Embedded World-Models

This post feels quite similar to things I have written in the past to justify my lack of enthusiasm about idealizations like AIXI and logically-omniscient Bayes. But I would go further: I think that grappling with embeddedness properly will inevitably make theories of this general type irrelevant or useless, so that "a theory like this, except for embedded agents" is not a thing that we can reasonably want. To specify what I mean, I'll use this paragraph as a jumping-off point:

Embedded agents don’t have the luxury of stepping outside of the universe to think about how to think. What we would like would be a theory of rational belief for situated agents which provides foundations that are similarly as strong as the foundations Bayesianism provides for dualistic agents.

Most "theories of rational belief" I have encountered -- including Bayesianism in the sense I think is meant here -- are framed at the level of an evaluator outside the universe, and have essentially no content when we try to transfer them to individual embedded agents. This is because these theories tend to be derived in the following way:

• We want a theory of the best possible behavior for agents.
• We have some class of "practically achievable" strategies $S$, which can actually be implemented by agents. We note that an agent's observations provide some information about the quality of different strategies $s\in S$. So if it were possible to follow a rule like $R\equiv$ "find the best $s\in S$ given your observations, and then follow that $s$," this rule would spit out very good agent behavior.
• Usually we soften this to a performance-weighted average rather than a hard argmax, but the principle is the same: if we could search over all of $S$, the rule $R$ that says "do the search and then follow what it says" can be competitive with the very best $s\in S$. (Trivially so, since it has access to the best strategies, along with all the others.)
• But usually $R\notin S$. That is, the strategy "search over all practical strategies and follow the best ones" is not a practical strategy. But we argue that this is fine, since we are constructing a theory of ideal behavior. It doesn't have to be practically implementable.

For example, in Solomonoff, $S$ is defined by computability while $R$ is allowed to be uncomputable. In the LIA construction, $S$ is defined by polytime complexity while $R$ is allowed to run slower than polytime. In logically-omniscient Bayes, finite sets of hypotheses can be manipulated in a finite universe but the full Boolean algebra over hypotheses generally cannot.

I hope the framework I've just introduced helps clarify what I find unpromising about these theories. By construction, any agent you can actually design and run is a single element of $S$ (a "practical strategy"), so every fact about rationality that can be incorporated into agent design gets "hidden inside" the individual $s\in S$, and the only things you can learn from the "ideal theory" $R$ are things which can't fit into a practical strategy.

For example, suppose (reasonably) that model averaging and complexity penalties are broadly good ideas that lead to good results. But all of the model averaging and complexity penalization that can be done computably happens inside some Turing machine or other, at the level "below" Solomonoff. Thus Solomonoff only tells you about the extra advantage you can get by doing these things uncomputably. Any kind of nice Bayesian average over Turing machines that can happen computably is (of course) just another Turing machine.

This also explains why I find it misleading to say that good practical strategies constitute "approximations to" an ideal theory of this type. Of course, since $R$ just says to follow the best strategies in $S$, if you are following a very good strategy in $S$ your behavior will tend to be close to that of $R$. But this cannot be attributed to any of the searching over $S$ that $R$ does, since you are not doing a search over $S$; you are executing a single member of $S$ and ignoring the others. Any searching that can be done practically collapses down to a single practical strategy, and any that doesn't is not practical. Concretely, this talk of approximations is like saying that a very successful chess player "approximates" the rule "consult all possible chess players, then weight their moves by past performance." Yes, the skilled player will play similarly to this rule, but they are not following it, not even approximately! They are only themselves, not any other player.

Any theory of ideal rationality that wants to be a guide for embedded agents will have to be constrained in the same ways the agents are. But theories of ideal rationality usually get all of their content by going to a level above the agents they judge. So this new theory would have to be a very different sort of thing.

Epistemic Status: Attempting to bridge what I see as a missing inferential link in the post / sequence.

(This is a point which I picked up on because I am familiar with what Abram was thinking about 3 years ago, and I was surprised it didn't get mentioned. Maybe it was assumed to be obvious, maybe it's not as relevant as I assumed, but I think some others will find the point worth a bit more explaining.)

The reason we care about the relative size of the world and the model is that we have a deep reason to think that a model smaller than the world cannot perform optimally - it's the Conant-Ashby Theorem, which states "every good regulator of a system must be a model of that system." For a great explanation of this idea, there is a paper that Abram pointed me to years ago, "Every good key must be a model of the lock it opens (The Conant & Ashby Theorem Revisited)" To quote from there:

"What all of this means, more or less, is that the pursuit of a goal by some dynamic agent (Regulator) in the face of a source of obstacles (System) places at least one particular and unavoidable demand on that agent, which is that the agent's behaviors must be executed in such a reliable and predictable way that they can serve as a representation (Model) of that source of obstacles."

To lay the connection out explicitly, if the agent model of the world is not isomorphic to the world, the actions chosen will be sub-optimal. This is bad if we assume the world is not isomorphic to a simple model (and this sequence is laying out reasons that for reflexive agents, there cannot be such a computational model.)

Abram has made a major update to the post above, adding material on self-reference and the grain of truth problem. The corresponding text on the MIRI Blog version has also been expanded, with some extra material on those topics plus logical uncertainty.

• New material on paradoxes of self-reference
• Revised material on logical uncertainty

Some of these issues (obviously) are not limited to AI. Specifically, the problem of how to deal with multi-level models and "composibility" was the subject of an applied research project for military applications by my dissertation chair, Paul Davis, here: https://www.rand.org/content/dam/rand/pubs/monographs/2004/RAND_MG101.pdf -

"The appealing imagery of arbitrary plug-and-play is fatally flawed for complex models... The more-complex [lower level] model components have typically been developed for particular purposes and depend on context-sensitive assumptions, some of which are tacit."

This issue has formed the basis of a fair amount of his later work as well, but this work focuses on practical advice, rather than conceptual understanding of the limitations. Still, that type of work may be useful as inspiration.

This article talks about multi-level models, where you somehow switch between cheaper models and more-accurate models depending on your needs.  Would it be useful to generalize this idea to switching between multiple "same-level" models that are differentiated by something other than cheap vs. accurate?

For example, one might have one model that groups individual people together into "families", another that groups them into "organizations", and a third that groups them into "ideologies".  None of those models seems to be strictly "higher" than another (e.g. neither families nor ideologies are composed of each other), and different models might be useful for different problems.

One could also imagine combining all of those into one unified model, of course.  But it might be wasteful to model all of them for problems where you only really care about one.

I feel like humans do something like this.

If multiple "same-level" models can coexist, then one strategy for holding onto your values while inventing new models might be to always hold onto whichever model the values were originally defined in, even if you add more models alongside it.

I think this article is too vague, because for almost almost claims in it I am not sure if I understand the author correctly. Below I am posting my notes. If you want to help me and others clarify understanding of this article, consider answering questions in bold, or, if you see a mistake in my notes, correcting it. Also I hope my notes help the author as a piece of feedback. I've only finished 2/3 of the article so far, but posting notes because I might become less interested in this later.

Also it's unfortunate that unlike in https://intelligence.org/2018/11/02/embedded-models/ version of this article we don't have hyperlinks to explanations of various concepts here. Perhaps you could add them under the corresponding images? Or have images themselves be hyperlinks or reference links (like in academic articles) to the bottom of the document where all relevant links would be stored grouped by image number.

The post says an embedded agent can't hold an exact model of the environment in its head, can't think through the consequences of every potential course of action, can't hold in its head every possible way the environment could be. I think this may not be necessarily true and I am not sure what assumptions the author used here.

It seems the whole article assumes countable probability spaces (even before the AIXI part). I wonder why and I wonder how realizability is defined for uncountable probability space.

--

Regarding relative bounded loss and what this bound is for, my best guess is as follows. Here I use non-conditional probability notation $p\left(x\right)$ instead of $\pi \left(x\right)$. Let $e$ be the elementary event that is actually true. Let "expert" $h$ be a (not necessarily elementary) event, such that $p(e,h)>0$. Then loss of the expert is $L_h=-\log p(e\mid h)$. Loss of the prior is $L=-\log p\left(e\right)$. For their difference it holds that $L-L_h=\ln \frac{p(e,h)}{p\left(e\right)p\left(h\right)}\le -\ln p\left(h\right)$.

Initially, $\pi \left(h\right)$ is your initial trust in expert h, and in each case where it is even a little bit more correct than you, you increase your trust accordingly; the way you do this ensures you assign an expert probability 1 and hence copy it precisely before you lose more than $\log \pi \left(h\right)$ compared to it.

Remember, $p(h\mid e)=\frac{p(e\mid h)p\left(h\right)}{p\left(e\right)}$. It follows that probability of h increases given evidence e if and only if $p(e\mid h)>p\left(e\right)$, i.e. h "is even a little bit more correct than you". But I don't understand the bit about copying the expert h precisely before losing more than $\log \pi \left(h\right)$. If the expert is an event, how can you "copy" it?

Combining this with the previous idea about viewing Bayesian learning as a way of allocating “trust” to “experts” which meets a bounded loss condition, we can see the Solomonoff prior as a kind of ideal machine learning algorithm which can learn to act like any algorithm you might come up with, no matter how clever.

It is assuming all possible algorithms are computable, not that the world is.

I don't understand this. Our probability space is the cartesian product of the set of all possible UTM programs and the set of all possible UTM working tape initial configurations. Or, equivalently, the set of outputs of UTM under these conditions. Hence our whole hypothesis space only includes computable worlds. What does "can learn to act like any algorithm" mean here? "It's getting bounded loss on its predictive accuracy as compared with any computable predictor." Huh? Does predictor here mean expert h? If yes, what does it mean that h is computable and why? All in all, is the author claiming it's impossible to have a better computable predictor than AIXI with Solomonoff prior, even if it has non-computable worlds in the probability space?

probabilities may not be calibrated identification of causal structure may not work

What do these mean? I only know informally what calibration means related to forecasting.

So, does AIXI perform well without a realizability assumption?

How is AIXI even defined without realizability, i.e. when the actual world isn't in the probability space, or it has zero prior probability?

This is fine if the world “holds still” for us; but because the map is in the world, it may implement some function.

Is this about the world changing because of the agent just thinking? Or something else?

It should be noted, though, that there are additional barriers to getting this property in a game-theoretic setting; so in their common usage cases, "grain of truth" is technically demanding while "realizability" is a technical convenience.

...

In game theory, on the other hand, the assumption itself may be inconsistent. This is because games commonly yield paradoxes of self-reference.

From the former paragraph I don't understand anything except that (the author claims) game theory has more problems with grain of truth / realizability, than AIXI. After the latter paragraph, my best guess is: for any game, if there is no pure strategy equilibrium in it, then we say it has no grain of truth, because for every possible outcome rational agents wouldn't choose it.

If we put weight in both places until a proof rules one out, the beliefs just oscillate forever rather than doing anything useful.

Weights represent possible worlds, therefore they are on the scales right from the beginning (the prior), we never put new weights on the scales. My probably incorrect guess of what the author is saying is some agent which acts like AIXI but instead of updating on pieces of evidence as soon as he receives it, he stockpiles it, and at some points he (boundedly) searches for proofs that these pieces of evidence are in favor of some hypothesis and performs update only when he finds them. But still, why oscillation?

Any computable beliefs about logic must have left out something, since the tree will grow larger than any container.

I interpret it as there are infinitely many theorems, hence an agent with finite amount of space or finite amount of computation steps can't process all of them.

Another consequence of the fact that the world is bigger than you is that you need to be able to use high-level world models: models which involve things like tables and chairs.

This is related to the classical symbol grounding problem; but since we want a formal analysis which increases our trust in some system, the kind of model which interests us is somewhat different. This also relates to transparency and informed oversight: world-models should be made out of understandable parts.

No idea what the second quoted paragraph means.

All in all, I doubt that high level world models are necessary. And it's very not clear what is meant by "high level" or "things" here. Perhaps embedded agents can (boundedly) reason about the world in other ways, e.g. by modeling only part of the world.

https://intelligence.org/files/OntologicalCrises.pdf explains the ontological crisis idea better. Suppose our AIXI-like agent thinks the world is an elementary outcome of some parameterized probability distribution with the parameter θ. θ is either 1 or 2. We call the set of elementary outcomes with θ=1 the first ontology (e.g. possible worlds running on classical mechanics), and the set of elementary outcomes with θ=2 the second ontology (e.g. possible worlds running on superstrings theory). The programmer has only programmed the agent's utility functiom for θ=1 part, i.e. a u function from ontology 1 to real numbers. The agent keeps count of which value of θ is more probable and chooses actions by considering only current ontology. If at some point he decides that the second ontology is more useful, he switches to it. The agent should extrapolate the utility function to θ=2 part. How can he do it?

I can follow most of this, but i'm confused about one part of the premise.

What if the agent created a low-resolution simulation of its behavior, called it Approximate Self, and used that in its predictions? Is the idea that this is doable, but represents a unacceptably large loss of accuracy? Are we in a 'no approximation' context where any loss of accuracy is to be avoided?

My perspective: It seems to me that humans also suffer from the problem of embedded self-reference. I suspect that humans deal with this by thinking about a highly approximate representation of their own behavior. For example, when i try to predict how a future conversation will go, i imagine myself saying things that a 'reasonable person' might say. Could a machine use a analogous form of non-self-referential approximation?

Great piece, thanks for posting.

What if the agent which is a quantum mechanical intelligence CAN temporarily tunnel out of the environment long enough to make certain key observations/measurements. It could be both in the embedded environment AND out at the same time as a hyper wavefunction or in the form of its own pilot wave? Thinking as a human is a quantum mechanical process to a degree. You cannot change a system from within it is a psychological norm, however if the agent is quantum mechanical in nature then it is likely neither particle nor wave but something undeterminable by other agents. The agent might be in quantum flux indefinately n'est pas? Hence incompleteness theorem in both physics and mathematics.