It sounds like we are not that far apart here. We've been doing some empirical work on toy systems to try to make the leap from mechanistic interpretability "stories" to semi-formal heuristic explanations. The max-of-k draft is an early example of this, and we have more ambitious work in progress along similar lines. I think of this work in a similar way to you: we are not trying to test empirical assumptions (in the way that some empirical work on frontier LLMs is, for example), but rather to learn from the process of putting our ideas into practice.

For those who are interested in the mathematical details, but would like something more accessible than the paper itself, see this talk I gave about the paper:

Thank you – this is probably the best critique of ARC's research agenda that I have read since we started working on heuristic explanations. This level of thoughtfulness in external feedback is very rare and I'm grateful for the detail and clarity you put into it. I don't think my response fully rebuts your central concern, but hopefully it gives a sense of my current thinking about it.

It sounds like we are in agreement that something very loosely heuristic explanation-flavored (interpreted so broadly as to include mechanistic interpretability, for example) can reasonably be placed at the root of the diagram, by which I mean that it's productive to try to explain neural network behaviors in this very loose sense, attempt to apply such explanations to downstream applications such as MAD/LPE/ELK etc. We begin to diverge, I think, about the extent to which ARC should focus on a more narrow conception of heuristic explanations. From least to most specific:

1. Any version that is primarily mathematical rather than "story-centric"
2. Some (mathematical) version that is consistent with our information-theoretic intuitions about what constitutes a valid explanation (i.e., in the sense of something like surprise accounting)
3. Some such version that is loosely based on independence assumptions
4. Some version that satisfies more specific desiderata for heuristic estimators (such as the ones discussed in the paper linked in (3), or in this more recent paper)

Opinions at ARC will differ, but (1) I feel pretty comfortable defending, (2) I think is quite a promising option to be considering, (3) seems like a reasonable best guess but I don't think we should be that wedded to it, and (4) I think is probably too specific (and with the benefit of hindsight I think we have focused too much on this in the past). ARC's research has actually been trending in the "less specific" direction over time, as should hopefully be evident from our most recent write-ups (with the exception of our recent paper on specific desiderata, which mostly covers work done in 2023), and I am quite unsure exactly where we should settle on this axis.

By contrast, my impression is that you would not really defend even (1) (although I am curious exactly where you come down this axis, if you want to clarify). So I'll give what I see as the basic case for searching for a mathematical rather than a "story-centric" approach:

• Mechanistic interpretability has so far yielded very little in the way of beating baselines at downstream tasks (this has been discussed at length elsewhere, see for example here, here and here), so I think it should still be considered a largely unproven approach (to be clear, this is roughly my view of all alignment approaches that aren't already in active use at labs, including ARC's, and I remain excited to see people's continued valiant attempts; my point is that the bar is low and a portfolio approach is appropriate).
• Relying purely on stories clearly doesn't work at sufficient scale under worst-case assumptions (because the AI will have concepts you don't have words for), and there isn't a lot of evidence that this isn't indeed already a bottleneck in practice (i.e., current AIs may well already have concepts you don't have words for).
• I think that ARC's worst-case, theoretical approach (described at zoom level 1) is an especially promising alternative to iterative, empirically-driven work. I think empirical approaches are more promising overall, but have correlated failure modes (namely, they could end up relying on correlated empirical contingencies that later turn out to be false), and have far more total effort going into them (arguably disproportionately so). Conditional on taking such an approach, story-centric methods don't seem super viable (how should one analyze stories theoretically?).
• I don't really buy the argument that because a system has a lot of complexity, it can only be analyzed in ad-hoc ways. It seems to me that an analogous argument would have failed to make good predictions about the bitter lesson (i.e., by arguing that a simple algorithm like SGD should not be capable of producing great complexity in a targeted way). Instead, because neural nets are trained in an incremental, automated way based on mathematical principles, it seems quite possible to me that we can find explanations for them in a similar way (which is not an argument that can be applied to biological brains).

This doesn't of course defend (2)–(4) (which I would only want to do more weakly in any case). We've tried to get our intuitions for those across in our write-ups (as linked in (2)–(4) above), but I'm not sure there's anything succinct I can add here if those were unconvincing. I agree that puts us in the rather unfortunate position of sharing a reference class with Stephen Wolfram to many external observers (although hopefully our claims are not quite so overstated).

I think it's important for ARC to recognize this tension, and to strike the right balance between making our work persuasive to external skeptics on the one hand, and having courage in our convictions on the other hand (I think both have been important virtues in scientific development historically). Concretely, my current best guess is that ARC should:

• (a) Avoid being too wedded to intuitive desiderata for heuristic explanations that we can't directly tie back to specific applications
• (b) Search for concrete cases that put our intuitions to the test, so that we can quickly reach a point where either we no longer believe in them, or they are more convincing to others
• (c) Also pursue research that is more agnostic to the specific form of explanation, such as work on low probability estimation or other applications
• (d) Stay on the lookout for ideas from alternative theoretical approaches (including singular learning theory, sparsity-based approaches, computational mechanics, causal abstractions, and neural net-oriented varieties of agent foundations), although my sense is that object-level intuitions here just differ enough that it's difficult to collaborate productively. (Separately, I'd argue that proponents of all these alternatives are in a similar predicament, and could generally be doing a better job on analogous versions of (a)–(c).)

I think we have been doing all of (a)–(d) to some extent already, although I imagine you would argue that we have not been going far enough. I'd be interested in more thoughts on how to strike the right balance here.

Yes, I think the most natural way to estimate total surprise in practice would be to use sampling like you suggest. You could try to find the best explanation for "the model does $bad_thing with probability less than 1 in a million" (which you believe based on sampling) and then see how unlikely $bad_thing is according to the resulting explanation. In the Boolean circuit worked example, the final 23-bit explanation is likely still the best explanation for why the model outputs TRUE on at least 99% of inputs, and we can use this explanation to see that the model actually outputs TRUE on all inputs.

Another possible approach is analogous to fine-tuning. You could start by using surprise accounting to find the best explanation for "the loss of the model is L" (where L is estimated during training), which should incentivize rich explanations of the model's behavior in general. Then to estimate the probability that model does some rare $bad_thing, you could "fine-tune" your explanation using an objective that encourages it to focus on the relevant tails of the distribution. We have more ideas about estimating the probability of events that are too rare to estimate via sampling, and have been considering objectives other than surprise accounting for this. We plan to share these ideas soon.