The Metaphysical Structure of Pearl's Theory of Time

Epistemic status: metaphysics

I was reading Factored Space Models (previously, Finite Factored Sets) and was trying to understand in what sense it was a Theory of Time.

Scott Garrabrant says [LW · GW] "[The Pearlian Theory of Time] ... is the best thing to happen to our understanding of time since Einstein". I read Pearl's book on Causality^[1], and while there's math, this metaphysical connection that Scott seems to make isn't really explicated. Timeless Causality [LW · GW] and Timeless Physics [LW · GW] is the only place I saw this view explained explicitly, but not at the level of math / language used in Pearl's book.

Here is my attempt at explicitly writing down what all of these views are pointing at (in a more rigorous language)—the core of the Pearlian Theory of Time, and in what sense FSM shares the same structure.

Causality leave a shadow of conditional independence relationships over the observational distribution. Here's an explanation providing the core intuition:

1. Suppose you represent the ground truth structure of [causality / determination] of the world via a Structural Causal Model over some variables, a very reasonable choice. Then, as you go down the Pearlian Rung: SCM $\to$^[2] Causal Bayes Net $\to$^[3] Bayes Net, theorems guarantee that the Bayes Net is still Markovian wrt the observational distribution.
   1. (Read Timeless Causality [LW · GW] for an intuitive example.)
2. Causal Discovery then (at least in this example) reduces to inferring the equation assignment directions of the SCM, given only the observational distribution.
3. The earlier result guarantees that all you have to do is find a Bayes Net that is Markovian wrt the observational distribution. Alongside the faithfulness assumption, this thus reduces to finding a Bayes Net structure G whose set of independencies (implied by d-separation) are identical to that of P (or, finding the Perfect Map of a distribution [LW · GW]^[4]).
4. Then, at least some of the edges of the Perfect Map will have its directions nailed down by the conditional independence relations.

The metaphysical claim is that, this direction is the definition of time^[5], morally so, based on the intuition provided by the example above.

So, the Pearlian Theory of Time is the claim that Time is the partial order over the variables of a Bayes Net corresponding to the perfect map of a distribution.

Abstracting away, the structure of any Theory of Time is then to:

1. find a mathematical structure [in the Pearlian Theory of Time, a Bayes Net]
2. ... that has gadgets [d-separation]
3. ... that are, in some sense, "equivalent" [soundness & completeness] to the conditional independence relations of the distribution the structure is modeling
4. ... while containing a notion of order [parenthood relationship of nodes in a Bayes Net]
5. ... while this order induced from the gadget coinciding to that of d-separation [trivially so here, because we're talking about Bayes Nets and d-separation] such that it captures the earlier example which provided the core intuition behind our Theory of Time.

This is exactly what Factored Space Model does:

1. find a mathematical structure [Factored Space Model]
2. ... that has gadgets [structural independence]
3. ... that are, in some sense, "equivalent" [soundness & completeness] to the conditional independence relations of the distribution the structure is modeling
4. ... while containing a notion of order [preorder relation induced by the subset relationship of the History]
5. ... while this order induced from the gadget coinciding to that of d-separation [by a theorem of FSM] such that it captures the earlier example which provided the core intuition behind our Theory of Time.
6. while, additionally, generalizing the scope of our Theory of Time from [variables that appear in the Bayes Net] to [any variables defined over the factored space].

... thus justifying calling FSM a Theory of Time in the same spirit that Pearlian Causal Discovery is a Theory of Time.

1. ^{^}

   Chapter 2, specifically, which is about Causal Discovery. All the other chapters are mostly irrelevant for this purpose.

2. ^{^}

   By (1) making a graph with edge direction corresponding to equation assignment direction, (2) pushforwarding uncertainties to endogenous variables, and (3) letting interventional distributions be defined by the truncated factorization formula.

3. ^{^}

   By (1) forgetting the causal semantics, i.e. no longer associating the graph with all the interventional distributions, and only the no intervention observational distribution.

4. ^{^}

   This shortform answers this question I had.

5. ^{^}

   Pearl comes very close. In his Temporal Bias Conjecture (2.8.2):

   "In most natural phenomenon, the physical time coincides with at least one statistical time."

   (where statistical time refers to the aforementioned direction.)

   But doesn't go as far as this ought to be the definition of Time.

↑ comment by cubefox · 2024-11-14T09:39:01.894Z · LW(p) · GW(p)

This approach goes back to Hans Reichenbach's book The Direction of Time. I think the problem is that the set of independencies alone is not sufficient to determine a causal and temporal order. For example, the same independencies between three variables could be interpreted as the chains $A\to B\to C$ and $A\leftarrow B\leftarrow C$. I think Pearl talks about this issue in the last chapter.

Replies from: Darcy

↑ comment by Dalcy (Darcy) · 2024-11-14T16:42:44.055Z · LW(p) · GW(p)

The critical insight is that this is not always the case!

Let's call two graphs I-equivalent if their set of independencies (implied by d-separation) are identical. A theorem of Bayes Nets say that two graphs are I-equivalent if they have the same skeleton and the same set of immoralities.

This last constraint, plus the constraint that the graph must be acyclic, allows some arrow directions to be identified - namely, across all I-equivalent graphs that are the perfect map of a distribution, some of the edges have identical directions assigned to them.

The IC algorithm (Verma & Pearl, 1990) for finding perfect maps (hence temporal direction) is exactly about exploiting these conditions to orient as many of the edges as possible:

More intuitively, (Verma & Pearl, 1992) and (Meek, 1995) together shows that the following four rules are necessary and sufficient operations to maximally orient the graph according to the I-equivalence (+ acyclicity) constraint:

Anyone interested in further detail should consult Pearl's Causality Ch 2. Note that for some reason Ch 2 is the only chapter in the book where Pearl talks about Causal Discovery (i.e. inferring time from observational distribution) and the rest of the book is all about Causal Inference (i.e. inferring causal effect from (partially) known causal structure).

Replies from: cubefox

↑ comment by cubefox · 2024-11-14T18:09:14.121Z · LW(p) · GW(p)

Ah yes, the fork asymmetry. I think Pearl believes that correlations reduce to causations, so this is probably why he wouldn't particularly try to, conversely, reduce causal structure to a set of (in)dependencies. I'm not sure whether the latter reduction is ultimately possible in the universe. Are the correlations present in the universe, e.g. defined via the Albert/Loewer Mentaculus probability distribution, sufficient to recover the familiar causal structure of the universe?

Thoughtdump on why I'm interested in computational mechanics:

• one concrete application to natural abstractions from here [LW · GW]: tl;dr, belief structures generally seem to be fractal shaped. one major part of natural abstractions is trying to find the correspondence between structures in the environment and concepts used by the mind. so if we can do the inverse of what adam and paul did, i.e. 'discover' fractal structures from activations and figure out what stochastic process they might correspond to in the environment, that would be cool
  • ... but i was initially interested in reading compmech stuff not with a particular alignment relevant thread in mind but rather because it seemed broadly similar in directions to natural abstractions.
• re: how my focus would differ from my impression of current compmech work done in academia: academia seems faaaaaar less focused on actually trying out epsilon reconstruction in real world noisy data. CSSR is an example of a reconstruction algorithm. apparently people did compmech stuff on real-world data, don't know how good, but effort-wise far too less invested compared to theory work
  • would be interested in these reconstruction algorithms, eg what are the bottlenecks to scaling them up, etc.
• tangent: epsilon transducers seem cool. if the reconstruction algorithm is good, a prototypical example i'm thinking of is something like: pick some input-output region within a model, and literally try to discover the hmm model reconstructing it? of course it's gonna be unwieldly large. but, to shift the thread in the direction of bright-eyed theorizing ...
• the foundational Calculi of Emergence paper talked about the possibility of hierarchical epsilon machines, where you do epsilon machines on top of epsilon machines and for simple examples where you can analytically do this, you get wild things like coming up with more and more compact representations of stochastic processes (eg data stream -> tree -> markov model -> stack automata -> ... ?)
  • this ... sounds like natural abstractions in its wildest dreams? literally point at some raw datastream and automatically build hierarchical abstractions that get more compact as you go up
  • haha but alas, (almost) no development afaik since the original paper. seems cool
• and also more tangentially, compmech seemed to have a lot to talk about providing interesting semantics to various information measures aka True Names [LW · GW], so another angle i was interested in was to learn about them.
  • eg crutchfield talks a lot about developing a right notion of information flow - obvious usefulness in eg formalizing boundaries?
  • many other information measures from compmech with suggestive semantics—cryptic order? gauge information? synchronization order? check ruro1 and ruro2 for more.

Epistemic status: literal shower thoughts, perhaps obvious in retrospect, but was a small insight to me.

I’ve been thinking about: “what proof strategies could prove structural selection theorems [LW · GW], and not just behavioral selection theorems [LW · GW]?”

Typical examples of selection theorems in my mind are: coherence theorems, good regulator theorem, causal good regulator theorem.

• Coherence theorem [? · GW]: Given an agent satisfying some axioms, we can observe their behavior in various conditions and construct $U$, and then the agent’s behavior is equivalent to a system that is maximizing $U$.
  • Says nothing about whether the agent internally constructs $U$ and uses them.
• (Little Less Silly version of the) Good regulator theorem [LW · GW]: A regulator $R$ that minimizes the entropy of a system variable $S$ (where there is an environment variable $X$ upstream of both $R$ and $S$) without unnecessary noise (hence deterministic) is behaviorally equivalent to a deterministic function of $S$ (despite being a function of $X$).
  • Says nothing about whether $R$ actually internally reconstructs $S$ and uses it to produce its output.
• Causal good regulator theorem (summary [LW(p) · GW(p)]): Given an agent achieving low regret across various environment perturbations, we can observe their behavior in specific perturbed-environments, and construct $G^{\prime }$ that is very similar to the true environment $G$. Then argue: “hence the agent must have something internally isomorphic to $G$”. Which is true, but …
  • says nothing about whether the agent actually uses those internal isomorphic-to-$G$ structures in the causal history of computing its output.

And I got stuck here wondering, man, how do I ever prove anything structural.

Then I considered some theorems that, if you squint really really hard, could also be framed in the selection theorem language in a very broad sense:

• SLT [? · GW]: Systems selected to get low loss are likely to be in a degenerate part of the loss landscape.^[1]
  • Says something about structure: by assuming the system to be a parameterized statistical model, it says the parameters satisfy certain conditions like degeneracy (which further implies e.g., modularity [LW · GW]).

This made me realize that to prove selection theorems on structural properties of agents, you should obviously give more mathematical structure to the “agent” in the first place:

• SLT represents a system as a parameterized function - very rich!
• In coherence theorem, the agent is just a single node that outputs decision given lotteries. In the good regulator theorem and the causal good regulator theorem, the agent is literally just a single node in a Bayes Net - very impoverished!

And recall, we actually have an agent foundations style selection theorem that does prove something structural about agent internals by giving more mathematical structure to the agent:

• Gooder regulator theorem [LW · GW]: A regulator is now two nodes instead of one, but the latter-in-time node gets an additional information about the choice of “game” it is being played against (thus the former node acts as a sort of information bottleneck). Then, given that the regulator makes $S$ take minimum entropy, the first node must be isomorphic to the likelihood function $s\mapsto P(S=s|X)$.
  • This does say something about structure, namely that an agent (satisfying certain conditions) with an internal information bottleneck (structural assumption) must have that bottleneck be behaviorally equivalent to a likelihood function, whose output is then connected to the second node. Thus it is valid to claim that (under our structural assumption) the agent internally reconstructs the likelihood values and uses it in its computation of the output.

So in short, we need more initial structure or even assumptions on our “agent,” at least more so than literally a single node in a Bayes Net, to expect to be able to prove something structural.

Here is my 5-minute attempt to put more such "structure" to the [agent/decision node] in the Causal good regulator theorem with the hopes that this would make the theorem more structural, and perhaps end up as a formalization of the Agent-like Structure Problem [LW · GW] (for World Models, at least), or very similarly the Approximate Causal Mirror hypothesis [LW · GW]:

• Similar setup to the Causal good regulator theorem, but instead of a single node representing an agent’s decision node, assume that the agent as a whole is represented by an unknown causal graph $G$, with a number of nodes designated as input and output, connected to the rest-of-the-world causal graph $E$. Then claim: Agents with low regret must have $G$ that admits an abstracting causal model map (summary [LW · GW]) from $E$, and (maybe more structural properties such as) the approximation error should roughly be lowest around the input/output & utility nodes, and increase as you move further away from it in the low-level graph. This would be a very structural claim!

1. ^{^}

   I'm being very very [imprecise/almost misleading] here—because I'm just trying to make a high-level point and the details don't matter too much—one of the caveats (among many) being that this statement makes the theoretically yet unjustified connection between SGD and Bayes.

Just read through Robust agents learn causal world models and man it is really cool! It proves a couple of bona fide selection theorems [? · GW], talking about the internal structure of agents selected against a certain criteria.

• Tl;dr, agents selected to perform robustly in various local interventional distributions must internally represent something isomorphic to a causal model of the variables upstream of utility, for it is capable of answering all causal queries for those variables.
  • Thm 1: agents achieving optimal policy (util max) across various local interventions must be able to answer causal queries for all variables upstream of the utility node
  • Thm 2: relaxation of above to nonoptimal policies, relating regret bounds to the accuracy of the reconstructed causal model
  • the proof is constructive - an algorithm that, when given access to regret-bounded-policy-oracle wrt an environment with some local intervention, queries them appropriately to construct a causal model
    • one implication is an algorithm for causal inference that converts black box agents to explicit causal models (because, y’know, agents like you and i are literally that aforementioned ‘regret-bounded-policy-oracle‘)
  • These selection theorems could be considered the converse of the well-known statement that given access to a causal model, one can find an optimal policy. (this and its relaxation to approximate causal models is stated in Thm 3)
• Thm 1 / 2 is like a ‘causal good regulator‘ theorem.
  • gooder regulator theorem [LW · GW] is not structural - as in, it gives conditions under which a model of the regulator must be isomorphic to the posterior of the system - a black box statement about the input-output behavior.
• theorem is limited. only applies to cases where the decision node is not upstream of the environment nodes (eg classification. a negative example would be an mdp). but authors claim this is mostly for simpler proofs and they think this can be relaxed.

Quick paper review of Measuring Goal-Directedness from the causal incentives group.

tl;dr, goal directedness of a policy wrt a utility function is measured by its min distance to one of the policies implied by the utility function, as per the intentional stance - that one should model a system as an agent insofar as doing so is useful.

Details

• how is "policies implied by the utility function" operationalized? given a value $u$, we define a set containing policies of maximum entropy (of the decision variable, given its parents in the causal bayes net) among those policies that attain the utility $u$.
• then union them over all the achievable values of $u$ to get this "wide set of maxent policies," and define goal directedness of a policy $\pi$ wrt a utility function $U$ as the maximum (negative) cross entropy between $\pi$ and an element of the above set. (actually we get the same result if we quantify the min operation over just the set of maxent policies achieving the same utility as $\pi$.)

Intuition

intuitively, this is measuring: "how close is my policy $\pi$ to being 'deterministic,' while 'optimizing $U$ at the competence level $u\left(\pi \right)$' and not doing anything else 'deliberately'?"

• "close" / "deterministic" ~ large negative $CE$ means small $CE(\pi ,{\pi }_{\text{maxent}})=H\left(\pi \right)+KL\left(\pi ||{\pi }_{\text{maxent}}\right)$
• "not doing anything else deliberately'" ~ because we're quantifying over maxent policies. the policy is maximally uninformative/uncertain, the policy doesn't take any 'deliberate' i.e. low entropy action, etc.
• "at the competence level $u\left(\pi \right)$" ~ ... under the constraint that it is identically competent to $\pi$

and you get the nice property of the measure being invariant to translation / scaling of $U$.

• obviously so, because a policy is maxent among all policies achieving $u$ on $U$ iff that same policy is maxent among all policies achieving $au+b$ on $aU+b$, so these two utilities have the same "wide set of maxent policies."

Critiques

I find this measure problematic in many places, and am confused whether this is conceptually correct.

• one property claimed is that the measure is maximum for uniquely optimal / anti-optimal policy.
  • it's interesting that this measure of goal-directedness isn't exactly an ~increasing function of $u\left(\pi \right)$, and i think it makes sense. i want my measure of goal-directedness to, when evaluated relative to human values, return a large number for both aligned ASI and signflip ASI.
• ... except, going through the proof one finds that the latter property heavily relies on the "uniqueness" of the policy.
  • My policy can get the maximum goal-directedness measure if it is the only policy of its competence level while being very deterministic. It isn't clear that this always holds for the optimal/anti-optimal policies or always relaxes smoothly to epsilon-optimal/anti-optimal policies.
• Relatedly, the fact that the quantification is only happening over policies of the same competence level, which feels problematic.
• minimum for uniformly random policy (this would've been a good property, but unless I'm mistaken I think the proof for the lower bound is incorrect, because negative cross entropy is not bounded below.)
• honestly the maxent motivation isn't super clear to me.
• not causal. the reason you need causal interventions is because you want to rule out accidental agency/goal-directedness, like a rock that happens to be the perfect size to seal a water bottle - does your rock adapt when I intervene to change the size of the hole? discovering agents is excellent in this regards.

↑ comment by mattmacdermott · 2024-08-25T21:01:11.514Z · LW(p) · GW(p)

Thanks for the feedback!

... except, going through the proof one finds that the latter property heavily relies on the "uniqueness" of the policy. My policy can get the maximum goal-directedness measure if it is the only policy of its competence level while being very deterministic. It isn't clear that this always holds for the optimal/anti-optimal policies or always relaxes smoothly to epsilon-optimal/anti-optimal policies.

Yeah, uniqueness definitely doesn't always hold for the optimal/anti-optimal policy. I think the way MEG works here makes sense: if you're following the unique optimal policy for some utility function, that's a lot of evidence for goal-directedness. If you're following one of many optimal policies, that's a bit less evidence -- there's a greater chance that it's an accident. In the most extreme case (for the constant utility function) every policy is optimal -- and we definitely don't want to ascribe maximum goal-directedness to optimal policies there.

With regard to relaxing smoothly to epsilon-optimal/anti-optimal policies, from memory I think we do have the property that MEG is increasing in the utility of the policy for policies with greater than the utility of the uniform policy, and decreasing for policies with less than the utility of the uniform policy. I think you can prove this via the property that the set of maxent policies is (very nearly) just Boltzman policies with varying temperature. But I would have to sit down and think about it properly. I should probably add that to the paper if it's the case.

minimum for uniformly random policy (this would've been a good property, but unless I'm mistaken I think the proof for the lower bound is incorrect, because negative cross entropy is not bounded below.)

Thanks for this. The proof is indeed nonsense, but I think the proposition is still true. I've corrected it to this.

↑ comment by habryka (habryka4) · 2024-07-30T20:36:50.698Z · LW(p) · GW(p)

Quick paper review of Measuring Goal-Directedness from the causal incentives group.

This link doesn't work for me: 

Replies from: Darcy

↑ comment by Dalcy (Darcy) · 2024-07-31T03:15:55.078Z · LW(p) · GW(p)

Thanks, it seems like the link got updated. Fixed!

EDIT: I no longer think this setup is viable, for reasons that connect to why I think Critch's operationalization is incomplete and why boundaries should ultimately be grounded in Pearlian Causality and interventions. Check update [LW(p) · GW(p)].

I believe there's nothing much in the way of actually implementing an approximation of Critch's boundaries [? · GW]^[1] using deep learning.

Recall, Critch's boundaries are:

• Given a world (markovian stochastic process) $W_t$, map its values $W$ (vector) bijectively using $f$ into 'features' that can be split into four vectors each representing a boundary-possessing system's Viscera, Active Boundary, Passive Boundary, and Environment.
• Then, we characterize boundary-ness (i.e. minimal information flow across features unmediated by a boundary) using two mutual information criterion each representing infiltration and exfiltration of information.
• And a policy of the boundary-posessing system (under the 'stance' of viewing the world implied by $f$) can be viewed as a stochastic map (that has no infiltration/exfiltration by definition) that best approximates the true $W_t$ dynamics.
  • The interpretation here (under low exfiltration and infiltration) is that $f$ can be viewed as a policy taken by the system in order to perpetuate its boundary-ness into the future and continue being well-described as a boundary-posessing system.

All of this seems easily implementable using very basic techniques from deep learning!

• Bijective feature map are implemented using two NN maps each way, with an autoencoder loss.
• Mutual information is approximated with standard variational approximations. Optimize $f$ to minimize it.
  • (the interpretation here being - we're optimizing our 'stance' towards the world in a way that best views the world as a boundary-possessing system)
• After you train your 'stance' using the above setup, learn the policy using an NN with standard SGD, with fixed $f$.

A very basic experiment would look something like:

• Test the above setup on two cellular automata (e.g., GoL, Lenia, etc) systems, one containing just random ash, and the other some boundary-like structure like noise-resistant glider structures found via optimization (there are a lot of such examples in the Lenia literature).^[2]
• Then (1) check if the infiltration/exfiltration values are lower for the latter system, and (2) do some interp to see if the V/A/P/E features or the learned policy NN have any interesting structures.

I'm not sure if I'd be working on this any time soon, but posting the idea here just in case people have feedback.

1. ^{^}

   I think research on boundaries - both conceptual work and developing practical algorithms for approximating them & schemes involving them - are quite important for alignment for reasons discussed earlier in my shortform [LW(p) · GW(p)].

2. ^{^}

   Ultimately we want our setup to detect boundaries that aren't just physically contiguous chunks of matter, like informational boundaries, so we want to make sure our algorithm isn't just always exploiting basic locality heuristics.

   I can't think of a good toy testbed (ideas appreciated!), but one easy thing to try is to just destroy all locality by mapping the automata lattice (which we were feeding as input) with the output of a complicated fixed bijective map over it, so that our system will have to learn locality if it turns out to be a useful notion in its attempt at viewing the system as a boundary.

Perhaps I should one day in the far far future write a sequence on bayes nets.

Some low-effort TOC (this is basically mostly koller & friedman):

• why bayesnets and markovnets? factorized cognition, how to do efficient bayesian updates in practice, it's how our brain is probably organized, etc. why would anyone want to study this subject if they're doing alignment research? explain philosophy behind them.
• simple examples of bayes nets. basic factorization theorems (the I-map stuff and separation criterion)
• tangent on why bayes nets aren't causal nets, though Zack M Davis had a good post [LW · GW] on this exact topic, comment threads there are high insight
• how inference is basically marginalization (basic theorems of: a reduced markov net represents conditioning, thus inference upon conditioning is the same as marginalization on a reduced net)
• why is marginalization hard? i.e. NP-completeness of exact and approximate inference worst-case
  what is a workaround? solve by hand simple cases in which inference can be greatly simplified by just shuffling in the order of sums and products, and realize that the exponential blowup of complexity is dependent on a graphical property of your bayesnet called the treewidth
• exact inference algorithms (bounded by treewidth) that can exploit the graph structure and do inference efficiently: sum-product / belief-propagation
• approximate inference algorithms (works in even high treewidth! no guarantee of convergence) - loopy belief propagation, variational methods, etc
• connections to neuroscience: "the human brain is just doing belief propagation over a bayes net whose variables are the cortical column" or smth, i just know that there is some connection

Formalizing selection theorems [? · GW] for abstractability

Tl;dr, Systems are abstractable to the extent they admit an abstracting causal model map with low approximation error. This should yield a pareto frontier of high-level causal models consisting of different tradeoffs between complexity and approximation error. Then try to prove a selection theorem for abstractability / modularity [? · GW] by relating the form of this curve and a proposed selection criteria.

Recall [LW · GW], an abstracting causal model (ACM)—exact transformations, $\tau$-abstractions, and approximations—is a map between two structural causal models satisfying certain requirements that lets us reasonably say one is an abstraction, or a high-level causal model of another.

• Broadly speaking, the condition is a sort of causal consistency requirement. It's a commuting diagram [LW · GW] that requires the "high-level" interventions to be consistent with various "low-level" ways of implementing that intervention. Approximation errors talk about how well the diagram commutes (given that the support of the variables in the high-level causal model is equipped with some metric)

Now consider a curve: x-axis is the node count, and y-axis is the minimum approximation error of ACMs of the original system with that node count (subject to some conditions^[1]). It would hopefully an decreasing one^[2].

• This curve would represent the abstractability of a system. Lower the curve, the more abstractable it is.
• Aside: we may conjecture that natural systems will have discrete jumps, corresponding to natural modules. The intuition being that, eg if we have a physics model of two groups of humans interacting, in some sense 2 nodes (each node representing the human-group) and 4 nodes (each node representing the individual-human) are the most natural, and 3 nodes aren't (perhaps the 2 node system with a degenerate node doing ~nothing, so it would have very similar approximation scores with the 2 node case).

Then, try hard to prove a selection theorem of the following form: given low-level causal model satisfying certain criteria (eg low regret over varying objectives [LW · GW], connection costs [LW · GW]), the abstractability curve gets pushed further downwards. Or conversely, find conditions that make this true.

I don't know how to prove this^[3], but at least this gets closer to a well-defined mathematical problem.

1. ^{^}

   I've been thinking about this for an hour now and finding the right definition here seems a bit non-trivial. Obviously there's going to be an ACM of zero approximation error for any node count, just have a single node that is the joint of all the low-level nodes. Then the support would be massive, so a constraint on it may be appropriate.

   Or instead we could fold it in to the x-axis—if there is perhaps a non ad-hoc, natural complexity measure for Bayes Nets that capture [high node counts => high complexity because each nodes represent stable causal mechanisms of the system, aka modules] and [high support size => high complexity because we don't want modules that are "contrived" in some sense] as special cases, then we could use this as the x-axis instead of just node count.

   Immediate answer: Restrict this whole setup into a prediction setting so that we can do model selection. Require on top of causal consistency that both the low-level and high-level causal model have a single node whose predictive distribution are similar. Now we can talk about eg the RLCT of a Bayes Net. I don't know if this makes sense. Need to think more.

2. ^{^}

   Or rather, find the appropriate setup to make this a decreasing curve.

3. ^{^}

   I suspect closely studying the robust agents learn causal world models paper would be fruitful, since they also prove a selection theorem over causal models. Their strategy is to (1) develop an algorithm that queries an agent with low regret to construct a causal model, (2) prove that this yields an approximately correct causal model of the data generating model, (3) then arguing that this implies the agent must internally represent something isomorphic to a causal world model.

Any thoughts on how to customize LessWrong to make it LessAddictive? I just really, really like the editor for various reasons, so I usually write a bunch (drafts, research notes, study notes, etc) using it but it's quite easy to get distracted.

moments of microscopic fun encountered while studying/researching:

• Quantum mechanics call vector space & its dual bra/ket because ... bra-c-ket. What can I say? I like it - But where did the letter 'c' go, Dirac?
• Defining cauchy sequences and limits in real analysis: it's really cool how you "bootstrap" the definition of Cauchy sequences / limit on real using the definition of Cauchy sequences / limit on rationals. basically:
  • (1) define Cauchy sequence on rationals
  • (2) use it to define limit (on rationals) using rational-Cauchy
  • (3) use it to define reals
  • (4) use it to define Cauchy sequence on reals
  • (5) show it's consistent with Cauchy sequence on rationals in both directions
    • a. rationals are embedded in reals hence the real-Cauchy definition subsumes rational-Cauchy definition
    • b. you can always find a rational number smaller than a given real number hence a sequence being rational-Cauchy means it is also real-Cauchy)
  • (6) define limit (on reals)
  • (7) show it's consistent with limit on rationals
  • (8) ... and that they're equivalent to real-Cauchy
  • (9) proceed to ignore the distinction b/w real-Cauchy/limit and their rational counterpart. Slick!

(will probably keep updating this in the replies)

Any advice on reducing neck and shoulder pain while studying? For me that's my biggest blocker to being able to focus longer (especially for math, where I have to look down at my notes/book for a long period of time). I'm considering stuff like getting a standing desk or doing regular back/shoulder exercises. Would like to hear what everyone else's setups are.

(Quality: Low, only read when you have nothing better to do—also not much citing)

30-minute high-LLM-temp stream-of-consciousness on "How do we make mechanistic interpretability work for non-transformers, or just any architectures?"

• We want a general way to reverse engineer circuits
  • e.g., Should be able to rediscover properties we discovered from transformers
• Concrete Example: we spent a bunch of effort reverse engineering transformer-type architectures—then boom, suddenly some parallel-GPU-friendly-LSTM architecutre turns out to have better scaling properties, and everyone starts using it. LSTMs have different inductive biases, like things in the same layer being able to communicate multiple times with each other (unlike transformers), which incentivizes e.g., reusing components (more search-y?).
• Formalize:
  • You have task X. You train a model A with inductive bias I_A. You also train a model B with inductive bias I_B. Your mechanistic interpretability techniques work well on deciphering A, but not B. You want your mechanistic interpretability techniques to work well for B, too.
• Proposal: Communication channel
  • Train a Transformer on task X
    • Existing Mechanistic interpretability work does well on interpreting this architecture
  • Somehow stitch the LSTM to the transformer (?)
    • I'm trying to get at to the idea of "interface conversion," that by the virtue of SGD being greedy, it will try to convert the outputs of transformer-friendly types
  • Now you can better understand the intermediate outputs of the LSTM by just running mechanistic interpretability on the transformer layers whose input are from the LSTM
  • (I don't know if I'm making any sense here, my LLM temp is > 1)
• Proposal: approximation via large models?
  • Train a larger transformer architecture to approximate the smaller LSTM model (either just input output pairs, or intermediate features, or intermediate features across multiple time-steps, etc):
    • the basic idea is that a smaller model would be more subject to following its natural gradient shaped by the inductive bias, while larger model (with direct access to the intermediate outputs of the smaller model) would be able to approximate it despite not having as much inductive bias incentive towards it.
      • probably false but illustrative example: Train small LSTM on chess. By the virtue of being able to run serial computation on same layers, it focuses on algorithms that have repeating modular parts. In contrast, a small Transformer would learn algorithms that don't have such repeating modular parts. But instead, train a large transformer to "approximate" the small LSTM—it should be able to do so by, e.g., inefficiently having identical modules across multiple layers. Now use mechanistic interpretability on that.
• Proposal: redirect GPS?
  • Thane's value formation picture says GPS should be incentivized to reverse-engineer the heuristics because it has access to inter-heuristic communication channel. Maybe, in the middle of training, gradually swap different parts of the model with those that have different inductive biases, see GPS gradually learn to reverse-engineer those, and mechanistically-interpret how GPS exactly does that, and reimplement in human code?
• Proposal: Interpretability techniques based on behavioral constraints
  • e.g., Discovering Latent Knowledge without Supervision, putting constraints?
• How to do we "back out" inductive biases, just given e.g., architecture, training setup? What is the type signature?
  • (I need to read more literature)

Discovering agents provide a genuine causal, interventionist account of agency and an algorithm to detect them, motivated by the intentional stance. I find this paper very enlightening from a conceptual perspective!

I've tried to think of problems that needed to be solved before we can actually implement this on real systems - both conceptual and practical - on approximate order of importance.

• There are no 'dynamics,' no learning. As soon as a mechanism node is edited, it is assumed that agents immediately change their 'object decision variable' (a conditional probability distribution given its object parent nodes) to play the subgame equilibria.
• Assumption of factorization of variables into 'object' / 'mechanisms,' and the resulting subjectivity. The paper models the process by which an agent adapts its policy given changes in the mechanism of the environment via a 'mechanism decision variable' (that depends on its mechanism parent nodes), which modulates the conditional probability distribution of its child 'object decision variable', the actual policy.
  • For example, the paper says a learned RL policy isn't an agent, because interventions in the environment won't make it change its already-learned policy - but that a human or a RL policy together with its training process is an agent, because it can adapt. Is this reasonable?
    1. Say I have a gridworld RL policy that's learned to get cheese (3 cell world, cheese always on left) by always going to the left. Clearly it can't change its policy when I change the cheese distribution to favor right, so it seems right to call this not an agent.
    2. Now, say the policy now has sensory access to the grid state, and correctly generalized (despite only being trained on left-cheese) to move in the direction where it sees the cheese, so when I change the cheese distribution, it adapts accordingly. I think it is right to call this an agent?
    3. Now, say the policy is an LLM agent (static weight) on an open world simulation which reasons in-context. I just changed the mechanism of the simulation by lowering the gravity constant, and the agent observes this, reasons in-context, and adapts its sensorimotor policy accordingly. This is clearly an agent?
  • I think this is because the paper considers, in the case of the RL policy alone, the 'object policy' to be the policy of the trained neural network (whose induced policy distribution is definitionally fixed), and the 'mechanism policy' to be a trivial delta function assigning the already-trained object policy. And in the case of the RL policy together with its training process, the 'mechanism policy' is now defined as the training process that assigns the fully-trained conditional probability distribution to the object policy.
  • But what if the 'mechanism policy' was the in-context learning process by which it induces an 'object policy'? Then changes in the environment's mechanism can be related to the 'mechanism policy' and thus the 'object policy' via in-context learning as in the second and third example, making them count as agents.
  • Ultimately, the setup in the paper forces us to factorize the means-by-which-policies-adapt into mechanism vs object variables, and the results (like whether a system is to be considered an agent) depends on this factorization. It's not always clear what the right factorization is, how to discover them from data, or if this is the right frame to think about the problem at all.
• Implicit choice of variables that are convenient for agent discovery. The paper does mention that the algorithm is dependent in the choice of the variable, as in: if the node corresponding to the 'actual agent decision' is missing but its children is there, then the algorithm will label its children to be the decision nodes. But this is already a very convenient representation!
  • Prototypical example: Minecraft world with RL agents interacting represented as a coarse-grained lattice (dynamical Bayes Net?) with each node corresponding to a physical location and its property, like color. Clearly no single node here is an agent, because agents move [LW · GW]! My naive guess is that in principle, everything will be labeled an agent.
  • So the variables of choice must be abstract variables of the underlying substrate, like functions over them. But then, how do you discover the right representation automatically, in a way that interventions in the abstract variable level can faithfully translate to actually performable interventions in the underlying substrate?
• Given the causal graph, even the slightest satisfaction of the agency-criterion labels the nodes as decision / utility. No "degree-of-agency" - maybe by summing over the extent to which the independencies fail to satisfy?
• Then different agents are defined as causally separated chunks (~connected component) of [set-of-decision-nodes / set-of-utility-nodes]. How do we accommodate hierarchical agency (like subagents), systems with different degrees of agency, etc?
• The interventional distribution on the object/mechanism variables are converted into a causal graph using the obvious [perform-do()-while-fixing-everything-else] algorithm. My impression is that causal discovery doesn't really work in practice, especially in noisy reality with a large number of variables via gazillion conditional independence tests.
• The correctness proof requires lots of unrealistic assumptions, e.g., agents always play subgame equilibria, though I think some of this can be relaxed.

I am curious as to how often the asymptotic results proven using features of the problem that seem basically practically-irrelevant become relevant in practice.

Like, I understand that there are many asymptotic results (e.g., free energy principle in SLT) that are useful in practice, but i feel like there's something sus about similar results from information theory or complexity theory where the way in which they prove certain bounds (or inclusion relationship, for complexity theory) seem totally detached from practicality?

• joint source coding theorem is often stated as why we can consider the problem of compression and redundancy separately, but when you actually look at the proof it only talks about possibility (which is proven in terms of insanely long codes) and thus not-at-all trivial that this equivalence is something that holds in the context of practical code-engineering
• complexity theory talks about stuff like quantifying some property over all possible boolean circuits of a given size which seems to me considering a feature of the problem just so utterly irrelevant to real programs that I'm suspicious it can say meaningful things about stuff we see in practice
  • as an aside, does the P vs NP distinction even matter in practice? we just ... seem to have very good approximation to NP problems by algorithms that take into account the structures specific to the problem and domains where we want things to be fast; and as long as complexity methods doesn't take into account those fine structures that are specific to a problem, i don't see how it would characterize such well-approximated problems using complexity classes.
  • Wigderson's book had a short section on average complexity which I hoped would be this kind of a result, and I'm unimpressed (the problem doesn't sound easier - now how do you specify the natural distribution??)

I recently learned about metauni, and it looks amazing. TL;DR, a bunch of researchers give out lectures or seminars on Roblox - Topics include AI alignment/policy, Natural Abstractions, Topos Theory, Singular Learning Theory, etc.

I haven't actually participated in any of their live events yet and only watched their videos, but they all look really interesting. I'm somewhat surprised that there hasn't been much discussion about this on LW!

Complaint with Pugh's real analysis textbook: He doesn't even define the limit of a function properly?!

It's implicitly defined together with the definition of continuity where $\forall ϵ>0\exists \delta >0|x-x_0|<\delta ⟹|f\left(x\right)-f\left(x_0\right)|<ϵ$, but in Chapter 3 when defining differentiability he implicitly switches the condition to $0<|x-x_0|<\delta$ without even mentioning it (nor the requirement that $x_0$ now needs to be an accumulation point!) While Pugh has its own benefits, coming from Terry Tao's analysis textbook background, this is absurd!

(though to be fair Terry Tao has the exact same issue in Book 2, where his definition of function continuity via limit in metric space precedes that of defining limit in general ... the only redeeming factor is that it's defined rigorously in Book 1, in the limited context of $R$)

*sigh* I guess we're still pretty far from reaching the Pareto Frontier of textbook quality, at least in real analysis.

... Speaking of Pareto Frontiers, would anyone say there is such a textbook that is close to that frontier, at least in a different subject? Would love to read one of those.

I used to try out near-random search on ideaspace, where I made a quick app that spat out 3~5 random words from a dictionary of interesting words/concepts that I curated, and I spent 5 minutes every day thinking very hard on whether anything interesting came out of those combinations.

Of course I knew random search on exponential space was futile, but I got a couple cool invention ideas (most of which turned out to already exist), like:

• infinite indoor rockclimbing: attach rocks to a vertical treadmill, and now you have an infinite indoor rock climbing wall (which is also safe from falling)! maybe add some fancy mechanism to add variations to the rocks + a VR headgear, I guess.
• clever crypto mechanism design (in the spirit of CO2 Coin) to incentivize crowdsourcing of age-reduction molecule design animal trials from the public. (I know what you're thinking)

You can probably do this smarter now if you wanted, with eg better GPT models.

Having lived ~19 years, I can distinctly remember around 5~6 times when I explicitly noticed myself experiencing totally new qualia with my inner monologue going “oh wow! I didn't know this dimension of qualia was a thing.” examples:

• hard-to-explain sense that my mind is expanding horizontally with fractal cube-like structures (think bismuth) forming around it and my subjective experience gliding along its surface which lasted for ~5 minutes after taking zolpidem for the first time to sleep (2 days ago)
• getting drunk for the first time (half a year ago)
• feeling absolutely euphoric after having a cool math insight (a year ago)
• ...

Reminds me of myself around a decade ago, completely incapable of understanding why my uncle smoked, being "huh? The smoke isn't even sweet, why would you want to do that?" Now that I have [addiction-to-X] as a clear dimension of qualia/experience solidified in myself, I can better model their subjective experiences although I've never smoked myself. Reminds me of the SSC classic.

Also one observation is that it feels like the rate at which I acquire these is getting faster, probably because of increase in self-awareness + increased option space as I reach adulthood (like being able to drink).

Anyways, I think it’s really cool, and can’t wait for more.

To me, the fact that the human brain basically implements SSL+RL [? · GW] is very very strong evidence that the current DL paradigm (with a bit of "engineering" effort, but nothing like fundamental breakthroughs) will kinda just keep scaling until we reach point-of-no-return. Does this broadly look correct to people here? Would really appreciate other perspectives.

I wonder if the following is possible to study textbooks more efficiently using LLMs:

• Feed the entire textbook to the LLM and produce a list of summaries that increases in granularity and length, covering all the material in the textbook just at a different depth (eg proofs omitted, further elaboration on high-level perspectives, etc)
• The student starts from the highest-level summary, and gradually moves to the more granular materials.

When I study textbooks, I spend a significant amount of time improving my mental autocompletion, like being able to familiarize myself with the terminologies, which words or proof-style usually come in which context, etc. Doing this seems to significantly improve my ability to read eg long proofs, since I can ignore all the pesky details (which I can trust my mental autocompletion to later fill in the details if needed) and allocate my effort in getting a high-level view of the proof.

Textbooks don't really admit this style of learning, because the students don't have prior knowledge of all the concept-dependencies of a new subject they're learning, and thus are forced to start at the lowest-level and make their way up to the high-level perspective.

Perhaps LLMs will let us reverse this direction, instead going from the highest to the lowest.

What's a good technical introduction to Decision Theory and Game Theory for alignment researchers? I'm guessing standard undergrad textbooks don't include, say, content about logical decision theory. I've mostly been reading posts on LW but as with most stuff here they feel more like self-contained blog posts (rather than textbooks that build on top of a common context) so I was wondering if there was anything like a canonical resource providing a unified technical / math-y perspective on the whole subject.

i absolutely hate bureaucracy, dumb forms, stupid websites etc. like, I almost had a literal breakdown trying to install Minecraft recently (and eventually failed). God.

God, I wish real analysis was at least half as elegant as any other math subject — way too much pathological examples that I can't care less about. I've heard some good things about constructivism though, hopefully analysis is done better there.

There were various notions/frames of optimization floating around, and I tried my best to distill them:

• Eliezer's Measuring Optimization Power [LW · GW] on unlikelihood of outcome + agent preference ordering
• Alex Flint's The ground of optimization [LW · GW] on robustness of system-as-a-whole evolution
• Selection vs Control [LW · GW] as distinguishing different types of "space of possibilities"
  • Selection as having that space explicitly given & selectable numerous times by the agent
  • Control as having that space only given in terms of counterfactuals, and the agent can access it only once.
  • These distinctions correlate with the type of algorithm being used & its internal structure, where Selection uses more search-like process using maps, while Control may just use explicit formula ... although it may very well use internal maps to Select on counterfactual outcomes!
    • In other words, the Selection vs Control may very well be viewed as a different cluster of Analysis. Example:
      • If we decide to focus our Analysis of "space of possibilities" on eg "Real life outcome," then a guided missile is always Control.
      • But if we decide to focus on "space of internal representation of possibilities," then a guided missle that uses internal map to search on becomes Selection.
• "Internal Optimization" vs "External Optimization" [LW(p) · GW(p)]
  • Similar to Selection vs Control, but the analysis focuses more on internal structure:
    • Why? Motivated by the fact that, as with the guided missile example, Control systems can be viewed as Selection systems depending on perspective
    • ... hence, better to focus on internal structures where it's much less ambiguous.
  • IO: Internal search + selection
  • EO: Flint's definition of "optimizing system"
    • IO is included in EO, if we assume accurate map-to-environment correspondence.
  • To me, this doesn't really get at what the internals of actually-control-like systems look like, which presumably a subset of EO - IO.
• Search-in-Territory vs Search-in-Map [LW · GW]
  • Greater emphasis on internal structure—specifically, "maps."
  • Maps are capital investment, allowing you to be able to optimize despite not knowing what to exactly optimize for (by compressing info)

I have several thoughts on these framings, but one trouble is the excessive usage of words to represent "clusters" i.e. terms to group a bunch of correlated variables. Selection vs Control, for example, doesn't have a clear definition/criteria but rather points at a number of correlated things, like internal structure, search, maps, control-like things, etc.

Sure, deconfusing and pointing out clusters is useful because clusters imply correlations and correlations perhaps imply hidden structure + relationships—but I think the costs from cluster-representing-words doing hidden inference [LW · GW] is much greater than the benefits, and it would be better to explicitly lay out the features-of-clusters that the one is referring to instead of just using the name of the cluster.

This is similar to the trouble I had with "wrapper-minds," which is yet another example of a cluster pointing at a bunch of correlated variables, and people using the same term to mean different things.

Anyways, I still feel totally confused about optimization—and while these clusters/frames are useful, I think thinking in terms of them would ensue even more confusion within myself. It's probably better to take the useful individual parts within the cluster and start deconfusing from the ground-up using those as the building blocks.

I find the intersection of computational mechanics, boundaries [LW · GW]/frames [LW · GW]/factored-sets [LW · GW], and some works from the causal incentives group - especially discovering agents and robust agents learn causal world model (review [LW(p) · GW(p)]) - to be a very interesting theoretical direction.

By boundaries [? · GW], I mean a sustaining/propagating system that informationally/causally insulates its 'viscera' from the 'environment,' and only allows relatively small amounts of deliberate information flow through certain channels in both directions. Living systems are an example of it (from bacteria to humans). It doesn't even have to be a physically distinct chunk of spacetime, they can be over more abstract variables like societal norms. Agents are an example of it.

I find them very relevant to alignment especially from the direction of detecting such boundary-possessing/agent-like structures embedded in a large AI system and backing out a sparse relationship between these subsystems, which can then be used to e.g., control the overall dynamic. Check out these [LW · GW] posts [LW · GW] for more.

A prototypical deliverable would be an algorithm that can detect such 'boundaries' embedded in a dynamical system when given access to some representation of the system, performs observations & experiments and returns a summary data structure of all the 'boundaries' embedded in a system and their desires/wants, how they game-theoretically relate to one another (sparse causal relevance graph?), the consequences of interventions performed on them, etc - that's versatile enough to detect e.g., gliders embedded in Game of Life / Particle Lenia, agents playing Minecraft while only given coarse grained access to the physical state of the world, boundary-like things inside LLMs, etc. (I'm inspired by this [LW(p) · GW(p)])

Why do I find the aforementioned directions relevant to this goal?

• Critch's Boundaries [LW · GW] operationalizes boundaries/viscera/environment as functions of the underlying variable that executes policies that continuously prevents information 'flow' ^[1] between disallowed channels, quantified via conditional transfer entropy.
• Relatedly, Fernando Rosas's paper on Causal Blankets operationalize boundaries using a similar but subtly different^[2] form of mutual information constraint on the boundaries/viscera/environment variables than that of Critch's. Importantly, they show that such blankets always exist between two coupled stochastic processes (using a similar style of future morph equivalence relation characterization from compmech, and also a metric they call "synergistic coefficient" that quantifies how boundary-like this thing is.^[3]
• More on compmech, epsilon transducers generalize epsilon machines to input-output processes. PALO (Perception Action Loops) and Boundaries as two epsilon transducers coupled together?
• These directions are interesting, but I find them still unsatisfactory because all of them are purely behavioral accounts of boundaries/agency. One of the hallmarks of agentic behavior (or some boundary behaviors) is adapting ones policy if an intervention changes the environment in a way that the system can observe and adapt to.^[4][5]
• (is there an interventionist extension of compmech?)
• Discovering agents provide a genuine causal, interventionist account of agency and an algorithm to detect them, motivated by the intentional stance. I think the paper is very enlightening from a conceptual perspective, but there are many problems yet to be solved before we can actually implement this. Here's [LW(p) · GW(p)] my take on it.
• More fundamentally, (this is more vibes, I'm really out of my depth here) I feel there is something intrinsically limiting with the use of Bayes Nets, especially with the fact that choosing which variables to use in your Bayes Net already encodes a lot of information about the specific factorization structure of the world. I heard good things about finite factored sets [LW · GW] and I'm eager to learn more about them.

1. ^{^}

   Not exactly a 'flow', because transfer entropy conflates between intrinsic information flow and synergistic information - a 'flow' connotes only the intrinsic component, while transfer entropy just measures the overall amount of information that a system couldn't have obtained on its own. But anyways, transfer entropy seems like a conceptually correct metric to use.

2. ^{^}

   Specifically, Fernando's paper criticizes blankets of the following form ($V$ for viscera, $A$ and $P$ for active/passive boundaries, $E$ for environment):

   • $V_t\to A_t,P_t\to E_t$
     • DIP implies $I(V_t;A_t,P_t)\ge I(V_t;E_t)$
   • This clearly forbids dependencies formed in the past that stays in 'memory'.

   but Critch instead defines boundaries as satisfying the following two criteria:

   • $V_{t+1},A_{t+1}\to V_t,A_t,P_t\to E_t$ (infiltration)
     • DIP implies $I(V_t;A_t,P_t)\ge I(V_t;E_t)$
   • $E_{t+1},P_{t+1}\to A_t,P_t,E_t\to V_t$ (exfiltration)
     • DIP implies $I(V_{t+1},A_{t+1};A_t,P_t)\ge I(V_{t+1},A_{t+1};E_t)$
   • and now that the independencies are entangled across different t, there is no longer a clear upper bound on $I(V_t;E_t)$, so I don't think the criticisms apply directly.
3. ^{^}

   My immediate curiosities are on how these two formalisms relate to one another. e.g., Which independency requirements are more conceptually 'correct'? Can we extend the future-morph construction to construct Boundaries for Critch's formalism? etc etc

4. ^{^}

   For example, a rock is very goal-directed relative to 'blocking-a-pipe-that-happens-to-exactly-match-its-size,' until one performs an intervention on the pipe size to discover that it can't adapt at all.

5. ^{^}

   Also, interventions are really cheap to run on digital systems (e.g., LLMs, cellular automata, simulated environments)! Limiting oneself to behavioral accounts of agency would miss out on a rich source of cheap information.

Does anyone know if Shannon arrive at entropy from the axiomatic definition first, or the operational definition first?

I've been thinking about these two distinct ways in which we seem to arrive at new mathematical concepts, and looking at the countless partial information decomposition measures in the literature all derived/motivated based on an axiomatic basis, and not knowing which intuition to prioritize over which, I've been assigning less premium on axiomatic conceptual definitions than i used to:

• decision theoretic justification of probability > Cox's theorem
• shannon entropy as min description length > three information axioms
• fernando's operational definition of synergistic information > rest of the literature with its countless non-operational PID measures

The basis of comparison would be its usefulness and ease-of-generalization to better concepts:

• at least in the case of fernando's synergistic information, it seems far more useful because i at least know what i'm exactly getting out of it, unlike having to compare between the axiomatic definitions based on handwavy judgements.
• for ease of generalization [LW · GW], the problem with axiomatic definitions is that there are many logically equivalent ways to state the initial axiom (from which they can then be relaxed), and operational motivations seem to ground these equivalent characterizations better, like logical inductors from the decision theoretic view of probability theory

(obviously these two feed into each other)

'Symmetry' implies 'redundant coordinate' implies 'cyclic coordinates in your Lagrangian / Hamiltonian' implies 'conservation of conjugate momentum'

And because the action principle (where the true system trajectory extremizes your action, i.e. integral of Lagrangian) works in various dynamical systems, the above argument works in non-physical dynamical systems.

Thus conserved quantities usually exist in a given dynamical system.

mmm, but why does the action principle hold in such a wide variety of systems though? (like how you get entropy by postulating something to be maximized in an equilibrium setting)

Mildly surprised how some verbs/connectives barely play any role in conversations, even in technical ones. I just tried directed babbling [LW · GW] with someone, and (I think?) I learned quite a lot about Israel-Pakistan relations with almost no stress coming from eg needing to make my sentences grammatically correct.

Example of (a small part of) my attempt to summarize my understanding of how Jews migrated in/out of Jerusalem over the course of history:

They here *hand gesture on air*, enslaved out, they back, kicked out, and boom, they everywhere.

(audience nods, given common knowledge re: gestures, meaning of "they," etc)

Why haven't mosquitos evolved to be less itchy? Is there just not enough selection pressure posed by humans yet? (yes probably) Or are they evolving towards that direction? (they of course already evolved towards being less itchy while biting, but not enough to make that lack-of-itch permanent)

this is a request for help i've been trying and failing to catch this one for god knows how long plz halp

tbh would be somewhat content coexisting with them (at the level of houseflies) as long as they evolved the itch and high-pitch noise away, modulo disease risk considerations.

Just noticing that the negation of a statement exists is enough to make meaningful updates.

e.g. I used to (implicitly) think "Chatbot Romance is weird" without having evaluated anything in-depth about the subject (and consequently didn't have any strong opinions about it)—probably as a result of some underlying cached belief. 

But after seeing this post [LW · GW], just reading the title was enough to make me go (1) "Oh! I just realized it is perfectly possible to argue in favor of Chatbot Romance ... my belief on this subject must be a cached belief!" (2) hence is probably by-default biased towards something like the consensus opinion, and (3) so I should update away from my current direction, even without reading the post.

(Note: This was a post, but in retrospect was probably better to be posted as a shortform)

(Epistemic Status: 20-minute worth of thinking, haven't done any builder/breaker on this yet although I plan to, and would welcome any attempts in the comment)

1. Have an algorithmic task whose input/output pair could (in reasonable algorithmic complexity) be generated using highly specific combination of modular components (e.g., basic arithmetic, combination of random NN module outputs, etc).
2. Train a small transformer (or anything, really) on the input/output pairs.
3. Take a large transformer that takes the activation/weights, and outputs a computational graph.
4. Train that large transformer over the small transformer, across a diverse set of such algorithmic tasks (probably automatically generated) with varying complexity. Now you have a general tool that takes in a set of high-dimensional matrices and backs-out a simple computational graph, great! Let's call it Inspector.
5. Apply the Inspector in real models and see if it recovers anything we might expect (like induction heads).
6. To go a step further, apply the Inspector to itself. Maybe we might back-out a human implementable general solution for mechanistic interpretability! (Or, at least let us build a better intuition towards the solution.)

(This probably won't work, or at least isn't as simple as described above. Again, welcome any builder/breaker attempts!)

People mean different things when they say "values" (object vs meta values)

I noticed that people often mean different things when they say "values," and they end up talking past each other (or convergence only happens after a long discussion). One of the difference is in whether they contain meta-level values.

• Some people refer to the "object-level" preferences that we hold.
  • Often people bring up the "beauty" of the human mind's capacity for its values to change, evolve, adopt, and grow—changing mind as it learns more about the world, being open to persuasion via rational argumentation, changing moral theories, etc.
• Some people include the meta-values (that are defined on top of other values, and the evolution of such values).
  • e.g., My "values" include my meta-values, like wanting to be persuaded by good arguments, wanting to change my moral theories when I get to know better, even "not wanting my values to be fixed"
    • example of this view: carado's post on you want what you want, and one of Vanessa Cosoy's shortform/comment (can't remember the link)

Is there a way to convert a LessWrong sequence into a single pdf? Should ideally preserve comments, latex, footnotes, etc.

I don't know if this is just me, but it took me an embarrassingly long time in my mathematical education to realize that the following three terminologies, which introductory textbooks used interchangeably without being explicit, mean the same thing. (Maybe this is just because English is my second language?)

X => Y means X is sufficient for Y means X only if Y

X <= Y means X is necessary for Y means X if Y

Unidimensional Continuity of Preference $\approx$ Assumption of "Resources"?

tl;dr, the unidimensional continuity of preference assumption in the money pumping argument used to justify the VNM axioms correspond to the assumption that there exists some unidimensional "resource" that the agent cares about, and this language is provided by the notion of "souring / sweetening" a lottery.

Various coherence theorems - or more specifically, various money pumping arguments generally have the following form:

If you violate this principle, then [you are rationally required] / [it is rationally permissible for you] to follow this trade that results in you throwing away resources [LW(p) · GW(p)]. Thus, for you to avoid behaving pareto-suboptimally by throwing away resources, it is justifiable to call this principle a 'principle of rationality,' which you must follow.

... where "resources" (the usual example is money) are something that, apparently, these theorems assume exist. They do, but this fact is often stated in a very implicit way. Let me explain.

In the process of justifying the VNM axioms using money pumping arguments, one of the three main mathematical primitives are: (1) lotteries (probability distribution over outcomes), (2) preference relation  (general binary relation), and (3) a notion of Souring/Sweetening of a lottery. Let me explain what (3) means.

• Souring of $A$ is denoted $A^{-}$, and a sweetening of $A$ is denoted $A^{+}$.
• $A^{-}$ is to be interpreted as "basically identical with A but strictly inferior in a single dimension that the agent cares about." Based on this interpretation, we assume $A>A^{-}$. Sweetening is the opposite, defined in the obvious way.

Formally, souring could be thought of as introducing a new preference relation $A{>}_{\text{uni}}B$, which is to be interpreted as "lottery B is basically identical to lottery A, but strictly inferior in a single dimension that the agent cares about".

• On the syntactic level, such $B$ is denoted as $A^{-}$.
• On the semantic level, based on the above interpretation, ${>}_{\text{uni}}$ is related to $>$ via the following: $A{>}_{\text{uni}}B⟹A>B$

This is where the language to talk about resources come from. "Something you can independently vary alongside a lottery A such that more of it makes you prefer that option compared to A alone" sounds like what we'd intuitively call a resource^[1].

Now that we have the language, notice that so far we haven't assumed sourings or sweetenings exist. The following assumption does it:

Unidimensional Continuity of Preference: If $X>Y$, then there exists a prospect $X^{-}$ such that 1) $X^{-}$ is a souring of X and 2) $X>X^{-}>Y$.

Which gives a more operational characterization of souring as something that lets us interpolate between the preference margins of two lotteries - intuitively satisfied by e.g., money due to its infinite divisibility.

So the above assumption is where the assumption of resources come into play. I'm not aware of any money pump arguments for this assumption, or more generally, for the existence of a "resource." Plausibly instrumental convergence [LW(p) · GW(p)].

1. ^{^}

   I don't actually think this + the assumption below fully capture what we intuitively mean by "resources", enough to justify this terminology. I stuck with "resources" anyways because others around here used that term to (I think?) refer to what I'm describing here.

Damn, why did Pearl recommend readers (in the preface of his causality book) to read all the chapters other than chapter 2 (and the last review chapter)? Chapter 2 is literally the coolest part - inferring causal structure from purely observational data! Almost skipped that chapter because of it ...

Bayes Net inference algorithms maintain its efficiency by using dynamic programming over multiple layers.

Level 0: Naive Marginalization

• No dynamic programming whatsoever. Just multiply all the conditional probability distribution (CPD) tables, and sum over the variables of non-interest.

Level 1: Variable Elimination

• Cache the repeated computations within a query.
• For example, given a chain-structured Bayes Net $A⟶B⟶C⟶D$, instead of doing $P\left(D\right)={\sum }_A{\sum }_B{\sum }_{C}P(A,B,C,D)$, we can do $P\left(D\right)={\sum }_{C}P\left(D|C\right){\sum }_{B}P\left(C|B\right){\sum }_{A}P\left(A\right)P\left(B|A\right)$. Check my post [LW · GW] for more.

Level 2: Clique-tree based algorithms — e.g., Sum-product (SP) / Belief-update (BU) calibration algorithms

• Cache the repeated computations across queries.
• Suppose you have a fixed Bayes Net, and you want to compute the marginalization not only $P\left(D\right)$, but also $P\left(A\right)$. Clearly running two instances of Variable Elimination as above is going to contain some overlapping computation.
• Clique-tree is a data structure where, given the initial factors (in this case the CPD tables), you "calibrate" a tree whose nodes correspond to a subset of the variables. Cost can be amortized by running many queries over the same Bayes Net.
  • Calibration can be done by just two passes across the tree, after which you have the joint marginals for all the nodes of the clique tree.
  • Incorporating evidence is equally simple. Just zero-out the entries of variables that you are conditioning on for some node, then "propagate" that information downwards via a single pass across the tree.

Level 3: Specialized query-set answering algorithms over a calibrated clique tree.

• Cache the repeated computations across a certain query-class
• e.g., computing $P(X,Y)$ for every pair of variables can be done by using yet another layer of dynamic programming by maintaining a table of $P(C_i,C_j)$ for each pair of clique-tree nodes ordered according to their distance in-between.

Man, deviation arguments are so cool:

• what are macrostates? Variables which are required to make your thermodynamics theory work! If they don't, add more macrostates!
• nonequilibrium? Define it as systems that don't admit a thermodynamic description!
• inductive biases? Define it as the amount of correction needed for a system to obey Bayesian updating, i.e. correction terms in the exponent of the Gibbs measure!
• coarse graining? Define the coarse-grained variables to keep the dynamics as close as possible to that of the micro-dynamics!
• or in a similar spirit - does your biological system deviate [LW(p) · GW(p)] from expected utility theory? Well, there's discovery (and money) to be made!

It's easy to get confused and think the circularity is a problem ("how can you define thermodynamics in terms of equilibriums, when equilibriums are defined using thermodynamics?"), but it's all about carving nature at the right joints—and a sign that you made the right carving is that the amount of corrections needed to be applied aren't too numerous, and they all seem "natural" (and of course, all of this while letting you make nontrivial predictions. that's what matters at the end of the day).

Then, it's often the case that those corrections also turn out to be meaningful and natural quantities of interest.

One of the rare insightful lessons from high school: Don't set your AC to the minimum temperature even if it's really hot, just set it to where you want it to be.

It's not like the air released gets colder with lower target temperature, because most ACs (according to my teacher, I haven't checked lol) are just a simple control system that turns itself on/off around the target temperature, meaning the time it takes to reach a certain temperature X is independent of the target temperature (as long it's lower than X)

... which is embarrassingly obvious in hindsight.

Quick thoughts on my plans:

1. I want to focus on having a better mechanistic picture of agent value formation & distinguishing between hypotheses (e.g., shard theory, Thane Ruthenis's value-compilation hypothesis, etc) and forming my own.
2. I think I have a specific but very high uncertainty baseline model of what-to-expect from agent value-formation using greedy search optimization. It's probably time to allocate more resources on reducing that uncertainty by touching reality [LW · GW] i.e. running experiments.
   1. (and also think about related theoretical arguments like Selection Theorem)
3. So I'll probably allocate my research time:
   1. Studying math (more linear algebra / dynamical systems / causal inference / statistical mechanics)
   2. Sketching a better picture of agent development, assigning confidence, proposing high-bit experiments (that might have the side-effect of distinguishing between different conflicting pictures), formalization, etc.
      1. and read relevant literature (eg ones on theoretic DL and inductive biases)
   3. Upskilling mechanistic interpretability to actually start running quick experiments
   4. Unguided research brainstorming (e.g., going through various alignment exercises, having a writeup of random related ideas, etc)
4. Possibly participate in programs like MATS? Probably the biggest benefit to me would be (1) commitment mechanism / additional motivation and (2) high-value conversations with other researchers.

Dunno, sounds pretty reasonable!

Useful perspective when thinking of mechanistic pictures of agent/value development is to take the "perspective" of different optimizers, consider their relative "power," and how they interact with each other.

E.g., early on SGD is the dominant optimizer, which has the property of (having direct access to feedback from U / greedy [LW · GW]). Later on early proto-GPS (general-purpose search [LW · GW]) forms, which is less greedy, but still can largely be swayed by SGD (such as having its problem-specification-input tweaked, having the overall GPS-implementation modified, etc). Much later, GPS becomes the dominant optimizing force "at run-time" which shortens the relevant time-scale and we can ignore the SGD's effect. This effect becomes much more pronounced after reflectivity + gradient hacking when the GPS's optimization target becomes fixed.

(very much inspired by reading Thane Ruthenis's value formation [LW · GW] post)

It seems like retrieval-based transformers like RETRO is "obviously" the way to go—(1) there's just no need to store all the factual information as fixed weights, (2) and it uses much less parameter/memory. Maybe mechanistic interpretability should start paying more attention to these type of architectures, especially since they're probably going to be a more relevant form of architecture.

They might also be easier to interpret thanks to specialization!

I've noticed during my alignment study that just the sheer amount of relevant posts out there is giving me a pretty bad habit of (1) passively engaging with the material and (2) not doing much independent thinking. Just keeping up to date & distilling the stuff in my todo read list takes up most of my time.

• I guess the reason I do it is because (at least for me) it takes a ton of mental effort to switch modes between "passive consumption" and "active thinking":
  • I noticed then when self-studying math; like, my subjective experience is that I enjoy both "passively listening lectures+taking notes" and "solving practice problems," the problem is that it takes a ton of mental energy to switch between the two equilibriums.
  • (This is actually still a problem—too much wide & passive consumption rather than actively practicing them and solving problems.)
• Also relevant is wanting to just progress/upskill as fast and wide of a subject as I can, sacrificing mastery for diversity. This probably makes sense to some degree (especially in the sense that having more frames is good), but I think I'm taking this wayyyy too far.
• My $r$ for opening new links far exceeds 1. This definitely helped me when I was trying to get a rapid overview of the entire field, but now it's just a bad adaptation + akrasia.

Okay, then, don't do that! Some directions to move towards:

• Independent brainstorming/investigation sessions to form concrete inside views
  • like the advice from the field guide [LW · GW] post, exercises [LW · GW] from the MATS model, et
• Commitment mechanisms, like making regular posts or shortforms (eg [? · GW])

Is there a case for AI gain-of-function [LW · GW] research?

(Epistemic Status: I don't endorse this yet, just thinking aloud. Please let me know if you want to act/research based on this idea)

It seems like it should be possible to materialize certain forms of AI alignment failure modes with today's deep learning algorithms, if we directly optimize for their discovery. For example, training a Gradient Hacker Enzyme [LW(p) · GW(p)].

A possible benefit of this would be that it gives us bits of evidence wrt how such hypothesized risks would actually manifest in real training environments. While the similarities would be limited because the training setups would be optimizing for their discovery, it should at least serve as a good lower bound for the scenarios in which these risks could manifest.

Perhaps having a concrete bound for when dangerous capabilities appear (eg a X parameter model trained in Y modality has Z chance of forming a gradient hacker) would make it easier for policy folks to push for regulations.

Is AI gain-of-function equally dangerous as biotech gain-of-function? Some arguments in favor (of the former being dangerous):

• The malicious actor argument is probably stronger for AI gain-of-function.
  • if someone publicly releases a Gradient Hacker Enzyme, this lowers the resource that would be needed for a malicious actor to develop a misaligned AI (eg plug in the misaligned Enzyme at an otherwise benign low-capability training run).
• Risky researcher incentive is equally strong.
  • e.g., a research lab carelessly pursuing gain-of-function research, deliberately starting risky training runs for financial/academic incentives, etc.

Some arguments against:

• Accident risks from financial incentives are probably weaker for AI gain-of-function.
  • The standard gain-of-function risk scenario is: research lab engineers a dangerous pathogen, it accidentally leaks, and a pandemic happens.
  • I don't see how these events would happen "accidentally" when dealing with AI programs; e.g., the researcher would have to deliberately cut parts of the network weights and replace it with the enzyme, which is certainly intentional.

Random alignment-related idea: train and investigate a "Gradient Hacker Enzyme"

TL;DR, Use meta-learning methods like MAML to train a network submodule i.e. circuit that would resist gradient updates in a wide variety of contexts (various architectures, hyperparameters, modality, etc), and use mechanistic interpretability to see how it works.

It should be possible to have a training setup for goals other than "resist gradient updates," such as restricting the meta-objective to a specific sub-sub-circuit. In that case, the outer circuit might (1) instrumentally resist updates, or (2) somehow get modified while keeping its original behavioral objective intact.

This setup doesn't have to be restricted to circuits of course; there was a previous work which did this on the level of activations [LW · GW], although iiuc the model found a trivial solution by exploiting relu—it would be interesting to extend this to more diverse setup.

Anyways, varying this "sub-sub-circuit/activation-to-be-preserved" over different meta-learning episodes would incentivize the training process to find "general" Gradient Hacker designs that aren't specific to a particular circuit/activation—a potential precursor for various forms of advanced Gradient Hackers (and some loose analogies to how enzymes accelerate reactions).

What is the Theory of Impact for training a "Gradient Hacker Enzyme"?

(note: while I think these are valid, they're generated post-hoc and don't reflect the actual process for me coming up with this idea)

• Estimating the lower-bound for the emergence Gradient Hackers.
  • By varying the meta-learning setups we can get an empirical estimate for the conditions in which Gradient Hackers are possible.
    • Perhaps gradient hackers are actually trivial to construct using tricks we haven't thought of before (like the relu example before). Maybe not! Perhaps they require [high-model-complexity/certain-modality/reflective-agent/etc].
  • Why lower-bound? In a real training environment, gradient hackers appear because of (presumably) convergent training incentives. Instead in the meta-learning setup, we're directly optimizing for gradient hackers.
• Mechanistically understanding how Gradient Hackers work.
  • Applying mechanistic interpretability here might not be too difficult, because the circuit is cleanly separated from the rest of the model.
  • There has been several [LW · GW] speculations [LW · GW] on [LW · GW] how such circuits might emerge. Testing them empirically sounds like a good idea!

This is just a random idea and I'm probably not going to work on it; but if you're interested, let me know. While I don't think this is capabilities-relevant, this probably falls under AI gain-of-function [LW · GW] research and should be done with caution.