NP-complete counterparts (e.g., query the oracle and then negate the output), but many believe that there isn't a corresponding Karp reduction (where we do a polynomial amount of computation before querying the oracle and returning its answer). Since we aren't considering complexity but instead whether it's enumerable at all, complementation is fine.

You're using the word "enumerable" in a nonstandard way here, which might indicate that you've missed something (and if not, then perhaps at least this will be useful for someone else reading this). "Enumerable" is not usually used as a synonym for computable. A set is computable if there is a program that determines whether or not its input is in the set. But a set is enumerable if there is a program that halts if its input is in the set, and does not halt otherwise. Every computable set is enumerable (since you can just use the output of the computation to decide whether or not to halt). But the halting set is an example of a set that is enumerable but not computable (it is enumerable because you can just run the program coded by your input, and halt if/when it halts). Enumerable sets are not closed under complementation; in fact, an enumerable set whose complement is enumerable is computable (because you can run the programs for the set and its complement in parallel on the same input; eventually one of them will halt, which will tell you whether or not the input is in the set).

The distinction between Cook and Karp reductions remains meaningful when "polynomial-time" is replaced by "Turing computable" in the definitions. Any set that an enumerable set is Turing-Karp reducible to is also enumerable, but an enumerable set is Turing-Cook reducible to its complement.

The reason "enumerable" is used for this concept is that a set is enumerable iff there is a program computing a sequence that enumerates every element of the set. Given a program that halts on exactly the elements of a given set, you can construct an enumeration of the set by running your program on every input in parallel, and adding an element to the end of your sequence whenever the program halts on that input. Conversely, given an enumeration of a set, you can construct a program that halts on elements of the set by going through the sequence and halting whenever you find your input.

Comment by alexmennen on An Extensive Categorisation of Infinite Paradoxes · 2018-12-18T07:33:58.989Z · LW · GW

I don't follow the analogy to 1/x being a partial function that you're getting at.

Maybe a better way to explain what I'm getting at is that it's really the same issue that I pointed out for the two-envelopes problem, where you know the amount of money in each envelope is finite, but the uniform distribution up to an infinite surreal would suggest that the probability that the amount of money is finite is infinitesimal. Suppose you say that the size of the ray is an infinite surreal number . The size of the portion of this ray that is distance at least from is when is a positive real, so presumably you would also want this to be so for surreal . But using, say, , every point in is within distance of , but this rule would say that the measure of the portion of the ray that is farther than from is ; that is, almost all of the measure of is concentrated on the empty set.

Comment by alexmennen on An Extensive Categorisation of Infinite Paradoxes · 2018-12-16T19:29:34.761Z · LW · GW

The latter. It doesn't even make sense to speak of maximizing the expectation of an unbounded utility function, because unbounded functions don't even have expectations with respect to all probability distributions.

There is a way out of this that you could take, which is to only insist that the utility function has to have an expectation with respect to probability distributions in some restricted class, if you know your options are all going to be from that restricted class. I don't find this very satisfying, but it works. And it offers its own solution to Pascal's mugging, by insisting that any outcome whose utility is on the scale of 3^^^3 has prior probability on the scale of 1/(3^^^3) or lower.

Comment by alexmennen on An Extensive Categorisation of Infinite Paradoxes · 2018-12-16T19:18:05.341Z · LW · GW

It's a bad bullet to bite. Its symmetries are essential to what makes Euclidean space interesting.

And here's another one: are you not bothered by the lack of countable additivity? Suppose you say that the volume of Euclidean space is some surreal number . Euclidean space is the union of an increasing sequence of balls. The volumes of these balls are all finite, in particular, less than , so how can you justify saying that their union has volume greater than ?

Comment by alexmennen on An Extensive Categorisation of Infinite Paradoxes · 2018-12-15T19:46:04.097Z · LW · GW

Why? Plain sequences are a perfectly natural object of study. I'll echo gjm's criticism that you seem to be trying to "resolve" paradoxes by changing the definitions of the words people use so that they refer to unnatural concepts that have been gerrymandered to fit your solution, while refusing to talk about the natural concepts that people actually care about.

I don't think think your proposal is a good one for indexed sequences either. It is pretty weird that shifting the indices of your sequence over by 1 could change the size of the sequence.

Comment by alexmennen on An Extensive Categorisation of Infinite Paradoxes · 2018-12-15T19:04:55.957Z · LW · GW

What about rotations, and the fact that we're talking about destroying a bunch of symmetry of the plane?

Comment by alexmennen on An Extensive Categorisation of Infinite Paradoxes · 2018-12-15T17:11:04.375Z · LW · GW

There are measurable sets whose volumes will not be preserved if you try to measure them with surreal numbers. For example, consider . Say its measure is some infinite surreal number . The volume-preserving left-shift operation sends to , which has measure , since has measure . You can do essentially the same thing in higher dimensions, and the shift operation in two dimensions () can be expressed as the composition of two rotations, so rotations can't be volume-preserving either. And since different rotations will have to fail to preserve volumes in different ways, this will break symmetries of the plane.

I wouldn't say that volume-preserving transformations fail to preserve volume on non-measurable sets, just that non-measurable sets don't even have measures that could be preserved or not preserved. Failing to preserve measures of sets that you have assigned measures to is entirely different. Non-measurable sets also don't arise in mathematical practice; half-spaces do. I'm also skeptical of the existence of non-measurable sets, but the non-existence of non-measurable sets is a far bolder claim than anything else I've said here.

Comment by alexmennen on An Extensive Categorisation of Infinite Paradoxes · 2018-12-15T02:05:38.665Z · LW · GW
Indeed Pascal's Mugging type issues are already present with the more standard infinities.

Right, infinity of any kind (surreal or otherwise) doesn't belong in decision theory.

"Surreal numbers are not the right tool for measuring the volume of Euclidean space or the duration of forever" - why?

How would you? If you do something like taking an increasing sequence of bounded subsets that fill up the space you're trying to measure, find a formula f(n) for the volume of the nth subset, and plug in , the result will be highly dependent on which increasing sequence of bounded subsets you use. Did you have a different proposal? It's sort of hard to explain why no method for measuring volumes using surreal numbers can possibly work well, though I am confident it is true. At the very least, volume-preserving transformations like shifting everything 1 meter to the left or rotating everything around some axis cease to be volume-preserving, though I don't know if you'd find this convincing.

Comment by alexmennen on An Extensive Categorisation of Infinite Paradoxes · 2018-12-15T01:39:51.969Z · LW · GW
You want to conceive of this problem as "a sequence whose order-type is ω", but from the surreal perspective this lacks resolution. Is the number of elements (surreal) ω, ω+1 or ω+1000? All of these are possible given that in the ordinals 1+ω=ω so we can add arbitrarily many numbers to the start of a sequence without changing its order type.

It seems to me that measuring the lengths of sequences with surreals rather than ordinals is introducing fake resolution that shouldn't be there. If you start with an infinite constant sequence 1,1,1,1,1,1,..., and tell me the sequence has size , and then you add another 1 to the beginning to get 1,1,1,1,1,1,1,..., and you tell me the new sequence has size , I'll be like "uh, but those are the same sequence, though. How can they have different sizes?"

Comment by alexmennen on An Extensive Categorisation of Infinite Paradoxes · 2018-12-14T20:50:11.702Z · LW · GW

Surreal numbers are useless for all of these paradoxes.

Infinitarian paralysis: Using surreal-valued utilities creates more infinitarian paralysis than it solves, I think. You'll never take an opportunity to increase utility by because it will always have higher expected utility to focus all of your attention on trying to find ways to increase utility by , since there's some (however small) probability that such efforts would succeed, so the expected utility of focusing your efforts on looking for ways to increase utility by will have expected utility , which is higher than . I think a better solution would be to note that for any person, a nonzero fraction of people are close enough to identical to that person that they will make the same decisions, so any decision that anyone makes affects a nonzero fraction of people. Measure theory is probably a better framework than surreal numbers for formalizing what is meant by "fraction" here.

Paradox of the gods: The introduction of surreal numbers solves nothing. Why wouldn't he be able to advance more than miles if no gods erect any barriers until he advances miles for some finite ?

Two-envelopes paradox: it doesn't make sense to model your uncertainty over how much money is in the first envelope with a uniform surreal-valued probability distribution on for an infinite surreal , because then the probability that there is a finite amount of money in the envelope is infinitesimal, but we're trying to model the situation in which we know there's a finite amount of money in the envelope and just have no idea which finite amount.

Sphere of suffering: Surreal numbers are not the right tool for measuring the volume of Euclidean space or the duration of forever.

Hilbert hotel: As you mentioned, using surreals in the way you propose changes the problem.

Trumped, Trouble in St. Petersburg, Soccer teams, Can God choose an integer at random?, The Headache: Using surreals in the way you propose in each of these changes the problems in exactly the same way it does for the Hilbert hotel.

St. Petersburg paradox: If you pay infinity dollars to play the game, then you lose infinity dollars with probability 1. Doesn't sound like a great deal.

Banach-Tarski Paradox: The free group only consists of sequences of finite length.

The Magic Dartboard: First, a nitpick: that proof relies on the continuum hypothesis, which is independent of ZFC. Aside from that, the proof is correct, which means any resolution along the lines you're imagining that imply that no magic dartboards exist is going to imply that the continuum hypothesis is false. Worse, the fact that for any countable ordinal, there are countably many smaller countable ordinals and uncountably many larger countable ordinals follows from very minimal mathematical assumptions, and is often used in descriptive set theory without bringing in the continuum hypothesis at all, so if you start trying to change math to make sense of "the second half of the countable ordinals", you're going to have a bad time.

Parity paradoxes: The lengths of the sequences involved here are the ordinal , not a surreal number. You might object that there is also a surreal number called , but this is different from the ordinal . Arithmetic operations act differently on ordinals than they do on the copies of those ordinals in the surreal numbers, so there's no reasonable sense in which the surreals contain the ordinals. Example: if you add another element to the beginning of either sequence (i.e. flip the switch at , or add a at the beginning of the sum, respectively), then you've added one thing, so the surreal number should increase by , but the order-type is unchanged, so the ordinal remains the same.

Comment by alexmennen on Safely and usefully spectating on AIs optimizing over toy worlds · 2018-08-21T21:11:56.038Z · LW · GW

The agent could be programmed to have a certain hard-coded ontology rather than searching through all possible hypotheses weighted by description length.

Comment by alexmennen on Safely and usefully spectating on AIs optimizing over toy worlds · 2018-08-15T20:53:35.731Z · LW · GW

I haven't heard the term "platonic goals" before. There's been plenty written on capability control before, but I don't know of anything written before on the strategy I described in this post (although it's entirely possible that there's been previous writing on the topic that I'm not aware of).

Comment by alexmennen on Safely and usefully spectating on AIs optimizing over toy worlds · 2018-08-09T00:17:28.157Z · LW · GW

Are you worried about leaks from the abstract computational process into the real world, leaks from the real world into the abstract computational process, or both? (Or maybe neither and I'm misunderstanding your concern?)

There will definitely be tons of leaks from the abstract computational process into the real world; just looking at the result is already such a leak. The point is that the AI should have no incentive to optimize such leaks, not that the leaks don't exist, so the existence of additional leaks that we didn't know about shouldn't be concerning.

Leaks from the outside world into the computational abstraction would be more concerning, since the whole point is to prevent those from existing. It seems like it should be possible to make hardware arbitrarily reliable by devoting enough resources to error detection and correction, which would prevent such leaks, though I'm not an expert, so it would be good to know if this is wrong. There may be other ways to get the AI to act similarly to the way it would in the idealized toy world even when hardware errors create small differences. This is certainly the sort of thing we would want to take seriously if hardware can't be made arbitrarily reliable.

Incidentally, that story about accidental creation of a radio with an evolutionary algorithm was part of what motivated my post in the first place. If the evolutionary algorithm had used tests of its oscillator design in a computer model, rather than in the real world, then it would have have built a radio receiver, since radio signals from nearby computers would not have been included in the computer model of the environment, even though they were present in the actual environment.

Comment by alexmennen on Probabilistic Tiling (Preliminary Attempt) · 2018-08-08T00:17:44.711Z · LW · GW
What I meant was that the computation isn't extremely long in the sense of description length, not in the sense of computation time. Also, we aren't doing policy search over the set of all turing machines, we're doing policy search over some smaller set of policies that can be guaranteed to halt in a reasonable time (and more can be added as time goes on)

Wouldn't the set of all action sequences have lower description length than some large finite set of policies? There's also the potential problem that all of the policies in the large finite set you're searching over could be quite far from optimal.