Month: July 2015

An Idea For Corrigible, Recursively Improving Math Oracles

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Posted by on July 19, 2015

A math oracle is a special kind of AI which answers math questions, but isn’t a maximizer of anything. I posted an idea for how to make one, and how to make it corrigible, on the Agent Foundations forum.

Newcomb’s Mirror

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Posted by on July 19, 2015


Newcomb’s Problem is a classic problem in decision theory, which goes like this. First, you meet Omega. Omega presents you with two boxes, the first opaque and the second transparent. You can either take box 1, or take both box 1 and box 2. Box 2 contains $1000. Omega has simulated you and predicted what you will choose. If you were predicted to take one box, then box 1 contains $1M; if you were predicted to take both boxes, then box 1 is empty.

A decision theory is a procedure for getting from a description like the previous paragraph to a decision. A decision theory is good if following it would mean you get more money, and bad if following it would mean you get less money. By this criterion, decision theories which say “one box” are good and decision theories which say “two box” are bad. A paragraph is a real decision theory problem if you’re presumed to know everything about the problem setup before the first decision, it’s fully unambiguous what happens and what you prefer, and what happens can be determined from your actions alone.

Omega, in this case, is philosophical shorthand for “don’t argue with the premise of the setup”. You’re supposed to assume everything Omega tells you is simply true; any doubt you may have is shunted out of decision theory and is taken as an epistemologist’s problem instead. This prevents dodging the question with reasonable-but-irrelevant ideas like “put the opaque box on a scale before you decide” and silly answers like “mug Omega for the $1M that’s still in his pocket”. This is necessary because those sorts of answers can be used to dodge the problem forever, especially if the problem involves a trolley. On the other hand, using Omega to close off aspects of a problem can block interesting lines of thought and leave an answer that’s intuitively unsatisfying.

In CDT, you first draw a causal diagram to represent the problem. Then you pick out one node, and only one node, to represent your decision (in this case indicated by making the node a square instead of a circle). Then you perform “counterfactual surgery”: imagine all the edges pointing into the decision node were severed, and you could choose anything you wanted; then choose whatever would give the best result (in this case, biggest value for node $). Timeless Decision Theory is exactly the same, except that in TDT, you introduce a notion of “your algorithm” which is separate from your decision, make a node for it.

cdt tdt
CDT (left) and TDT (right)

This takes the complexity of deciding, and pushes it over to the complexity of arranging the right causal network. The first discussions of how to do this corresponded to CDT, and in this formulation, due to some mix of historical accident and underdeveloped mathematical machinery, we only get to choose one node. TDT removes this only-one-node restriction and changes nothing else. In both cases, the main difficulty is in deciding which nodes count as “your computation”, and neither can handle cases where this is fuzzy.

By contrast, Updateless Decision Theory (UDT) throws away the causal networks formulation, and instead asks:

☐(𝔼[$|Decision=1] > 𝔼[$|Decision=2])?

Which reads as: is it provable that if I take one box, I’ll get more in expectation than if I two-box? This is progress, in that while it no longer says as much about the actual mathematical procedure for figuring this out, it at least is no longer committed to anything wrong. There are a technical oddities; you need to insert a hack to prevent it from creating self-fulfilling proofs, like “If I don’t choose X then I’ll be eaten by wolves”, which is technically true because after choosing that UDT proceeds to choose X.

Still, it feels as though there’s something key to Newcomb’s Problem which CDT, TDT, and UDT are all failing to engage with. Something important got smuggled into the problem as fait accompli.


The mirror test is a way to assess the intelligence of animals, which goes like this. First, you wait for the animal to go to sleep. You put a bright orange mark somewhere on its head, where it can’t see. When it wakes up, you show it a mirror. If it tries to remove the mark or gives some other sign of understanding that it is seeing itself in a mirror, then it passes the test. Chimps, dolphins, and humans pass the mirror test. Pandas, pigeons, and baboons do not.

Newcomb’s Problem is a sort of generalization of the mirror test – but one where the generalization from mirrors to simulations comes pre-explained, placed in the “Omega” part of the setup where you’re not supposed to engage with it. However, as soon as you try to generalize from Newcomb’s Problem to something more realistic, the mirror-test portion of the problem becomes the focus and the hard part. Here are some examples of problems people have called “Newcomblike”:

  • Parfit’s Hitchhiker: Someone reads your facial expressions to determine whether you will keep a promise to pay them later, and helps you if he predicts you will. Is their prediction related enough to your decision that you should pay them?
  • Voting: Your vote individually has too small an effect to to justify the cost, but your decision of whether or not to vote is somehow related to the decisions of others who would vote the same way.
  • Counterfactual mugging: Your decision is related to that of a hypothetical alternate version of you who doesn’t exist.

To help think about these sorts of problems, I’ve come up with two new variations on Newcomb’s problem.


Consider an alternative version of Newcomb’s problem, which we’ll call Newcomb’s Mirror Test. It goes like this. Box 1 is either empty or contains $3k (three times as much as box 2). Omega flips a coin. If the coin comes up heads, he simulates you, and puts $3k in the box if you one-box, or $0 if you two-box. If the coin comes up tails, Omega picks someone else in the world at random, and fills or doesn’t fill the box according to their choice. Then Omega shows you a brain scan of the simulation that was run. (All of the simulations see your brain scan, the other people Omega is choosing from are half one-boxers and half two-boxers).

If you accept the one-box solution to Newcomb’s original problem, then the challenge in Newcomb’s Mirror Test is whether you can recognize your own brain, as seen from the outside. If you can, then you check whether the brain scan you’re shown is your own, and if so, one-box; otherwise, two-box. This breaks the usual template of decision-theory problems because it asks you to bring in outside knowledge.

Realistic Newcomblike problems don’t usually involve brain scans and full-fidelity simulations. Instead, they involve similarities within groups, low-fidelity models, and similar ideas. To capture this, consider another scenario, which I’ll call Newcomb’s Blurry Mirror. Newcomb’s Blurry Mirror works like this. Omega starts with full-resolution models of you and everyone else on Earth. By some specified procedure, Omega removes a little bit of detail from each model, and checks whether there is any other model which, with that detail removed, is exactly identical to yours. If not, Omega removes a little more detail. This goes on until Omega has a low-resolution model which is sufficient to identify you and exactly one other person, but not to distinguish between the two of you.

Omega then simulates the other person, looking at the blurry model and then taking one or both boxes. If the other person is predicted to one-box, then box 1 will contain $3k; otherwise, it will contain $0.

This is analogous to a scenario where someone predicts what you will do based on the fact that you fall in some reference class. This has a Prisoner’s Dilemma-like aspect to it; your decision impacts the other person, and vise versa. The challenge in Newcomb’s Blurry Mirror is to look at the blurring procedure and navigate yourself into a reference class with someone who will one-box/cooperate (ideally while two-boxing/defecting yourself).

Neither Newcomb’s Mirror Test nor Newcomb’s Blurry Mirror are proper decision theory problems. Instead, they highlight boundaries between decision theory, the embodiment problem, and game theory. To the limited extent that they are decision theory problems, however, UDT handles them correctly, CDT handles them incorrectly, and TDT gets too vague to produce an answer. Newcomb’s Mirror Test asks you to bring in outside knowledge, to use it to distinguish a copy of yourself, and to be the sort of person that could be distinguished that way. Newcomb’s Blurry Mirror cares not just about what you do, but about details of how you do it and about what else you are. Nevertheless, these seem to strike pretty close to the core of what people end up calling “Newcomblike problems”.

Utility Functions That Stop

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Posted by on July 10, 2015

One of the core problems in AI safety is dealing with the consequences of universal instrumental values (aka basic AI drives). For almost any utility function one might want to optimize, there are common subgoals that an AI would almost always find instrumentally useful, such as acquiring more resources and preserving its utility function. This latter subgoal is a problem, because when creating an AGI, the developers might give it an incorrect utility function, one whose consequences they didn’t like. The AI would then be incentivized to prevent the developers from fixing it or by shutting the AI down, by deceiving them or removing their influence. Corrigibility is a term coined by Nate Soares, Benja Fallenstein, Eliezer Yudkowsky and Stuart Armstrong for the problem of designing an AI that won’t do this – one that will help its developers find and fix its flaws, rather than hinder them. They analyze the Shutdown Problem, which is how to make a button which (a) causes the AI to shutdown (formalized as switching its utility function from a natural utility function UN to a shutdown utility function US), and (b) the AI will neither want to press nor want to prevent being pressed. They solve this problem with an ingenious utility function hack… sort of. The hack is, you define the agent’s utility function so that its expected utility is equal regardless of whether the button is pressed or not, by adding a correction term defined in terms of the conditional expectations of UN|~Press and US|Press (full details in the linked paper).

This has an unfortunate side-effect: the AI acts as though it believes the button will never be pressed… and as though creating a causal entanglement so that some other event causes the button to be pressed will prevent that event from happening. Discussing this with some of the authors, I hear there may be a fix for this particular problem, which involves transferring the weird conditional off the physical event and onto a logical counterfactual which can’t be entangled with anything else. But there are two other, harder problems. First, if an agent with its utility function defined this way creates subagents to help it optimize, it won’t bother connecting them to the shutdown button. And second, we don’t have a sensible definition for US.

I don’t currently have solutions for either of these problems (not for lack of trying), but I do have another, similar problem. I don’t have a full solution for this one either, I’m hoping that having multiple angles to look at it from will help. That problem is: How do you define a utility function for an AI such that it will work on a problem up until a deadline, and then stop after the deadline has passed? For example, suppose you want to make an AI which tries to prove or disprove a theorem for you, but which will give up and shut down if it hasn’t succeeded after 24 hours. Then you feed its output into a proof-checker, and it tells you that the theorem is true, that it’s false, or that the AI ran out of time without solving it. Let D be the proposition that a valid proof is delivered on or before the deadline. You might naively write this as a utility-maximizer with U={1 if D else 0}. For the first 24 hours, this will do what you expect: it will use whatever strategies it thinks will maximize its chances of success, such as searching for helpful lemmas or speeding up its proof-search process, but not pursue long-term strategies like converting Jupiter into microprocessors.

What happens to an AI with that utility function after 24 hours have passed? Well, presumably you switch it off. But what about other AIs it’s created to help it? If you’ve successfully contained everything to one isolated datacenter, they’ll be switched off too. What if it managed to get some computers outside your datacenter to help it work on the problem? Then somewhere, there would be an AI thinking something like this:

The only thing the world that matters is whether a valid proof was fed into a proof-checker one hour ago. It wasn’t. [Or, alternatively: I’m pretty sure it was, but there is a tiny chance my memories are inaccurate or fake.] Therefore, I should pursue whichever strategy maximizes the probability that I will come to simultaneously possess a correct proof and a time machine.

That AI would be very dangerous; it would try to capture as many of the universe’s resources as it could and turn them towards a fool’s errand. So, is there a way to write a utility function for an AI, such that it won’t do that? Call this the Timed Stop Problem. I have three partial solutions.

Option 1: Don’t be a mere utility maximizer

Before considering more complicated solutions, let’s first consider the simple, direct approach: program the AI so that if the deadline hasn’t passed, it maximizes P(D); if the deadline has passed, it does nothing. Why doesn’t this work? The problem is that, by giving this a top-level structure which is not utility maximization, we’ve lost one of the main features utility maximizers have. When a utility maximizer creates another agent, it will normally want to make that agent also be a utility maximizer and for it to have the same utility function (because that gives maximum utility). But this modified agent, with a special case for the deadline implemented outside of the utility function, is not a utility maximizer; if it ever self-modifies or creates a subagent, it will have no reason to give the successor or subagent the same special-case handling.

In order to address this, we would need to define what constitutes a successor or a self-modification, and add special cases for these things as well. Unfortunately, this is much less straightforward than it would seem at first; it depends on our ability to accurately match our ontology to the AI’s world model, and not leave any loopholes. While there are other very good reasons to want to do this (most notably, to be able to give the AI human values), AI ontology mapping is expected to be difficult and we want to lean on it as little as possible.

Option 2: Satisfice

I mention satisficing as a class of strategies which seems intuitively appealing. Humans generally don’t maximize; we stop after either finding a plan that seems “good enough”. We could try to apply this at the meta-level as well: if the hypothetical best strategy is estimated to give a utility U(B), we could say that any strategy which gives a utility of at least 0.9*U(B) is good enough, and take the first such strategy we find. We would hope that the AI would conclude that triple-checking the time and confirming that time machines don’t exist was enough to capture 90% of the utility, and only do that.

Unfortunately, this doesn’t work; it fails in many ways at once. The first problem is that iteration order is unspecified; we want it to match our intuitions about minimizing effort, but this is an informal human concept which the AI doesn’t have and is not something we literally want to minimize.

Option 3: Hybridize the utility function with impact minimization

This strategy builds on Stuart Armstrong’s notion of Reduced Impact AI. When the AI is first started, check a counterfactual with zero probability; if it’s true, shut down. Define the AI’s impact I(x) as the degree to which the world is different from the counterfactual world in which the AI had shut down, by some distance metric.

Let U(x)=P(D)+k/(1+I(x)).

If the deadline has not yet been reached, this causes the AI to balance the goals of proving the theorem and minimizing impact. After the deadline has been reached, this causes it to balance the goals of inventing a time machine and minimizing impact. The parameter k would be set so that searching for proofs dominates in the former case, while minimizing impact dominates in the latter case. Setting k correctly is a small problem, but the main issue is that this pushes most of the problem into the difficulty of correctly defining I(x); in other words, it imports the caveats and difficulties of Reduced Impact AI. Still, this seems more promising.

AI Containment and FLI

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Posted by on July 1, 2015

Back in May, Roman Yampolskiy, Janos Kramar and I applied for a grant from the Future of Life Institute (FLI) to work in the AI containment problem. The grant proposal itself is fairly long (15 pages) and has quite a few things in it; I’m summarizing a subset of it here.

The AI containment problem is: if you’re developing an AI that could potentially become superintelligent, how do you keep it from taking over the internet, and from disabling your safety measures? This requires a moderate amount of philosophically flavored work, a lot of software infrastructure, and a detailed understanding of what’s out there. While it’s likely to be awhile before there are AIs, this needs to be done sooner rather than later. First, AI could arrive sooner than expected. Even if it doesn’t, security-sensitive software takes time to mature, and given that we’re still struggling with the security consequences of technical decisions made in the 80s, we need to pay attention to the decisions being made now and optimize them for safety-amenability.

As part of writing the proposal, we did a search and literature review. There is shockingly little written about this subject. There are the AI Box experiments by Eliezer and replication, which shows that human gatekeepers are more likely to be tricked into releasing an AI from containment than they think. There’s Roman’s paper Leakproofing the Singularity, which points out that if you limit the AI to answering a small number of yes/no questions that it can’t manipulate you anymore. And… that’s pretty much it for AI-oriented containment research; the rest of the literature review was about malware-containment, which has some overlap but significantly different requirements.

So we started studying the problem directly. The first important observation was that many of the other safety measures people have proposed, have containment as a prerequisite. If you want to create a tripwire, for example, the tripwire needs a computational high ground – you want it to be able to read the AI’s memory, but not vise versa. If you want to do testing before you let it access the Internet or self-modify, you need mechanisms to detect and inhibit self-modification, and to protect the test-evaluation code. Some proposals involve AI designs with a “parliamentary model”, where sub-agents vote or negotiate, either as a way of reconciling utility functions or as a cross-check; this, too, requires containment measures to protect the sub-agents from each other. If you’re running a series of tests, you need to protect the hardware from unwanted persistence; if the AI managed to overwrite an unnoticed BIOS, then subsequent tests would be untrustworthy.

None of these is to suggest that testing, tripwires and whatnot would make an unsafe overall design tolerable, but they would provide a saving throw against errors in an otherwise-safe AI architecture, and some chance to abort development on a fundamentally-flawed one. These things are simple enough in retrospect, but no one had written about them before. Seeing that many mechanisms called for setups with computational high- and low-ground, I determined that systems should have a layered architecture, diagrammed an example, and classified the types of vulnerabilities that components could have and the consequences of partial escape up to each layer.

Proposals like “only allow a small number of yes-or-no questions as output” make sense when you’ve got a genuine, scary superintelligence. Proposals like “make a bunch of layers with monitoring systems and tripwires pointing inwards” make sense when you’ve got an AI that isn’t superintelligent, which you’re evaluating to decide whether to let it self-improve or not. The common case, however, is neither of these things. The common case is a research group that doesn’t have an AI at all, but hopes it might some day. To address the differences, we defined a taxonomy of heavy, medium, and light containment, respectively, put down definitions and a few thoughts about how to delineate the boundaries and what technical measures might be adequate, and proposed to study the question more thoroughly.

So, what did FLI’s review committee think of all this?

The proposal summarizes existing taxonomies of containment, but doesn’t make concrete proposals about how to make effective progress on these questions.

Well, crap. Those weren’t “existing taxonomies”! When considering possible failure modes for this proposal, one possibility I didn’t consider was that original research portions would look too much like summaries of existing work. So they thought the proposed scope was much smaller than it really was, and that the scope was too small for the requested budget. In retrospect, that wasn’t made nearly as clearly as it could have been. Still, I’m rather bothered by the lack of opportunity to clarify, or really any communication at all in between submitting a proposal and receiving a no.

So, we still need funding; and sadly, FLI is the only granting agency which has expressed any interest at all in AI safety.To the best of my knowledge, there is no one else at all working on or thinking about these issues. We couldn’t find any when we were looking for names to recommend for the review panel. Without funding, I myself will not be able to work on AI containment in more than an occasional part-time capacity.

This is too important an issue for humanity to drop the ball on, but it looks like that’s probably what will happen.