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Yeah, I completely agree this is a good research direction! My only caveat is I don’t think this is a silver bullet in the same way capabilities benchmarks are (not sure if you’re arguing this, just explaining my position here). The inevitable problem with interpretability benchmarks (which to be clear, your paper appears to make a serious effort to address) is that you either:
- Train the model in a realistic way - but then you don’t know if the model really learned the algorithm you expected it to
- Train the model to force it to learn a particular algorithm - but them you have to worry about how realistic that training method was or the algorithm you forced it to learn
This doesn’t seem like an unsolvable problem to me, but it does mean that (unlike capabilities benchmarks) you can’t have the level of trust in your benchmarks that allow you to bypass traditional scientific hygiene. In other words, there are enough subtle strings attached here that you can’t just naively try to “make number go up” on InterpBench in the same way you can with MMLU.
I probably should emphasize I think trying to bring interpretability into contact with various “ground truth” settings seems like a really high value research direction, whether that be via modified training methods, toy models where possible algorithms are fairly simple (e.g. modular addition), etc. I just don’t think it changes my point about methodological standards.
Understanding deep learning isn’t a leaderboard sport - handle with care.
Saliency maps, neuron dissection, sparse autoencoders - each surged on hype, then stalled[1] when follow‑up work showed the insight was mostly noise, easily spoofed, or valid only in cherry‑picked settings. That risks being negative progress: we spend cycles debunking ghosts instead of building cumulative understanding.
The root mismatch is methodological. Mainstream ML capabilities research enjoys a scientific luxury almost no other field gets: public, quantitative benchmarks that tie effort to ground truth. ImageNet accuracy, MMLU, SWE‑bench - one number silently kills bad ideas. With that safety net, you can iterate fast on weak statistics and still converge on something useful. Mechanistic interpretability has no scoreboard for “the network’s internals now make sense.” Implicitly inheriting benchmark‑reliant habits from mainstream ML therefore swaps a ruthless filter for a fog of self‑deception.
How easy is it to fool ourselves? Recall the “Could a neuroscientist understand a microprocessor?” study: standard neuroscience toolkits - ablation tests, tuning curves, dimensionality reduction - were applied to a 6502 chip whose ground truth is fully known. The analyses produced plausible‑looking stories that entirely missed how the processor works. Interpretability faces the same trap: shapely clusters or sharp heat‑maps can look profound until a stronger test dissolves them.
What methodological standard should replace the leaderboard? Reasonable researchers will disagree[2]. Borrowing from mature natural sciences like physics or neuroscience seems like a sensible default, but a proper discussion is beyond this note. The narrow claim is simpler:
Because no external benchmark will catch your mistakes, you must design your own guardrails. Methodology for understanding deep learning is an open problem, not a hand‑me‑down from capabilities work.
So, before shipping the next clever probe, pause and ask: Where could I be fooling myself, and what concrete test would reveal it? If you don’t have a clear answer, you may be sprinting without the safety net this methodology assumes - and that’s precisely when caution matters most.
- ^
This is perhaps a bit harsh - I think SAEs for instance still might hold some promise, and neuron-based analysis still has its place, but I think it's fair to say the hype got quite ahead of itself.
- ^
Exactly how much methodological caution is warranted here will obviously be a point of contention. Everyone thinks the people going faster than them are reckless and the people going slower are needlessly worried. My point here is just to think actively about the question - don't just blindly inherit standards from ML.
4
4It's not relevant to predictions about how well the learning algorithm generalises. And that's the vastly more important factor for general capabilities.
Quite tangential to your point, but the problem with the universal approximation theorem is not just "it doesn't address generalization" but that it doesn't even fulfill its stated purpose: it doesn't answer the question of why neural networks can space-efficiently approximate real-world functions, even with arbitrarily many training samples. The statement "given arbitrary resources, a neural network can approximate any function" is actually kind of trivial - it's true not only of polynomials, sinusoids, etc, but even just a literal interpolated lookup table (if you have an astronomical space budget). It turns out the universal approximation theorem requires exponentially many neurons (in the size of the input dimension) to work, far too much to be practical - in fact this is actually the same amount of resources a lookup table would cost. This is fine if you want to approximate a 2D function or something, but this goes nowhere to explaining why, like, even a space-efficient MNIST classifier is possible. The interesting question is, why can neural networks efficiently approximate the functions we see in practice?
(It's a bit out of scope to fully dig into this, but I think a more sensible answer is something in the direction of "well, anything you can do efficiently on a computer, you can do efficiently in a neural network" - i.e. you can always encode polynomial-size Boolean circuits into a polynomial-size neural network. Though there are some subtleties here that make this a little more complicated than that.)
After convergence, the samples should be viewed as drawn from the stationary distribution, and ideally they have low autocorrelation, so it doesn't seem to make sense to treat them as a vector, since there should be many equivalent traces.
This is a very subtle point theoretically, so I'm glad you highlighted this. Max may be able to give you a better answer here, but I'll try my best to attempt one myself.
I think you may be (understandably) confused about a key aspect of the approach. The analysis isn't focused on autocorrelation within individual traces, but rather correlations between different input traces evaluated on the same parameter samples from SGLD.
What this approach is actually doing, from a theoretical perspective, is estimating the expected correlation of per-datapoint losses over the posterior distribution of parameters. SGLD serves purely as a mechanism for sampling from this posterior (as it ordinarily does).
When examining correlations between and across different inputs but identical parameter samples , the method approximates the posterior expectation
.
Assuming that SGLD is taking unbiased IID samples from the posterior, the dot product of traces is an unbiased estimator of this correlation. The vectorization of traces is therefore an efficient mechanism for computing these expectations in parallel, and representing the correlation structure geometrically.
At the risk of being repetitive, at an intuitive level, this is designed to detect when different inputs respond similarly to the same parameter perturbations. When inputs and share functional circuitry within the model, they'll likely show correlated loss patterns when that shared circuitry is disrupted at parameter . When two trace vectors are orthogonal, that means the per-sample losses for and are uncorrelated across the posterior. The presence or absence of synchronization reveals functional similarities between inputs that might not be apparent through other means.
What all this means is that we do in fact want to use SGLD for plain old MCMC sampling - we are genuinely attempting to use random samples from the posterior rather than e.g. examining the temporal behavior of SGLD. Ideally, we want all the usual desiderata of MCMC sampling algorithms here, like convergence, unbiasedness, low autocorrelation, etc. You're completely correct that if SGLD is properly converged and sampling ideally, it has fairly boring autocorrelation within a trace - but this is exactly what we want.
1IIRC @jake_mendel and @Kaarel have thought about this more, but my rough recollection is: a simple story about the regularization seems sufficient to explain the training dynamics, so a fancier SLT story isn't obviously necessary. My guess is that there's probably something interesting you could say using SLT, but nothing that simpler arguments about the regularization wouldn't tell you also. But I haven't thought about this enough.
1Good catch, thanks! Fixed now.
It's worth noting that Jesse is mostly following the traditional "approximation, generalization, optimization" error decomposition from learning theory here - where "generalization" specifically refers to finite-sample generalization (gap between train/test loss), rather than something like OOD generalization. So e.g. a failure of transformers to solve recursive problems would be a failure of approximation, rather than a failure of generalization. Unless I misunderstood you?
Repeating a question I asked Jesse earlier, since others might be interested in the answer: how come we tend to hear more about PAC bounds than MAC bounds?
Note that in the SLT setting, "brains" or "neural networks" are not the sorts of things that can be singular (or really, have a certain ) on their own - instead they're singular for certain distributions of data.
This is a good point I often see neglected. Though there's some sense in which a model can "be singular" independent of data: if the parameter-to-function map is not locally injective. Then, if a distribution minimizes the loss, the preimage of in parameter space can have non-trivial geometry.
These are called "degeneracies," and they can be understood for a particular model without talking about data. Though the actual that minimizes the loss is determined by data, so it's sort of like the "menu" of degeneracies are data-independent, and the data "selects one off the menu." Degeneracies imply singularities, but not necessarily vice-versa, so they aren't everything. But we do think that degeneracies will be fairly important in practice.
I’ve been trying to understand modules for a long time. They’re a particular algebraic structure in commutative algebra which seems to show up everywhere any time you get anywhere close to talking about rings - and I could never figure out why. Any time I have some simple question about algebraic geometry, for instance, it almost invariably terminates in some completely obtuse property of some module. This confused me. It was never particularly clear to me from their definition why modules should be so central, or so “deep.”
I’m going to try to explain the intuition I have now, mostly to clarify this for myself, but also incidentally in the hope of clarifying this for other people. I’m just a student when it comes to commutative algebra, so inevitably this is going to be rather amateur-ish and belabor obvious points, but hopefully that leads to something more understandable to beginners. This will assume familiarity with basic abstract algebra. Unless stated otherwise, I’ll restrict to commutative rings because I don’t understand much about non-commutative ones.
The typical motivation for modules: "vector spaces but with rings"
The typical way modules are motivated is simple: they’re just vector spaces, but you relax the definition so that you can replace the underlying field with a ring. That is, an R-module M is a ring R and an abelian group M, and an operation ⋅:R×M→M that respects the ring structure of R, i.e. for all r,s∈R and x,y∈M:
This is literally the definition of a vector space, except we haven’t required our scalars R to be a field, only a ring (i.e. multiplicative inverses don’t have to always exist). So, like, instead of multiplying vectors by real numbers or something, you multiply vectors by integers, or a polynomial - those are your scalars now. Sounds simple, right? Vector spaces are pretty easy to understand, and nobody really thinks about the underlying field of a vector space anyway, so swapping the field out shouldn’t be much of a big deal, right?
As you might guess from my leading questions, this intuition is profoundly wrong and misleading. Modules are far more rich and interesting than vector spaces. Vector spaces are quite boring; two vector spaces over the same field are isomorphic if and only if they have the same dimension. This means vector spaces are typically just thought of as background objects, a setting which you quickly establish before building the interesting things on top of. This is not the role modules play in commutative algebra - they carry far more structure intrinsically.
A better motivation for modules: "ring actions"
Instead, the way you should think about a module is as a representation of a ring or a ring action, directly analogous to group actions or group representations.[1] Modules are a manifestation of how the ring acts on objects in the “real world.”[2] Think about how group actions tell you how groups manifest in the real world: the dihedral group D4 is a totally abstract object, but it manifests concretely in things like “mirroring/rotating my photo in Photoshop,” where implicitly there is now an action of D4 on the pixels of my photo.
In fact, we can define modules just by generalizing the idea of group actions to ring actions. How might we do this? Well, remember that a group action of a group G on a set M is just a group homomorphism G→End(M), where End(M) is the set of all endomorphisms on M (maps from M to itself). In other words, for every element of G, we just have to specify how it transforms elements of M. Can we get a “ring action” of a ring R on a set M just by looking at a ring homomorphism R→End(M)?
That doesn’t quite work, because End(M) is in general a group, not a ring - what would the “addition” operation be? But if we insist that M is a group, giving it an addition operation, that turns End(M) into a ring, with the operation (f+g)(x)=f(x)+g(x). (This addition rule only works to produce another endomorphism if M is abelian, which is a neat little proof in itself!). Now our original idea works: we can define a “ring action” of a ring R on an abelian group M as a ring homomorphism R→End(M).
It turns out that this is equivalent to our earlier definition of a module: our “ring action” of R on M is literally just the definition of an R-module M. We got to the definition of a module just by taking the definition of a group action and generalizing to rings in the most straightforward possible way.
To make this more concrete, let’s look at a simple illustrative example, which connects modules directly to linear algebra.
A module structure is a ring homomorphism ϕ:C[x]→End(C2). The ring End(C2) is just the ring of 2x2 complex matrices, M2(C).
Now, a homomorphism out of a polynomial ring like C[x] is completely determined by one thing: where you send the variable x. So, to define a whole module structure, all we have to do is choose a single 2x2 matrix T to be the image of x.
T=ϕ(x)
The action of any other polynomial, say p(x)=anxn+⋯+a0, on a vector v is then automatically defined: p(x)⋅v=(anTn+⋯+a0I)v. In short, the study of C[x]-modules on C2 is literally the same thing as the study of 2x2 matrices.
Thinking this way immediately clears up some confusing “module” properties. For instance, in a vector space, if a scalar c times a vector v is zero (c⋅v=0), then either c=0 or v=0. This fails for modules. In our C[x]-module defined by the matrix T=(0100), the polynomial x2 is not zero, but for any vector v, x2⋅v=T2v=0v=0. This phenomenon, called torsion, feels strange from a vector space view, but from the “ring action” view, it's natural: some of our operators (like T2) can just happen to be the zero transformation.
Why are modules so deep?
Seeing modules as “ring actions” starts to explain why you should care about them, but it still doesn’t explain why the theory of modules is so deep. After all, if we compare to group actions, the theory of group actions is in some sense just the theory of groups - there’s really not much to say about a faithful group action beyond the underlying group itself. Group theorists don’t spend much time studying group actions in their own right - why do commutative algebraists seem so completely obsessed with modules? Why are modules more interesting?
The short answer is that the theory of modules is far richer: even when it comes to “faithful” modules, there may be multiple, non-isomorphic modules for the same ring R and abelian group M. In other words, unlike group actions, the ways in which a given ring can manifest concretely are not unique, even when acting faithfully on the same object.
To show this concretely, let’s return to our C[x] module example. Remember that a C[x]-module with structure ϕ:C[x]→End(C2) is determined just by ϕ(x), where we choose to send the element x. We can define two different modules here which are not isomorphic:
These are radically different modules.
The underlying abelian group C2 is the same in both cases. The ring C[x] is the same. But the "action" - which is determined by our choice of the matrix ϕ(x) - creates completely different algebraic objects.
The analogy: a group is a person, a ring is an actor
This was quite a perspective shift for me, and something I had trouble thinking about intuitively. A metaphor I found helpful is to think of it like this: a group is like a person, while a ring is like an actor.
Using our C[x] example: The actor is the polynomial ring, C[x]. This actor has immense range.
The "actor" is the same. The "stage" is the same. But the "performances" are fundamentally different. To understand the actor C[x], you cannot just watch one performance. You gain a more complete appreciation for their range, subtlety, and power by seeing all the different roles they can inhabit.
Why can rings manifest in multiple ways?
It’s worth pausing for a moment to notice the suprising fact that modules can have this sort of intricate structure, when group actions don’t. What is it about rings that lets them act in so many different “roles”, in a way that groups can’t?
In short, the ring axioms force the action of a ring’s elements to behave like linear transformations rather than simple permutations.
The object is its category of actions
Part of the reason why I like this metaphor is because it points towards a deeper understanding of how we should think about rings in the first place. This leads to, from my position, one of the most fascinating ideas in modern algebra:
The "identity" of the ring is not its list of elements, but the totality of all the ways it can manifest - all the performances it can give.
So, why care?
Alright, so hopefully some of that makes sense. We started with the (in my opinion) unhelpful idea of a module as a "vector space over a ring" and saw that it's much more illuminating to see it as a "ring action" - the way a ring shows up in the world. The story got deeper when we realized that, unlike groups, rings can show up in many fundamentally different ways, analogous to an actor playing a vast range of roles.
So where does this leave us? For me, at least, modules don't feel as alien anymore. At the risk of being overly glib, the reason modules seem to show up everywhere is because they are everywhere. They are the language we use to talk about the manifestations of rings. When a property of a geometric space comes down to a property of a module, it's because the ring of functions on that space is performing one of its many roles, and the module is that performance.
There’s further you could take this (structure of modules over PIDs, homological algebra, etc), but this seems like a good place to stop for now. Hopefully this was helpful!
In fact, modules literally generalize group representation theory!
Sometimes in a quite literal way. In quantum mechanics, the ring is an algebra of observables, and a specific physical system is defined by choosing a specific operator for the Hamiltonian, H. This is directly analogous to how we chose a specific matrix T for x to define our module, as in the example of C[x]-modules we study later.