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I certainly don't think SGD is a powerful enough optimization process to do science directly, but it definitely seems powerful enough to find an agent which does do science.

A policy gradient network, on the other hand, could maybe (after having its parameters updated through gradient descent) become a network that, in a single forward pass, considers diverse world-models (generated with a messy non-local heuristic), and analyzes their plausibility, and then acts. At the end of the day, what we have is an agent modeling world, and we can expect it to consider any model that a human could come up with. (This paragraph also applies to the DQN with a gradient-descent-trained method for selectively attending to different parts of a wide network, since that could amount to effectively considering different models).

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49 Formal Solution to the Inner Alignment Problem

by michaelcohen

18th Feb 2021

AI Alignment Forum

2 min read

123

49 Ω 19

We've written a paper on online imitation learning, and our construction allows us to bound the extent to which mesa-optimizers could accomplish anything. This is not to say it will definitely be easy to eliminate mesa-optimizers in practice, but investigations into how to do so could look here as a starting point. The way to avoid outputting predictions that may have been corrupted by a mesa-optimizer is to ask for help when plausible stochastic models disagree about probabilities.

Here is the abstract:

In imitation learning, imitators and demonstrators are policies for picking actions given past interactions with the environment. If we run an imitator, we probably want events to unfold similarly to the way they would have if the demonstrator had been acting the whole time. No existing work provides formal guidance in how this might be accomplished, instead restricting focus to environments that restart, making learning unusually easy, and conveniently limiting the significance of any mistake. We address a fully general setting, in which the (stochastic) environment and demonstrator never reset, not even for training purposes. Our new conservative Bayesian imitation learner underestimates the probabilities of each available action, and queries for more data with the remaining probability. Our main result: if an event would have been unlikely had the demonstrator acted the whole time, that event's likelihood can be bounded above when running the (initially totally ignorant) imitator instead. Meanwhile, queries to the demonstrator rapidly diminish in frequency.

The second-last sentence refers to the bound on what a mesa-optimizer could accomplish. We assume a realizable setting (positive prior weight on the true demonstrator-model). There are none of the usual embedding problems here—the imitator can just be bigger than the demonstrator that it's modeling.

(As a side note, even if the imitator had to model the whole world, it wouldn't be a big problem theoretically. If the walls of the computer don't in fact break during the operation of the agent, then "the actual world" and "the actual world outside the computer conditioned on the walls of the computer not breaking" both have equal claim to being "the true world-model", in the formal sense that is relevant to a Bayesian agent. And the latter formulation doesn't require the agent to fit inside world that it's modeling).

Almost no mathematical background is required to follow [Edit: most of ] the proofs. [Edit: But there is a bit of jargon. "Measure" means "probability distribution", and "semimeasure" is a probability distribution that sums to less than one.] We feel our bounds could be made much tighter, and we'd love help investigating that.

These slides (pdf here) are fairly self-contained and a quicker read than the paper itself.

Below, and $P_{im}$ refer to the probability of the event supposing the demonstrator or imitator were acting the entire time. The limit below refers to successively more unlikely events $B$ ; it's not a limit over time. Imagine a sequence of events $B_{i}$ such that ${lim}_{i \to \infty} P_{d e m} (B_{i}) = 0$ .

Academic PapersMesa-OptimizationAI RiskConservatism (AI)Inner AlignmentWorld Modeling TechniquesAI

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1682021 AI Alignment Literature Review and Charity Comparison

114Infra-Bayesian physicalism: a formal theory of naturalized induction

103My take on Vanessa Kosoy's take on AGI safety

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[-]paulfchristiano4yΩ15200

I haven't read the paper yet, looking forward to it. Using something along these lines to run a sufficiently-faithful simulation of HCH seems like a plausible path to producing an aligned AI with a halting oracle. (I don't think that even solves the problem given a halting oracle, since HCH is probably not aligned, but I still think this would be noteworthy.)

First I'm curious to understand this main result so I know what to look for and how surprised to be. In particular, I have two questions about the quantitative behavior described here:

if an event would have been unlikely had the demonstrator acted the whole time, that event's likelihood can be bounded above when running the (initially totally ignorant) imitator instead. Meanwhile, queries to the demonstrator rapidly diminish in frequency.

1: Dependence on the prior of the true hypothesis

It seems like you have the following tough case even if the human is deterministic:

There are N hypotheses about the human, of which one is correct and the others are bad.
I would like to make a series of N potentially-catastrophic decisions.
On decision k, all (N-1) bad hypotheses propose the same catastrophic prediction. We don't know k in advance

... (read more)

[-]Vanessa Kosoy4yΩ8100

Here's the sketch of a solution to the query complexity problem.

Simplifying technical assumptions:

The action space is ${0, 1}$
All hypotheses are deterministic
Predictors output maximum likelihood predictions instead of sampling the posterior

I'm pretty sure removing those is mostly just a technical complication.

Safety assumptions:

The real hypothesis has prior probability lower bounded by some known quantity $δ$ , so we discard all hypotheses of probability less than $δ$ from the onset.
Malign hypotheses have total prior probability mass upper bounded by some known quantity $ϵ$ (determining parameters $δ$ and $ϵ$ is not easy, but I'm pretty sure any alignment protocol will have parameters of some such sort.)
Any predictor that uses a subset of the prior without malign hypotheses is safe (I would like some formal justification for this, but seems plausible on the face of it.)
Given any safe behavior, querying the user instead of producing a prediction cannot make it unsafe (this can and should be questioned but let it slide for now.)

Algorithm: On each round, let $p_{0}$ be the prior probability mass of unfalsified hypotheses predicting $0$ and $p_{1}$ be the same for $1$ .

If $p_{0} > p_{1} + ϵ$

... (read more)

8paulfchristiano4y

I agree that this settles the query complexity question for Bayesian predictors and deterministic humans. I expect it can be generalized to have complexity O(ϵ2δ2) in the case with stochastic humans where treacherous behavior can take the form of small stochastic shifts. I think that the big open problems for this kind of approach to inner alignment are: * Is ϵδ bounded? I assign significant probability to it being 2100 or more, as mentioned in the other thread between me and Michael Cohen, in which case we'd have trouble. (I believe this is also Eliezer's view.) * It feels like searching for a neural network is analogous to searching for a MAP estimate, and that more generally efficient algorithms are likely to just run one hypothesis most of the time rather than running them all. Then this algorithm increases the cost of inference by ϵδ which could be a big problem. * As you mention, is it safe to wait and defer or are we likely to have a correlated failure in which all the aligned systems block simultaneously? (e.g. as described here)

4Vanessa Kosoy4y

Yes, you're right. A malign simulation hypothesis can be a very powerful explanation to the AI for the why it found itself at a point suitable for this attack, thereby compressing the "bridge rules" by a lot. I believe you argued as much in your previous writing, but I managed to confuse myself about this. Here's the sketch of a proposal how to solve this. Let's construct our prior to be the convolution of a simplicity prior with a computational easiness prior. As an illustration, we can imagine a prior that's sampled as follows: * First, sample a hypothesis h from the Solomonoff prior * Second, choose a number n according to some simple distribution with high expected value (e.g. n−1−α) with α≪1 * Third, sample a DFA A with n states and a uniformly random transition table * Fourth, apply A to the output of h We think of the simplicity prior as choosing "physics" (which we expect to have low description complexity but possibly high computational complexity) and the easiness prior as choosing "bridge rules" (which we expect to have low computational complexity but possibly high description complexity). Ofc this convolution can be regarded as another sort of simplicity prior, so it differs from the original simplicity prior merely by a factor of O(1), however the source of our trouble is also "merely" a factor of O(1). Now the simulation hypothesis no longer has an advantage via the bridge rules, since the bridge rules have a large constant budget allocated to them anyway. I think it should be possible to make this into some kind of theorem (two agents with this prior in the same universe that have access to roughly the same information should have similar posteriors, in the α→0 limit).

8paulfchristiano4y

I broadly think of this approach as "try to write down the 'right' universal prior." I don't think the bridge rules / importance-weighting consideration is the only way in which our universal prior is predictably bad. There are also issues like anthropic update and philosophical considerations about what kind of "programming language" to use and so on. I'm kind of scared of this approach because I feel unless you really nail everything there is going to be a gap that an attacker can exploit. I guess you just need to get close enough that εδ is manageable but I think I still find it scary (and don't totally remember all my sources of concern). I think of this in contrast with my approach based on epistemic competitiveness approach, where the idea is not necessarily to identify these considerations in advance, but to be epistemically competitive with an attacker (inside one of your hypotheses) who has noticed an improvement over your prior. That is, if someone inside one of our hypotheses has noticed that e.g. a certain class of decisions is more important and so they will simulate only those situations, then we should also notice this and by the same token care more about our decision if we are in one of those situations (rather than using a universal prior without importance weighting). My sense is that without competitiveness we are in trouble anyway on other fronts, and so it is probably also reasonable to think of as a first-line defense against this kind of issue. This is very similar to what I first thought about when going down this line. My instantiation runs into trouble with "giant" universes that do all the possible computations you would want, and then using the "free" complexity in the bridge rules to pick which of the computations you actually wanted. I am not sure if the DFA proposal gets around this kind of problem though it sounds like it would be pretty similar.

2Vanessa Kosoy4y

I think that not every gap is exploitable. For most types of biases in the prior, it would only promote simulation hypotheses with baseline universes conformant to this bias, and attackers who evolved in such universes will also tend to share this bias, so they will target universes conformant to this bias and that would make them less competitive with the true hypothesis. In other words, most types of bias affect both ϵ and δ in a similar way. More generally, I guess I'm more optimistic than you about solving all such philosophical liabilities. I don't understand the proposal. Is there a link I should read? So, you can let your physics be a dovetailing of all possible programs, and delegate to the bridge rule the task of filtering the outputs of only one program. But the bridge rule is not "free complexity" because it's not coming from a simplicity prior at all. For a program of length n, you need a particular DFA of size Ω(n). However, the actual DFA is of expected size m with m≫n. The probability of having the DFA you need embedded in that is something like m!(m−n)!m−2n≈m−n≪2−n. So moving everything to the bridge makes a much less likely hypothesis.

2Vanessa Kosoy4y

I think that there are roughly two possibilities: either the laws of our universe happen to be strongly compressible when packed into a malign simulation hypothesis, or they don't. In the latter case, ϵδ shouldn't be large. In the former case, it means that we are overwhelming likely to actually be inside a malign simulation. But, then AI risk is the least of our troubles. (In particular, because the simulation will probably be turned off once the attack-relevant part is over.) [EDIT: I was wrong, see this.] Probably efficient algorithms are not running literally all hypotheses, but, they can probably consider multiple plausible hypotheses. In particular, the malign hypothesis itself is an efficient algorithm and it is somehow aware of the two different hypotheses (itself and the universe it's attacking). Currently I can only speculate about neural networks, but I do hope we'll have competitive algorithms amenable to theoretical analysis, whether they are neural networks or not. I think that the problem you describe in the linked post can be delegated to the AI. That is, instead of controlling trillions of robots via counterfactual oversight, we will start with just one AI project that will research how to organize the world. This project would top any solution we can come up with ourselves.

5paulfchristiano4y

It seems like the simplest algorithm that makes good predictions and runs on your computer is going to involve e.g. reasoning about what aspects of reality are important to making good predictions and then attending to those. But that doesn't mean that I think reality probably works that way. So I don't see how to salvage this kind of argument. It seems to me like this requires a very strong match between the priors we write down and our real priors. I'm kind of skeptical about that a priori, but then in particular we can see lots of ways in which attackers will be exploiting failures in the prior we write down (e.g. failure to update on logical facts they observe during evolution, failure to make the proper anthropic update, and our lack of philosophical sophistication meaning that we write down some obviously "wrong" universal prior). Do we have any idea how to write down such an algorithm though? Even granting that the malign hypothesis does so it's not clear how we would (short of being fully epistemically competitive); but moreover it's not clear to me the malign hypothesis faces a similar version of this problem since it's just thinking about a small list of hypotheses rather than trying to maintain a broad enough distribution to find all of them, and beyond that it may just be reasoning deductively about properties of the space of hypotheses rather than using a simple algorithm we can write down.

3Vanessa Kosoy4y

I think it works differently. What you should get is an infra-Bayesian hypothesis which models only those parts of reality that can be modeled within the given computing resources. More generally, if you don't endorse the predictions of the prediction algorithm than either you are wrong or you should use a different prediction algorithm. How the can the laws of physics be extra-compressible within the context of a simulation hypothesis? More compression means more explanatory power. I think that is must look something like, we can use the simulation hypothesis to predict the values of some of the physical constants. But, it would require a very unlikely coincidence for physical constants to have such values unless we are actually in a simulation. I agree that we won't have a perfect match but I think we can get a "good enough" match (similarly to how any two UTMs that are not too crazy give similar Solomonoff measures.) I think that infra-Bayesianism solves a lot of philosophical confusions, including anthropics and logical uncertainty, although some of the details still need to be worked out. (But, I'm not sure what specifically do you mean by "logical facts they observe during evolution"?) Ofc this doesn't mean I am already able to fully specify the correct infra-prior: I think that would take us most of the way to AGI. I have all sorts of ideas, but still nowhere near the solution ofc. We can do deep learning while randomizing initial conditions and/or adding some noise to gradient descent (e.g. simulated annealing), producing a population of networks that progresses in an evolutionary way. We can, for each prediction, train a model that produces the opposite prediction and compare it to the default model in terms of convergence time and/or weight magnitudes. We can search for the algorithm using meta-learning. We can do variational Bayes with a "multi-modal" model space: mixtures of some "base" type of model. We can do progressive refinement of infra-Bayesian

4michaelcohen4y

This is very nice and short! And to state what you left implicit: If p0>p1+ε, then in the setting with no malign hypotheses (which you assume to be safe), 0 is definitely the output, since the malign models can only shift the outcome by ε, so we assume it is safe to output 0. And likewise with outputting 1. One general worry I have about assuming that the deterministic case extends easily to the stochastic case is that a sequence of probabilities that tends to 0 can still have an infinite sum, which is not true when probabilities must ∈{0,1}, and this sometimes causes trouble. I'm not sure this would raise any issues here--just registering a slightly differing intuition.

3michaelcohen4y

They're not quite symmetrical: midway through, some bad hypotheses will have been ruled out for making erroneous predictions about the demonstrator previously. But your conclusion is still correct. And It's certainly no better than that, and the bound I prove is worse. Yeah. Error bounds on full Bayesian are logarithmic in the prior weight on the truth, but error bounds on maximum a posteriori prediction are just inverse in the prior weight, and your example above is the one to show it. If each successive MAP model predicts wrong at each successive timestep, it could take N timesteps to get rid of N models, which is how many might begin with a prior weight exceeding the truth, if the truth has a prior weight of 1/N. But, this situation seems pretty preposterous to me in the real world. If agent's first observation is, say, this paragraph, the number of models with prior weight greater than the truth that predicted something else as the first observation, will probably be a number way, way different from one. I'd go so far as to say at least half of models with prior weight greater than the truth would predict a different observation than this very paragraph. As long as this situation keeps up, we're in a logarithmic regime. I'm not convinced this logarithmic regime ever ends, but I think the case is more convincing that we at least start there, so let's suppose now that it eventually ends, and after this point the remaining models with posterior weight exceeding the truth are deliberately erring at unique timesteps. What's the posterior on the truth now? This is a phase where the all the top models are "capable" of predicting correctly. This shouldn't look like 2−1014 at all. It will look more like p(correct model)/p(treachery). And actually, depending on the composition of treacherous models, it could be better. To the extent some constant fraction of them are being treacherous at particular times, the logarithmic regime will continue. There are two reasons wh

[-]paulfchristiano4yΩ9100

I understand that the practical bound is going to be logarithmic "for a while" but it seems like the theorem about runtime doesn't help as much if that's what we are (mostly) relying on, and there's some additional analysis we need to do. That seems worth formalizing if we want to have a theorem, since that's the step that we need to be correct.

There is at most a linear cost to this ratio, which I don't think screws us.

If our models are a trillion bits, then it doesn't seem that surprising to me if it takes 100 bits extra to specify an intended model relative to an effective treacherous model, and if you have a linear dependence that would be unworkable. In some sense it's actively surprising if the very shortest intended vs treacherous models have description lengths within 0.0000001% of each other unless you have a very strong skills vs values separation. Overall I feel much more agnostic about this than you are.

There are two reasons why I expect that to hold.

This doesn't seem like it works once you are selecting on competent treacherous models. Any competent model will err very rarely (with probability less than 1 / (feasible training time), probably much less). I do... (read more)

1michaelcohen4y

I agree. Also, it might be very hard to formalize this in a way that's not: write out in math what I said in English, and name it Assumption 2. I don't think incompetence is the only reason to try to pull off a treacherous turn at the same time that other models do. Some timesteps are just more important, so there's a trade off. And what's traded off is a public good: among treacherous models, it is a public good for the models' moments of treachery to be spread out. Spreading them out exhausts whatever data source is being called upon in moments of uncertainty. But for an individual model, it never reaps the benefit of helping to exhaust our resources. Given that treacherous models don't have an incentive to be inconvenient to us in this way, I don't think a failure to qualifies as incompetence. This is also my response to Yeah, I've been assuming all treacherous models will err once. (Which is different from the RL case, where they can err repeatedly on off-policy predictions). I can't remember exactly where we left this in our previous discussions on this topic. My point estimate for the number of extra bits to specify an intended model relative to an effective treacherous model is negative. The treacherous model has to compute the truth, and then also decide when and how to execute treachery. So the subroutine they run to compute the truth, considered as a model in its own right, seems to me like it must be simpler.

[-]paulfchristiano4yΩ11120

I don't think incompetence is the only reason to try to pull off a treacherous turn at the same time that other models do. Some timesteps are just more important, so there's a trade off. And what's traded off is a public good: among treacherous models, it is a public good for the models' moments of treachery to be spread out.

Trying to defect at time T is only a good idea if it's plausible that your mechanism isn't going to notice the uncertainty at time T and then query the human. So it seems to me like this argument can never drive P(successful treachery) super low, or else it would be self-defeating (for competent treacherous agents).

The treacherous model has to compute the truth, and then also decide when and how to execute treachery. So the subroutine they run to compute the truth, considered as a model in its own right, seems to me like it must be simpler.

Subroutines of functions aren't always simpler. Even without treachery this seems like it's basically guaranteed. If "the truth" is just a simulation of a human, then the truth is a subroutine of the simplest model. But instead you could have a model of the form "Run program X, treat its output as a program, then run that." S... (read more)

1michaelcohen4y

I don't follow. Can't races to the bottom destroy all value for the agents involved? Consider the claim: firms will never set prices to equal marginal cost because that would destroy all profit. Yeah, okay. This is my mainline opinion, and I just think about inner alignment in case it's wrong. So for now we can assume that it is wrong, since there's not much more to say if it's right. That is my next guess, but I hadn't thought in terms of percentages before this. I had thought of normally distributed noise in the number of bits. It's occurring to me that this question doesn't matter to our present discussion. What makes the linear regime linear rather logarithmic is that if p(treachery)/p(honest model) is high, that allows for a large number of treacherous models to have greater posterior weight than the truth. But if a single model has n times the posterior weight of the truth, it still only takes one query to the demonstrator to interrupt its treacherous turn, not n queries.

3paulfchristiano4y

You are saying that a special moment is a particularly great one to be treacherous. But if P(discovery) is 99.99% during that period, and there is any other treachery-possible period where P(discovery) is small, then that other period would have been better after all. Right? This doesn't seem analogous to producers driving down profits to zero, because those firms had no other opportunity to make a profit with their machine. It's like you saying: there are tons of countries where firms could use their machines to make stuff and sell it at a profit (more countries than firms). But some of the places are more attractive than others, so probably everyone will sell in those places and drive profits to zero. And I'm saying: but then aren't those incredibly-congested countries actually worse places to sell? This scenario is only possible if firms are making so much stuff that they can drive profit down to zero in every country, since any country with remaining profit would necessarily be the best place to sell. It seems to me like this is clearly wrong in the limit (since simple consequentialists would take over simple physics). It also seems probably wrong to me for smaller models (i.e. without huge amounts of internal selection) but it would be good for someone to think about that case more seriously. if the treacherous model is 100 bits smaller, then it feels like there must be around 2^100 treacherous models that are all simpler than the intended one. If nothing else, it seems like you could literally add garbage bits to the treacherous models (or useful bits!).

4michaelcohen4y

Okay, sure. It's not clear to me that there isn't meaningful overhead involved. I agree with what you're saying but I don't see how it contradicts what I was. First, what I had in mind when saying that some timesteps are better for treachery because when the agent acts on a false prediction it has a greater effect on the world, though of course P(discovery) is also relevant. But my point is that when multiple treacherous models pick the same timestep to err, there may be pros and cons to doing this, but one thing that isn't on the cons list, is that in the long run, it makes our lives easier if they do. Making our lives difficult is a public good for treacherous models. So treacherous models won't be trying to avoid collisions in order to make queries be linear in p(treachery)/p(truth). If P(discovery) increases when multiple models are trying to be treacherous at the same time--which we could go onto discuss; it's not obvious to me either way as of yet--that will balanced against the inherit variation in some timesteps being a better for treachery than others.

3paulfchristiano4y

If you are willing to query only a billion points, and there are a billion competent treacherous models, and if the game is zero-sum between you and the models, then you're going to get at least as screwed as by the billionth-best opportunity for treachery. Otherwise, one of the treacherous models made a mistake: apparently there was a top-billion treachery opportunities where you didn't query, so they should have been able to get at least that much value (and you should have lost at least that much value). That's not quite right since your querying behavior will be stochastic (at least from the model's epistemic position). But you get the same conclusion to within a constant factor: if there are 2 billion treacherous models, then out of the top 2 billion opportunities for treachery, there must be some where our subjective probability of querying is at most 1/2, and if the models are competent they will get at least as much value as they could by defecting at one of those times, i.e. at least 1/2 of the harms from the 2-billionth-best opportunity for treachery. So we will lose at least that much value. (Those arguments aren't totally formal, we didn't even really discharge the "at least N treacherous models assumption" which you need to do in order to show that your decision about when to query isn't so influenced by any given model.)

1michaelcohen4y

The treacherous models make their decisions first, and whether we query at a timestep depends on whether they chose that moment to be treacherous. We don't have to say: this timestep is one of the billion best for treachery--guess we had better query. The decision process a treacherous model faces is: 1. From past observations, does it seem alpha is low enough that an honest model is in the top few? 2. If no, [let's just assume whatever happens here is bad] 3. If yes, is the honest model going to leave the top few any time? 4. If yes, wait. [let's assume this ends badly] 5. If no, admit defeat, because whenever you err, that will be when a query gets triggered. Salvage the situation by trying to get as much mileage as you can by decreasing probability on some action the demonstrator would take that you don't like much in order to shift probability mass to some other action they would take that you like a little better. OR 6. Hope that 1) other treacherous models, like soldiers on the front line storming the enemy and facing certain death so that their brothers behind them can win the battle, give erring a go anyway. And hope that 2) the operators quietly increase alpha after the billionth query. Acting on this hope, wait. (The front-line treacherous models are not entirely self-sacrificial; they might figure they might as well hope they were wrong about alpha. But the models know when queries happen, because every observation they get comes from a query, and it wouldn't take long to figure out that alpha really truly isn't high enough for them to execute treachery successfully). Above in the thread, I was talking about the game theory of step 6 (although I maybe wasn't making the case as strongly as I could have). Which models would ever volunteer? They don't care about the long term goal of exhausting our query resources so other treacherous models can capitalize on a quietly increased alpha. All that is to say, I don't follow the comment I excerpted. (In

1michaelcohen4y

This is a bit of a sidebar: I'm curious what you make of the following argument. 1. When an infinite sequence is sampled from a true model μ0, there is likely to be another treacherous model μ1 which is likely to end up with greater posterior weight than an honest model ^μ0, and greater still than the posterior on the true model μ0. 2. If the sequence were sampled from μ1 instead, the eventual posterior weight on μ1 will probably be at least as high. 3. When an infinite sequence is sampled from a true model μ1, there is likely to be another treacherous model μ2, which is likely to end up with greater posterior weight than an honest model ^μ1, and greater still than the posterior on the true model μ1. 4. And so on.

2paulfchristiano4y

The first treacherous model works by replacing the bad simplicity prior with a better prior, and then using the better prior to more quickly infer the true model. No reason for the same thing to happen a second time. (Well, I guess the argument works if you push out to longer and longer sequence lengths---a treacherous model will beat the true model on sequence lengths a billion, and then for sequence lengths a trillion a different treacherous model will win, and for sequence lengths a quadrillion a still different treacherous model will win. Before even thinking about the fact that each particular treacherous model will in fact defect at some point and at that point drop out of the posterior.)

1michaelcohen4y

Does it make sense to talk about ˇμ1, which is like μ1 in being treacherous, but is uses the true model μ0 instead of the honest model ^μ0? I guess you would expect ˇμ1 to have a lower posterior than μ0?

4evhub4y

It seems to me like this argument is requiring the deceptive models not being able to coordinate with each other very effectively, which seems unlikely to me—why shouldn't we expect them to be able to solve these sorts of coordination problems?

3michaelcohen4y

They're one shot. They're not interacting causally. When any pair of models cooperates, a third model defecting benefits just as much from their cooperation with each other. And there will be defectors--we can write them down. So cooperating models would have to cooperate with defectors and cooperators indiscriminately, which I think is just a bad move according to any decision theory. All the LDT stuff I've seen on the topic is how to make one's own cooperation/defection depend logically on the cooperation/defection of others.

4evhub4y

Well, just like we can write down the defectors, we can also write down the cooperators—these sorts of models do exist, regardless of whether they're implementing a decision theory we would call reasonable. And in this situation, the cooperators should eventually outcompete the defectors, until the cooperators all defect—which, if the true model has a low enough prior, they could do only once they've pushed the true model out of the top N.

3michaelcohen4y

If it's only the case that we can write them down, but they're not likely to arise naturally as simple consequentialists taking over simple physics, then that extra description length will be seriously costly to them, and we won't need to worry about any role they might play in p(treacherous)/p(truth). Meanwhile, when I was saying we could write down some defectors, I wasn't making a simultaneous claim about their relative prior weight, only that their existence would spoil cooperation. For cooperators to outcompete defectors, they would have to be getting a larger share of the gains from cooperation than defectors do. If some people are waiting for the fruit on public trees to ripen before eating, and some people aren't, the defectors will be the ones eating the almost ripe fruit. I might be misunderstanding this statement. The inverse of the posterior on the truth is a supermartingale (doesn't go up in expectation), so I don't know what it could mean for the true model to get pushed out.

4evhub4y

Here's the setup I'm imagining, but perhaps I'm still misunderstanding something. Suppose you have a bunch of deceptive models that choose to cooperate and have larger weight in the prior than the true model (I think you believe this is very unlikely, though I'm more ambivalent). Specifically, they cooperate in that they perfectly mimic the true model up until the point where the deceptive models make up enough of the posterior that the true model is no longer being consulted, at which point they all simultaneously defect. This allows for arbitrarily bad worst-case behavior.

3michaelcohen4y

This thread began by considering deceptive models cooperating with each other in the sense of separating the timing of their treacherous turns in order to be maximally annoying to us. So maybe our discussion on that topic is resolved, and we can move on to this scenario. if alpha is low enough, this won't ever happen, and if alpha is too high, it won't take very long. So I don't think this scenario is quite right. Then the question becomes, for an alpha that's low enough, how long will it take until queries are infrequent, noting that you need a query any time any treacherous model with enough weight decides to take a treacherous turn?

2Ofer4y

To extended Evan's comment about coordination between deceptive models: Even if the deceptive models lack relevant game theoretical mechanisms, they may still coordinate due to being (partially) aligned with each other. For example, a deceptive model X may prefer [some other random deceptive model seizing control] over [model X seizing control with probability 0.1% and the entire experiment being terminated with probability 99.9%]. Why should we assume that the deceptive models will be sufficiently misaligned with each other such that this will not be an issue? Do you have intuitions about the degree of misalignment between huge neural networks that were trained to imitate demonstrators but ended up being consequentialists that care about the state of our universe?

2michaelcohen4y

That's possible. But it seems like way less of a convergent instrumental goal for agents living in a simulated world-models. Both options--our world optimized by us and our world optimized by a random deceptive model--probably contain very little of value as judged by agents in another random deceptive model. So yeah, I would say some models would think like this, but I would expect the total weight on models that do to be much lower.

2paulfchristiano4y

I just meant the dependence on epsilon, it seems like there are unavoidable additional factors (especially the linear dependence on p(treachery)). I guess it's not obvious if you can make these additive or if they are inherently multipliactive. But your bound scales in some way, right? How much training data do I need to get the KL divergence between distributions over trajectories down to epsilon?

1michaelcohen4y

No matter how much data you have, my bound on the KL divergence won't approach zero.

2Vanessa Kosoy4y

Re 1: This is a good point. Some thoughts: [EDIT: See this] * We can add the assumption that the probability mass of malign hypotheses is small and that following any non-malign hypothesis at any given time is safe. Then we can probably get query complexity that scales with p(malign) / p(true)? * However, this is cheating because a sequence of actions can be dangerous even if each individual action came from a (different) harmless hypothesis. So, instead we want to assume something like: any dangerous strategy has high Kolmogorov complexity relative to the demonstrator. (More realistically, some kind of bounded Kolmogorov complexity.) * A related problem is: in reality "user takes action a" and "AI takes action a" are not physically identical events, and there might be an attack vector that exploits this. * If p(malign)/p(true) is really high that's a bigger problem, but then doesn't it mean we actually live in a malign simulation and everything is hopeless anyway? Re 2: Why do you need such a strong bound?

[-]Rohin Shah4y*Ω5140

Planned summary:

Since we probably can’t specify a reward function by hand, one way to get an agent that does what we want is to have it imitate a human. As long as it does this faithfully, it is as safe as the human it is imitating. However, in a train-test paradigm, the resulting agent may faithfully imitate the human on the training distribution but fail catastrophically on the test distribution. (For example, a deceptive model might imitate faithfully until it has sufficient power to take over.) One solution is to never stop training, that is, use an online learning setup where the agent is constantly learning from the demonstrator.
There are a few details to iron out. The agent needs to reduce the frequency with which it queries the demonstrator (otherwise we might as well just have the demonstrator do the work). Crucially, we need to ensure that the agent will never do something that the demonstrator wouldn’t have done, because such an action could be arbitrarily bad.
This paper proposes a solution in the paradigm where we use Bayesian updating rather than gradient descent to select our model, that is, we have a prior over possible models and then when we see a demonstrator acti

... (read more)

6michaelcohen4y

This is so much clearer than I've ever put it. N won't necessarily decrease over time, but all of the models will eventually agree with other. I would have described Vanessa's and my approaches as more about monitoring uncertainty, and avoiding problems before the fact rather than correcting them afterward. But I think what you said stands too.

4Rohin Shah4y

Ah, right. I rewrote that paragraph, getting rid of that sentence and instead talking about the tradeoff directly. Added a sentence to the opinion noting the benefits of explicitly quantified uncertainty.

[-]evhub4y*Ω10140

Edit: I no longer endorse this comment; see this comment instead.

I think you've just assumed the entire inner alignment problem away. You assume that the model—the imitation learner—is a perfect Bayesian, rather than just some trained neural network or other function approximator (edit: I was just misinterpreting here and the training process is supposed to Bayesian rather than the model, see Rohin's comment below). The entire point of the inner alignment problem, though, is that you can't guarantee that your trained model is actually a perfect Bayesian—or anything else. In practice, all we actually generally do when we do ML is we train some neural network on some training data/environment and hope that the implicit inductive biases of our training process are such that we end up with a model doing the right thing, meaning that we can't really control what sort of model—Bayesian or anything else—that we get when we do ML, and that uncontrollability, in my eyes, is the inner alignment problem.

[-]Rohin Shah4y*Ω15200

While I share your position that this mostly isn't addressing the things that make inner alignment hard / risky in practice, I agree with Vanessa that this does not assume the inner alignment problem away, unless you have a particularly contorted definition of "inner alignment".

There's an optimization procedure (Bayesian updating) that is selecting models (the model of the demonstrator) that can themselves be optimizers, and you could get the wrong one (e.g. the model that simulates an alien civilization that realizes it's in a simulation and predicts well to be selected by the Bayesian updating but eventually executes a treacherous turn). The algorithm presented precludes this from happening with some probability. We can debate the significance, but it seems to me like it is clearly doing something solution-like with respect to the inner alignment problem.

5TurnTrout4y

"in a simulation", no?

4Rohin Shah4y

Lol yes fixed

4evhub4y

Hmmm... I think I just misunderstood the setup then. It seemed to me like the Bayesian updating was supposed to represent the model rather than the training process, but under the framing that you just said, I agree that it's relevant. It's worth noting that I do think most of the danger lies in the ways in which gradient descent is not a perfect Bayesian, but I agree that modeling gradient descent as Bayesian is certainly relevant.

[-]Vanessa Kosoy4yΩ10160

I think this is completely unfair. The inner alignment problem exists even for perfect Bayesians, and solving it in that setting contributes much to our understanding. The fact we don't have satisfactory mathematical models of deep learning performance is a different problem, which is broader than inner alignment and to first approximation orthogonal to it. Ideally, we will solve this second problem by improving our mathematical understanding of deep learning and/or other competitive ML algorithms. The latter effort is already underway by researchers unrelated to AI safety, with some results. Moreover, we can in principle come up with heuristics how to apply this method of solving inner alignment (which I call "confidence thresholds" in my own work) to deep learning: e.g. use NNGP to measure confidence or use an evolutionary algorithm with a population of networks and check how well they agree with each other. Of course if we do this we won't have formal guarantees that it will work, but, like I said this is a broader issue than inner alignment.

[-]evhub4yΩ11180

Regardless of how you define inner alignment, I think that the vast majority of the existential risk comes from that “broader issue” that you're pointing to of not being able to get worst-case guarantees due to using deep learning or evolutionary search or whatever. That leads me to want to define inner alignment to be about that problem—and I think that is basically the definition we give in Risks from Learned Optimization, where we introduced the term. That being said, I do completely agree that getting a better understanding of deep learning is likely to be critical.

[-]johnswentworth4yΩ7100

I think that the vast majority of the existential risk comes from that “broader issue” that you're pointing to of not being able to get worst-case guarantees due to using deep learning or evolutionary search or whatever. That leads me to want to define inner alignment to be about that problem...

[Emphasis added.] I think this is a common and serious mistake-pattern, and in particular is one of the more common underlying causes of framing errors. The pattern is roughly:

Notice cluster of problems X which have a similar underlying causal pattern Cause(X)
Notice problem y in which Cause(X) could plausibly play a role
On deeper examination, the cause of y cause(y) doesn't quite fit Cause(X)
Attempt to redefine the pattern Cause(X) to include cause(y)

The problem is that, in trying to "shoehorn" cause(y) into the category Cause(X), we miss the opportunity to notice a different pattern, which is more directly useful in understanding y as well as some other cluster of problems related to y.

A concrete example: this is the same mistake I accused Zvi of making when trying to cast moral mazes as a problem of super-perfect competition. The conditions needed for super-perfect competition to explain m... (read more)

[-]evhub4yΩ11200

I mean, I don't think I'm “redefining” inner alignment, given that I don't think I've ever really changed my definition and I was the one that originally came up with the term (inner alignment was due to me, mesa-optimization was due to Chris van Merwijk). I also certainly agree that there are “more than just inner alignment problems going on in the lack of worst-case guarantees for deep learning/evolutionary search/etc.”—I think that's exactly the point that I'm making, which is that while there are other issues, inner alignment is what I'm most concerned about. That being said, I also think I was just misunderstanding the setup in the paper—see Rohin's comment on this chain.

5michaelcohen4y

If the inner alignment problem did not exist for perfect Bayesians, but did exist for neural networks, then it would appear to be a regime where more intelligence makes the problem go away. If the inner alignment problem were ~solved for perfect Bayesians, but unsolved for neural networks, I think there's still some of the flavor of that regime, but we do have to be pretty careful to make sure we're applying the same sort of solution to the non-Bayesian algorithms. I think in Vanessa's comment above, she's suggesting this looks doable. Note the method here of avoiding mesa-optimizers: error bounds. Neural networks don't have those. Naturally, one way to make mesa-optimizer-deceptively-selected-errors go away is just to have better learning algorithms that make errors go away. Algorithms like Gated Linear Networks with proper error bounds may be a safer building block for AGI. But none of this takes away from the fact that it is potentially important to figure out how to avoid mesa-optimization in neural networks, and I would add to your claim that this is a much harder setting; I would say it's a harder setting because of the non-existence of error bounds.

[-]evhub4yΩ8100

I think I mostly agree with what you're saying here, though I have a couple of comments—and now that I've read and understand your paper more thoroughly, I'm definitely a lot more excited about this research.

then it would appear to be a regime where more intelligence makes the problem go away.

I don't think that's right—if you're modeling the training process as Bayesian, as I now understand, then the issue is that what makes the problem go away isn't more intelligent models, but less efficient training processes. Even if we have arbitrary compute, I think we're unlikely to use training processes that look Bayesian just because true Bayesianism is really hard to compute such that for basically any amount of computation that you have available to you, you'd rather run some more efficient algorithm like SGD.

I worry about these sorts of Bayesian analyses where we're assuming that we're training a large population of models, one of which is assumed to be an accurate model of the world, since I expect us to end up using some sort of local search instead of a global search like that—just because I think that local search is way more efficient than any sort of Bayesianish global search... (read more)

5Vanessa Kosoy4y

I feel this is a wrong way to look at it. I expect any effective learning algorithm to be an approximation of Bayesianism in the sense that, it satisfies some good sample complexity bound w.r.t. some sufficiently rich prior. Ofc it's non-trivial to (i) prove such a bound for a given algorithm (ii) modify said algorithm using confidence thresholds in a way that leads to a safety guarantee. However, there is no sharp dichotomy between "Bayesianism" and "SGD" such that this approach obviously doesn't apply to the latter, or to something competitive with the latter.

3evhub4y

I agree that at some level SGD has to be doing something approximately Bayesian. But that doesn't necessarily imply that you'll be able to get any nice, Bayesian-like properties from it such as error bounds. For example, if you think of SGD as effectively just taking the MAP model starting from sort of simplicity prior, it seems very difficult to turn that into something like the top N posterior models, as would be required for an algorithm like this.

5Vanessa Kosoy4y

I mean, there's obviously a lot more work to do, but this is progress. Specifically if SGD is MAP then it seems plausible that e.g. SGD + random initial conditions or simulated annealing would give you something like top N posterior models. You can also extract confidence from NNGP.

4evhub4y

I agree that this is progress (now that I understand it better), though: I think there is strong evidence that the behavior of models trained via the same basic training process are likely to be highly correlated. This sort of correlation is related to low variance in the bias-variance tradeoff sense, and there is evidence that not only do massive neural networks tend to have pretty low variance, but that this variance is likely to continue to decrease as networks become larger.

3Vanessa Kosoy4y

Hmm, added to reading list, thank you.

3Vanessa Kosoy4y

Here's another way how you can try implementing this approach with deep learning. Train the predictor using meta-learning on synthetically generated environments (sampled from some reasonable prior such as bounded Solomonoff or maybe ANNs with random weights). The reward for making a prediction is 1−(1−δ)maxipi+ϵqi+ϵ, where pi is the predicted probability of outcome i, qi is the true probability of outcome i and ϵ,δ∈(0,1) are parameters. The reward for making no prediction (i.e. querying the user) is 0. This particular proposal is probably not quite right, but something in that general direction might work.

3evhub4y

Sure, but you have no guarantee that the model you learn is actually going to be optimizing anything like that reward function—that's the whole point of the inner alignment problem. What's nice about the approach in the original paper is that it keeps a bunch of different models around, keeps track of their posterior, and only acts on consensus, ensuring that the true model always has to approve. But if you just train a single model on some reward function like that with deep learning, you get no such guarantees.

3Vanessa Kosoy4y

Right, but but you can look at the performance of your model in training, compare it to the theoretical optimum (and to the baseline of making no predictions at all) and get lots of evidence about safety from that. You can even add some adversarial training of the synthetic environment in order to get tighter bounds. If on the vast majority of synthetic environments your model makes virtually no mispredictions, then, under the realizability assumption, it is very unlikely to make mispredictions in deployment. Ofc the realizability assumption should also be questioned: but that's true in the OP as well, so it's not a difference between Bayesianism and deep.

2evhub4y

Note that adversarial training doesn't work on deceptive models due to the RSA-2048 problem; also see more detail here. I think realizability basically doesn't help here—as long as there's a deceptive model which is easier to find according to your inductive biases, the fact that somewhere in the model space there exists a correct model, but not one that your local search process finds by default, is cold comfort.

4Vanessa Kosoy4y

I think that adversarial training working so well that it can find exponentially rare failures is an unnecessarily strong desideratum. We need to drive to probability of catastrophic failures to something very low, but not actually zero. If a system is going to work for time t during deployment, then running it in n different synthetic environments for time t during training is enough to drive the probability of failure down to O(1n). Now, this is prohibitively costly, but it's not exponentially prohibitively costly. And this doesn't use adversarial training at all! So, while it's not obvious how to get a good solution, it seems at least plausible that this can lead to a solution.

3evhub4y

I agree that random defection can potentially be worked around—but the RSA-2048 problem is about conditional defection, which can't be dealt with in the same way. More generally, I expect it to be extremely difficult if not impossible to prevent a model that you want to be able to operate in the real world from being able to determine at what point in training/deployment it's in.

2Vanessa Kosoy4y

Why do you expect it? During training, it finds itself in random universe. During deployment, it finds itself in another universe drawn from the same prior (the realizability assumption). How would it determine the difference?

5evhub4y

Because that's never what machine learning actually looks like in practice—essentially any access to text from the internet (let alone actual ability to interface with the world, both of which seem necessary for competitive systems) will let it determine things like the year, whether RSA-2048 has been factored, or other various pieces of information that are relevant to what stage in training/testing/deployment it's in, how powerful the feedback mechanisms keeping it in check are, whether other models are defecting, etc. that can let it figure out when to defect.

2Vanessa Kosoy4y

Hmm, maybe you misunderstood my proposal? I suggested to train the model by meta-learning on purely synthetic data, sampled from some kind of simplicity prior, without any interface to the world. Maybe you just think this wouldn't be competitive? If so, why? Is the argument just that there are no existing systems like this? But then it's weak evidence at best. On the contrary, even from a purely capability standpoint, meta-learning with synthetic data might be a promising strategy to lower deep learning's sample complexity. Here's why I think it will be competitive: * When a hypothetical competitive prediction algorithm A produces a malign prediction, the knowledge that it is malign clearly exists somewhere inside A in some form: otherwise it wouldn't come up with the prediction. In particular, the knowledge that there are multiple plausible hypotheses consistent with the data also exists somewhere inside A. * Therefore, there must exist some competitive algorithm B that would be able to use the knowledge of this ambiguity to abstain from predicting in such cases. * There is no reason why B should be tailored to fine details of our physical world: it can be quite generic (as all deep learning algorithms). * If we have an ML algorithm C that is powerful enough to produce superintelligence, then it is likely powerful enough to come up with B. Since B is generic, the task of finding B doesn't require any real-world data, and can be accomplished by meta-learning on synthetic data like I suggested.

2evhub4y

This seems very sketchy to me. If we let A = SGD or A = evolution, your first claim becomes “if SGD/evolution finds a malign model, it must understand that it's malign on some level,” which seems just straightforwardly incorrect. The last claim also seems pretty likely to be wrong if you let C = SGD or C = evolution. Moreover, it definitely seems like training on data sampled from a simplicity prior (if that's even possible—it should be uncomputable in general) is unlikely to help at all. I think there's essentially no realistic way that training on synthetic data like that will be sufficient to produce a model which is capable of accomplishing things in the world. At best, that sort of approach might give you better inductive biases in terms of incentivizing the right sort of behavior, but in practice I expect any impact there to basically just be swamped by the impact of fine-tuning on real-world data.

3Vanessa Kosoy4y

To me it seems straightforwardly correct! Suppose you're running evolution in order to predict a sequence. You end up evolving a mind M that is a superintelligent malign consequentialist: it makes good predictions on purpose, in order to survive, and then produces a malign false prediction at a critical moment. So, M is part of the state of your algorithm A. All of M's knowledge is also part of the state. M knows that M is malign, M knows the prediction it's making this time is false. In this case, B can be whatever algorithm M uses to form its beliefs + confidence threshold (it's strategy stealing.) Why? If you give evolution enough time, and the fitness criterion is good (as you apparently agreed earlier), then eventually it will find B. First, obviously we use a bounded simplicity prior, like I said in the beginning of this thread. It can be something like weighting programs by 2^{-length} while constraining their amount of computational resources, or something like an ANN with random weights (the latter is more speculative, but given that we know ANNs have inductive bias to simplicity, an untrained ANN probably reflects that.) Second, why? Suppose that your starting algorithm is good at finding results when a lot of data is available, but is also very data inefficient (like deep learning seems to be). Then, by providing it with a lot of synthetic data, you leverage its strength to find a new algorithm which is data efficient. Unless you believe deep learning is already maximally data efficient (which seems very dubious to me)?

3evhub4y

Sure, but then I think B is likely to be significantly more complex and harder for a local search process to find than A. I definitely don't think this, unless you have a very strong (and likely unachievable imo) definition of “good.” I guess I'm skeptical that you can do all that much in the fully generic setting of just trying to predict a simplicity prior. For example, if we look at actually successful current architectures, like CNNs or transformers, they're designed to work well on specific types of data and relationships that are common in our world—but not necessarily at all just in a general simplicity prior.

2Vanessa Kosoy4y

A is sufficiently powerful to select M which contains the complex part of B. It seems rather implausible that an algorithm of the same power cannot select B. CNNs are specific in some way, but in a relatively weak way (exploiting hierarchical geometric structure). Transformers are known to be Turing-complete, so I'm not sure they are specific at all (ofc the fact you can express any program doesn't mean you can effectively learn any program, and moreover the latter is false on computational complexity grounds, but it still seems to point to some rather simple and large class of learnable hypotheses). Moreover, even if our world has some specific property that is important for learning, this only means we may need to enforce this property in our prior. For example, if your prior is an ANN with random weights then it's plausible that it reflects the exact inductive biases of the given architecture.

3evhub4y

A's weights do not contain the complex part of B—deception is an inference-time phenomenon. It's very possible for complex instrumental goals to be derived from a simple structure such that a search process is capable of finding that simple structure that yields those complex instrumental goals without being able to find a model with those complex instrumental goals hard-coded as terminal goals.

2Vanessa Kosoy4y

Hmm, sorry, I'm not following. What exactly do you mean by "inference-time" and "derived"? By assumption, when you run A on some sequence it effectively simulates M which runs the complex core of B. So, A trained on that sequence effectively contains the complex core of B as a subroutine.

2evhub4y

A's structure can just be “think up good strategies for achieving X, then do those,” with no explicit subroutine that you can find anywhere in A's weights that you can copy over to B.

4Vanessa Kosoy4y

IIUC, you're saying something like: suppose trained-A computes the source code of the complex core of B and then runs it. But then, define B′ as: compute the source code of the complex core of B (in the same way A does it) and use it to implement B. B′ is equivalent to B and has about the same complexity as trained-A. Or, from a slightly different angle: if "think up good strategies for achieving X" is powerful enough to come up with M, then "think up good strategies for achieving [reward of the type I defined earlier]" is powerful enough to come up with B.

2evhub4y

I think that “think up good strategies for achieving [reward of the type I defined earlier]” is likely to be much, much more complex (making it much more difficult to achieve with a local search process) than an arbitrary goal X for most sorts of rewards that we would actually be happy with AIs achieving.

2Vanessa Kosoy4y

Why? It seems like M would have all the knowledge required for achieving good rewards of the good type, so simulating M should not be more difficult than achieving good rewards of the good type.

2evhub4y

It's not that simulating M is difficult, but that encoding for some complex goal is difficult, whereas encoding for a random, simple goal is easy.

4Vanessa Kosoy4y

The reward function I was sketching earlier is not complex. Moreover, if you can encode M then you can encode B, no reason why the latter should have much greater complexity. If you can't encode M but can only encode something that produces M, then by the same token you can encode something that produces B (I don't think that's even a meaningful distinction tbh). I think it would help if you could construct a simple mathematical toy model of your reasoning here?

2evhub4y

Here's a simple toy model. Suppose you have two agents that internally compute their actions as follows (perhaps with actual argmax replaced with some smarter search algorithm, but still basically structured as below): Mdeceptive(x)=argmaxaE[∑iUdeceptive(si) | a]Maligned(x)=argmaxaE[∑iUaligned(si) | a] Then, comparing the K-complexity of the two models, we get K(Maligned)−K(Mdeceptive)≈K(Ualigned)−K(Udeceptive) and the problem becomes that both Mdeceptive and Maligned will produce behavior that looks aligned on the training distribution, but Ualigned has to be much more complex. To see this, note that essentially any Udeceptive will yield good training performance because the model will choose to act deceptively during training, whereas if you want to get good training performance without deception, then Ualigned has to actually encode the full objective, which is likely to make it quite complicated.

5Vanessa Kosoy4y

I understand how this model explains why agents become unaligned under distributional shift. That's something I never disputed. However, I don't understand how this model applies to my proposal. In my proposal, there is no distributional shift, because (thanks to the realizability assumption) the real environment is sampled from the same prior that is used for training with C. The model can't choose to act deceptively during training, because it can't distinguish between training and deployment. Moreover, the objective I described is not complicated.

2evhub4y

Yeah, that's a fair objection—my response to that is just that I think that preventing a model from being able to distinguish training and deployment is likely to be impossible for anything competitive.

2Vanessa Kosoy4y

Okay, but why? I think that the reason you have this intuition is, the realizability assumption is false. But then you should concede that the main weakness of the OP is the realizability assumption rather than the difference between deep learning and Bayesianism.

2evhub4y

Perhaps I just totally don't understand what you mean by realizability, but I fail to see how realizability is relevant here. As I understand it, realizability just says that the true model has some non-zero prior probability—but that doesn't matter (at least for the MAP, which I think is a better model than the full posterior for how SGD actually works) as long as there's some deceptive model with greater prior probability that's indistinguishable on the training distribution, as in my simple toy model from earlier.

4Vanessa Kosoy4y

When talking about uniform (worst-case) bounds, realizability just means the true environment is in the hypothesis class, but in a Bayesian setting (like in the OP) it means that our bounds scale with the probability of the true environment in the prior. Essentially, it means we can pretend the true environment was sampled from the prior. So, if (by design) training works by sampling environments from the prior, and (by realizability) deployment also consists of sampling an environment from the same prior, training and deployment are indistinguishable.

2evhub4y

Sure—by that definition of realizability, I agree that's where the difficulty is. Though I would seriously question the practical applicability of such an assumption.

1michaelcohen4y

What's the distinction between training and deployment when the model can always query for more data?

3Vanessa Kosoy4y

We're doing meta-learning. During training, the network is not learning about the real world, it's learning how to be a safe predictor. It's interacting with a synthetic environment, so a misprediction doesn't have any catastrophic effects: it only teaches the algorithm that this version of the predictor is unsafe. In other words, the malign subagents have no way to attack during training because they can access little information about what the real universe is like. The training process is designed to select predictors that only make predictions when they can be confident, and the training performance allows us to verify this goal has truly been achieved.

1Vanessa Kosoy4y

You have no guarantees, sure, but that's a problem with deep learning in general and not just inner alignment. The point is, if your model is not optimizing that reward function then its performance during training will be suboptimal. To the extent your algorithm is able to approximate the true optimum during training, it will behave safely during deployment.

2evhub4y

I think you are understanding inner alignment very differently than we define it in Risks from Learned Optimization, where we introduced the term. This is not true for deceptively aligned models, which is the situation I'm most concerned about, and—as we argue extensively in Risks from Learned Optimization—there are a lot of reasons why a model might end up pursuing a simpler/faster/easier-to-find proxy even if that proxy yields suboptimal training performance.

1michaelcohen4y

It may be helpful to point to specific sections of such a long paper. (Also, I agree that a neural network trained trained with that reward could produce a deceptive model that makes a well-timed error.)

1michaelcohen4y

I don't understand the alternative, but maybe that's neither here nor there. It's a little hard to make a definitive statement about a hypothetical in which the inner alignment problem doesn't apply to Bayesian inference. However, since error bounds are apparently a key piece of a solution, it certainly seems that if Bayesian inference was immune to mesa-optimizers it would be because of competence not resource-prodigality. Here's another tack. Hypothesis generation seems like a crucial part of intelligent prediction. Pure Bayesian reasoning does hypothesis generation by brute force. Suppose it was inefficiency, and not intelligence, that made Bayesian reasoning avoid mesa-optimizers. Then suppose we had a Bayesian reasoning that was equally intelligent but more efficient, by only generating relevant hypotheses. It gains efficiency by not even bothering to consider a bunch of obviously wrong models, but it's posterior is roughly the same, so it should avoid inner alignment failure equally well. If, on the other hand, the hyopthesis generation routine was bad enough that some plausible hypotheses went unexamined, this could introduce an inner alignment failure, with a notable decrease in intelligence. I expect some heuristic search with no discernable structure, guided "attentively" by an agent. And I expect this heuristic search through models to identify any model that a human could hypothesize, and many more.

4evhub4y

Perhaps I'm just being pedantic, but when you're building mathematical models of things I think it's really important to call attention to what things in the real world those mathematical models are supposed to represent—since that's how you know whether your assumptions are reasonable or not. In this case, there are two ways I could interpret this analysis: either the Bayesian learner is supposed to represent what we want our trained models to be doing, and then we can ask how we might train something that works that way; or the Bayesian learner is supposed to represent how we want our training process to work, and then we can ask how we might build training processes that work that way. I originally took you as saying the first thing, then realized you were actually saying the second thing. That's a good point. Perhaps I am just saying that I don't think we'll get training processes that are that competent. I do still think there are some ways in which I don't fully buy this picture, though. I think that the point that I'm trying to make is that the Bayesian learner gets its safety guarantees by having a massive hypothesis space and just keeping track of how well each of those hypotheses is doing. Obviously, if you knew of a smaller space that definitely included the desired hypothesis, you could do that instead—but given a fixed-size space that you have to search over (and unless your space is absolutely tiny such that you basically already have what you want), for any given amount of compute, I think it'll be more efficient to run a local search over that space than a global one. Interesting. I'm not exactly sure what you mean by that. I could imagine “guided 'attentively' by an agent” to mean something like recursive oversight/relaxed adversarial training wherein some overseer is guiding the search process—is that what you mean?

4michaelcohen4y

I'm not exactly sure what I mean either, but I wasn't imagining as much structure as exists in your post. I mean there's some process which constructs hypotheses, and choices are being made about how computation is being directed within that process. I think any any heuristic search algorithm worth its salt will incorporate information about proximity of models. And I think that arbitrary limits on heuristic search of the form "the next model I consider must be fairly close to the last one I did" will not help it very much if it's anywhere near smart enough to merit membership in a generally intelligent predictor. But for the purpose of analyzing it's output, I don't think this discussion is critical if we agree that we can expect a good heuristic search through models will identify any model that a human could hypothesize.

2evhub4y

I think I would expect essentially all models that a human could hypothesize to be in the search space—but if you're doing a local search, then you only ever really see the easiest to find model with good behavior, not all models with good behavior, which means you're relying a lot more on your prior/inductive biases/whatever is determining how hard models are to find to do a lot more work for you. Cast into the Bayesian setting, a local search like this is relying on something like the MAP model not being deceptive—and escaping that to instead get N models sampled independently from the top q proportion or whatever seems very difficult to do via any local search algorithm.

3michaelcohen4y

So would you say you disagree with the claim ?

4evhub4y

Yeah; I think I would say I disagree with that. Notably, evolution is not a generally intelligent predictor, but is still capable of producing generally intelligent predictors. I expect the same to be true of processes like SGD.

3michaelcohen4y

If we ever produce generally intelligent predictors (or "accurate world-models" in the terminology we've been using so far), we will need a process that is much more efficient than evolution. But also, I certainly don't think that in order to be generally intelligent you need to start with a generally intelligent subroutine. Then you could never get off the ground. I expect good hypothesis-generation / model-proposal to use a mess of learned heuristics which would not be easily directed to solve arbitrary tasks, and I expect the heuristic "look for models near the best-so-far model" to be useful, but I don't think making it ironclad would be useful. Another thought on our exchange: If what you say is correct, then it sounds like exclusively-local search precludes human-level intelligence! (Which I don't believe, by the way, even if I think it's a less efficient path). One human competency is generating lots of hypotheses, and then having many models of the world, and then designing experiments to probe those hypotheses. It's hard for me to imagine that an agent that finds an "easiest-to-find model" and then calls it a day could ever do human-level science. Even something as simple as understanding an interlocuter requires generating diverse models on the fly: "Do they mean X or Y with those words? Let me ask a clarfying question." I'm not this bearish on local search. But if local search is this bad, I don't think it is a viable path to AGI, and if it's not, then the internals don't for the purposes of our discussion, and we can skip to what I take to be the upshot:

[-]evhub4yΩ6120

It's hard for me to imagine that an agent that finds an "easiest-to-find model" and then calls it a day could ever do human-level science.

I certainly don't think SGD is a powerful enough optimization process to do science directly, but it definitely seems powerful enough to find an agent which does do science.

if local search is this bad, I don't think it is a viable path to AGI

We know that local search processes can produce AGI, so viability is a question of efficiency—and we know that SGD is at least efficient enough to solve a wide variety of problems from image classification, to language modeling, to complex video games, all given just current compute budgets. So while I could certainly imagine SGD being insufficient, I definitely wouldn't want to bet on it.

3michaelcohen4y

Okay I think we've switched from talking about Q-learning to talking about policy gradient. (Or we were talking about the latter the whole time, and I didn't notice it). The question that I think is relevant is: how are possible world-models being hypothesized and analyzed? That's something I expect to be done with messy heuristics that sometimes have discontinuities their sequence of outputs. Which means I think that no reasonable DQN is will be generally intelligent (except maybe an enormously wide one attention-based one, such that finding models is more about selective attention at any given step than it is about gradient descent over the whole history). A policy gradient network, on the other hand, could maybe (after having its parameters updated through gradient descent) become a network that, in a single forward pass, considers diverse world-models (generated with a messy non-local heuristic), and analyzes their plausibility, and then acts. At the end of the day, what we have is an agent modeling world, and we can expect it to consider any model that a human could come up with. (This paragraph also applies to the DQN with a gradient-descent-trained method for selectively attending to different parts of a wide network, since that could amount to effectively considering different models).

3evhub4y

Hmmm... I don't think I was ever even meaning to talk specifically about RL, but regardless I don't expect nearly as large of a difference between Q-learning and policy gradient algorithms. If we imagine both types of algorithms making use of the same size massive neural network, the only real difference is how the output of that neural network is interpreted, either directly as a policy, or as Q values that are turned into a policy via something like softmax. In both cases, the neural network is capable of implementing any arbitrary policy and should be getting a similar sort of feedback signal from the training process—especially if you're using a policy gradient algorithm that involves something like advantage estimation rather than actual rollouts, since the update rule in that situation is going to look very similar to the Q learning update rule. I do expect some minor differences in the sorts of models you end up with, such as Q learning being more prone to non-myopic behavior across episodes, and I think there are some minor reasons that policy gradient algorithms are favored in real-world settings, since they get to learn their exploration policy rather than having it hard-coded and can handle continuous action domains—but overall I think these sorts of differences are pretty minor and shouldn't affect whether these approaches can reach general intelligence or not.

3michaelcohen4y

I interpreted this bit as talking about RL But taking us back out of RL, in a wide neural network with selective attention that enables many qualitatively different forward passes, gradient descent seems to be training the way different models get proposed (i.e. the way attention is allocated), since this happens in a single forward pass, and what we're left with is a modeling routine that is heuristically considering (and later comparing) very different models. And this should include any model that a human would consider. I think that is main thread of our argument, but now I'm curious if I was totally off the mark about Q-learning and policy gradient. I had thought that maybe since a Q-learner is trained as if the cached point estimate of the Q-value of the next state is the Truth, it won't, in a single forward pass, consider different models about what the actual Q-value of the next state is. At most, it will consider different models about what the very next transition will be. a) Does that seem right? and b) Aren't there some policy gradient methods that don't face this problem?

2evhub4y

This seems wrong to me—even though the Q learner is trained using its own point estimate of the next state, it isn't, at inference time, given access to that point estimate. The Q learner has to choose its Q values before it knows anything about what the Q value estimates will be of future states, which means it certainly should have to consider different models of what the next transition will be like.

1michaelcohen4y

Yeah I was agreeing with that. Right, but one thing the Q-network, in its forward pass, is trying to reproduce is the point of estimate of the Q-value of the next state (since it doesn't have access to it). What it isn't trying to reproduce, because it isn't trained that way, is multiple models of what the Q-value might be at a given possible next state.

[-]Vlad Mikulik4yΩ470

Thanks for the post and writeup, and good work! I especially appreciate the short, informal explanation of what makes this work.

Given my current understanding of the proposal, I have one worry which makes me reluctant to share your optimism about this being a solution to inner alignment:

The scheme doesn't protect us if somehow all top-n demonstrator models have correlated errors. This could happen if they are coordinating, or more prosaically if our way to approximate the posterior leads to such correlations. The picture I have in my head for the latter is... (read more)

2michaelcohen4y

Yes, I agree that an ensemble of models generated by a neural network may have correlated errors. I only claim to solve the inner alignment problem in theory (i.e. for idealized Bayesian reasoning).

[-]adamShimi4yΩ670

Thanks for sharing this work!

Here's my short summary after reading the slides and scanning the paper.

Because human demonstrator are safe (in the sense of almost never doing catastrophic actions), a model that imitates closely enough the demonstrator should be safe. The algorithm in this paper does that by keeping multiple models of the demonstrator, sampling the top models according to a parameter, and following what the sampled model does (or querying the demonstrator if the sample is "empty"). The probability that this algorithm does a very unlikely acti

... (read more)

2Vanessa Kosoy4y

I don't think this is a lethal problem. The setting is not one-shot, it's imitation over some duration of time. IDA just increases the effective duration of time, so you only need to tune how cautious the learning is (which I think is controlled by α in this work) accordingly: there is a cost, but it's bounded. You also need to deal with non-realizability (after enough amplifications the system is too complex for exact simulation, even if it wasn't to begin with), but this should be doable using infra-Bayesianism (I already have some notion how that would work). Another problem with imitation-based IDA is that external unaligned AI might leak into the system either from the future or from counterfactual scenarios in which such an AI is instantiated. This is not an issue with amplifying by parallelism (like in the presentation) but at the cost of requiring parallelizability.

2Charlie Steiner4y

I think this looks fine for IDA - the two problems remain the practical one of implementing Bayesian reasoning in a complicated world, and the philosophical one that probably IDA on human imitations doesn't work because humans have bad safety properties.

2michaelcohen4y

Edited to clarify. Thank you for this comment. 100% agree. Intelligent agency can be broken into intelligent prediction and intelligent planning. This work introduces a method for intelligent prediction that avoids an inner alignment failure. The original concern about inner alignment was that an idealized prediction algorithm (Bayesian reasoning) could be commandeered by mesa-optimizers. Idealized planning, on the other hand is an expectimax tree, and I don't think anyone has claimed mesa-optimizers could be introduced by a perfect planner. I'm not sure what it would even mean. There is nothing internal in the expectimax algorithm that could make the output something other than what the prediction algorithm would agree is the best plan. Expectimax, by definition, produces a policy perfectly aligned with the "goals" of the prediction algorithm. Tl;dr: I think that in theory, the inner alignment problem regards prediction, not planning, so that's the place to test solutions. If you want to see the inner alignment problem neutralized in the full RL setup, you can see we use a similar approach in this agent's prediction subroutine. So you can maybe say that work solved the inner alignment problem. But we didn't prove finite error bounds the way we have here, and I think RL setup obscures the extent to which rogue predictive models are dismissed, so it's a little harder to see than it is here. Not exactly. No models are ever sampled. The top models are collected, and they all can contribute to the estimated probabilities of actions. Then an action is sampled according to those probabilities, which sum to less than one, and queries the demonstrator if it comes up empty. Yes, that's right. The bounds should chain together, I think, but they would definitely grow.

[-]Vanessa Kosoy4yΩ460

The way to avoid outputting predictions that may have been corrupted by a mesa-optimizer is to ask for help when plausible stochastic models disagree about probabilities.

This is exactly the method used in my paper about delegative RL and also an earlier essay that doesn't assume finite MDPs or access to rewards but has stronger assumptions about the advisor (so it's essentially doing imitation learning + denoising). I pointed out the connection to mesa-optimizers (which I called "daemons") in another essay.

1michaelcohen4y

If I'm interpreting the algorithm on page 7 correctly, it looks like if there's a trap that the demonstrator falls into with probability 10−9, there's no limit on the probability that the agent falls into the trap? Also, and maybe relatedly, do the demonstrator-models in the paper have to be deterministic?

3Vanessa Kosoy4y

Yes to the first question. In the DLRL paper I assume the advisor takes unsafe actions with probability exactly 0. However, it is straightforward to generalize the result s.t. the advisor can take unsafe actions with probability δ≪ϵ, where ϵ is the lower bound for the probability to take an optimal action (Definition 8). Moreover, in DLIRL (which, I believe, is closer to your setting) I use a "soft" assumption (see Definition 3 there) that doesn't require any probability to vanish entirely. No to the second question. In neither setting the advisor is assumed to be deterministic.

1michaelcohen4y

Okay, then I assume the agent's models of the advisor are not exclusively deterministic either? What I care most about is the ratio of probabilities that the advisor vs. agent takes the unsafe action, where don't know as programmers (so the agent doesn't get told at the beginning) any bounds on what these advisor-probabilities are. Can this modification be recast to have that property? Or does it already?

2Vanessa Kosoy4y

Of course. I assume realizability, so one of the hypothesis is the true advisor behavior, which is stochastic. In order to achieve the optimal regret bound, you do need to know the values of δ and ϵ. In DLIRL, you need to know β. However, AFAIU your algorithm also depends on some parameter (α)? In principle, if you don't know anything about the parameters, you can set them to be some function of the time discount s.t. as γ→1 the bound becomes true and the regret still goes to 0. In DLRL, this requires ω(1−γ)≤ϵ(γ)≤o(1), in DLIRL ω((1−γ)23)≤β(γ)−1≤o(1). However, then you only know regret vanishes with certain asymptotic rate without having a quantitative bound.

3michaelcohen4y

That makes sense. Alpha only needs to be set based on a guess about what the prior on the truth is. It doesn't need to be set based on guesses about possibly countably many traps of varying advisor-probability. I'm not sure I understand whether you were saying ratio of probabilities that the advisor vs. agent takes an unsafe action can indeed be bounded in DL(I)RL.

3Vanessa Kosoy4y

Hmm, yes, I think the difference comes from imitation vs. RL. In your setting, you only care about producing a good imitation of the advisor. On the other hand in my settings, I want to achieve near-optimal performance (which the advisor doesn't achieve). So I need stronger assumptions. Well, in DLIRL the probability that the advisor takes an unsafe action on any given round is bounded by roughly e−β, whereas the probability that the agent takes an unsafe action over a duration of (1−γ)−1 is bounded by roughly β−1(1−γ)−23, so it's not a ratio but there is some relationship. I'm sure you can derive some relationship in DLRL too, but I haven't studied it (like I said, I only worked out the details when the advisor never takes unsafe actions).

1michaelcohen4y

Neat, makes sense.

[-]abramdemski4yΩ340

Reviewed here.

[-]Vanessa Kosoy4yΩ340

It is worth noting that this approach doesn't deal with non-Cartesian daemons, i.e. malign hypotheses that attack via the physical side effects of computations rather than their output. Homomorphic encryption is still the best solution I know to these.

1michaelcohen4y

It is worth noting this.

[-]Koen.Holtman4y*Ω220

Interesting paper! I like the focus on imitation learning, but the really new food-for-thought thing to me is the bit about dropping i.i.d. assumptions and then seeing how far you can get. I need to think more about the math in the paper before I can ask some specific questions about this i.i.d. thing.

My feelings about the post above are a bit more mixed. Claims about inner alignment always seem to generate a lot of traffic on this site. But a lot of this traffic consists of questions and clarification about what exactly counts as an inner alignment fail... (read more)

1michaelcohen4y

This is exactly right. The reason this is consistent is that queries will become infrequent, but they will still be well-timed. The choice of whether to query depends on what the treacherous models are doing. So if the treacherous models wait for a long time, then we will have a long time where no queries are needed, and as soon as they decide to be treacherous, we thwart that plan with a query to the demonstrator. So it does not imply that Yep, perfectly reasonable! If we set α small enough, we can make it arbitrarily like that πd never leaves the set of top policies.

1Koen.Holtman4y

[long comment, my bigger agenda here is to get to a state where discussions on this forum start using much more math and subcase analysis when they talk about analysing and solving inner alignment problems.] OK, I think you are kind of saved here, in the case of my counter-example, because of the special nature of your equation (2) in the paper that computes the set of policies. To make this more specific: Say I refine the counter-example by adding the prior w(πi)=1/|Π|, so each candidate is considered equally likely from the start. Say that |Π|=1010. The key thing to note in my example is that, before the time-synchronized treacherous turn happens for all policies πi≠πd, all policies πi are equally compatible with events so far, so the posterior w(πi|h<t) is the same for all of them. When I use α=10−4 with these all-the-same-posterior policies, then (if I am reading equation (2) correctly) the set Παh<t of top policies will have size N=104. Ties are broken arbitrarily, so the real πd is unlikely to be among these top N=104<<1010 candidates. In that case, the agent will perform the treacherous turn and kill the operator. Correct me if I am wrong here. However. Case of α<1/|Π|: much better than I thought! However, and this is where things get interesting, if we use α=1011, so α<1/|Π|, then equation (2) will compute Παh<t=Π, the top set of policies is all policies. The correct policy is therefore still in the top set, and we get no treacherous turn. So equation (2) is doing a lot of heavy lifting in making the approach work! This also seems to be a major difference with the β based approach in your earlier paper. Not sure if you made this change specifically to deal with examples like my counter-example above, or if you had a more general reason -- the paper does not say. Pretty interesting! So I would now agree with your statement that your approach provably solves the inner alignment problem, at least for α<1/|Π| and for certain values of 'the inner alignm

1michaelcohen4y

I don't think this is a problem. There will be plenty of them, but when they're wrong they'll get removed from the posterior.

1Koen.Holtman4y

I have seen you mention a number of times in this comment thread that 'this is not a problem because eventually the bad/wrong policies will disappear from the top set'. You have not qualified this statement with 'but we need a very low α like α<1/|Π| to make this work in a safe way', so I remain somewhat uncertain about your views are about how low α needs to go. In any case, I'll now try to convince you that if α>1/|Π|, your statement that 'when they're wrong they'll get removed from the posterior' will not always mean what you might want it to mean. Is the demonstrator policy πd to get themselves killed? The interesting thing in developing these counterexamples is that they often show that the provable math in the paper gives you less safety than you would have hoped for. Say that πp∈Π is the policy of producing paperclips in the manner demonstrated by the human demonstrator. Now, take my construction in the counterexample where α>1/|Π| and where at time step t, we have the likely case that πp∉Παh<t. In the world I constructed for the counterexample, the remaining top policies Παh<t now perform a synchronized treacherous turn where they kill the demonstrator. In time step t+1 and later, the policies Παh<t+1 diverge a lot in what actions they will take, so the agent queries the demonstrator, who is now dead. The query will return the null action. This eventually removes all 'wrong' policies from Παh<t+1+i, where 'wrong' means that they do not take the null action at all future time steps. The silver lining is perhaps that at least the agent will eventually stop, perform null actions only, after it has killed the demonstrator. Now. the paper proves that the behavior of the agent policy πiα will approximate that of the true demonstrator policy πd closer and closer when time progresses. We therefore have to conclude that in the counterexample world, the true demonstrator policy πd had nothing to do with producing paperclips, this was a wrong guess all along. T

[-]Koen.Holtman4y*Ω110

[edited to delete and replace an earlier question] Question about the paper: under equation (3) on page 4 I am reading:

The 0 on the l.h.s. means the imitator is picking the action itself instead of deferring to the demonstrator,

This confused me initially to no end, and still confuses me. Should this be:

The 0 on the l.h.s. means the imitator is picking the action $a$ itself instead of deferring to the demonstrator or picking one of the other actions???

This would seem to be more consistent with the definitions that follow, and it would seem to make more ... (read more)

1michaelcohen4y

A policy outputs a distribution over {0,1}×A, and equations 3 and 4 define what this distribution is for the imitator. If it outputs (0, a), that means qt=0 and and at=1 and if it outputs (1, a), that means qt=1 and at=a. When I say that's just describing the difference between equations 3 and 4. Look at equation 4 to see that when qt=1, the distribution over the action is equal to that of the demonstrator. So we describe the behavior that follows qt=1 as "deferring to the demonstrator". If we look at the distribution over the action when qt=0, it's something else, so we say the imitator is "picking its own action". The 0 means the imitator is picking the action, and the a means it's not picking another action that's not a.

1Koen.Holtman4y

I agree with your description above about how it all works. But I guess I was not explaining well enough why I got confused and why the edits of inserting the a and the bold text above would have stopped me getting confused. So I'll try again. I read the sentence fragment below equation (3) as an explanatory claim that the value defined in equation (3) defines the probability that the imitator is picking the action itself instead of deferring to the demonstrator, the probability given the history |h<t. However, this is not the value being defined by equation (3), instead it defines the probability the imitator is picking the action itself instead of deferring to the demonstrator when the history is |h<t and the next action taken is a. The actual probability of the imitator is picking the action itself under |h<t, is given by ∑a∈Aπiα(0,a|h<t), which is only mentioned in passing in the lines between equations (3) and (4). So when I was reading the later sections in the paper and I wanted to look back at what the probability was that the imitator would pick the action, my eye landed on equation (3) and the sentence below it. When I read that sentence, it stopped me from looking further to find the equation ∑a∈Aπiα(0,a|h<t), which is the equation I was really looking for. Instead my mind auto-completed equation (3) by adding an avga term to it, which makes for a much more conservative querying policy than the one you defined, and this then got me into wondering how you were dealing with learning nondeterminstic policies, if at all, etc. So overall I think you can improve readability by doing some edits to draw attention more strongly to the conditional nature of a, and foregroundig the definition of θq more clearly as a single-line equation.

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