Executive director at Timaeus. Working on singular learning theory and developmental interpretability.
Website: jessehoogland.com
Twitter: @jesse_hoogland
Jesse Hoogland(Jesse Hoogland)
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Right. SLT tells us how to operationalize and measure (via the LLC) basin volume in general for DL. It tells us about the relation between the LLC and meaningful inductive biases in the particular setting described in this post. I expect future SLT to give us meaningful predictions about inductive biases in DL in particular.
The post is live here.
If we actually had the precision and maturity of understanding to predict this “volume” question, we’d probably (but not definitely) be able to make fundamental contributions to DL generalization theory + inductive bias research.
Obligatory singular learning theory plug: SLT can and does make predictions about the “volume” question. There will be a post soon by @Daniel Murfet that provides a clear example of this.
You can find a v0 of an SLT/devinterp reading list here. Expect an updated reading list soon (which we will cross-post to LW).
Our work on the induction bump is now out. We find several additional “hidden” transitions, including one that splits the induction bump in two: a first part where previous-token heads start forming, and a second part where the rest of the induction circuit finishes forming.
The first substage is a type-B transition (loss changing only slightly, complexity decreasing). The second substage is a more typical type-A transition (loss decreasing, complexity increasing). We’re still unclear about how to understand this type-B transition structurally. How is the model simplifying? E.g., is there some link between attention heads composing and the basin broadening?
As a historical note / broader context, the worry about model class over-expressivity has been there in the early days of Machine Learning. There was a mistrust of large blackbox models like random forest and SVM and their unusually low test or even cross-validation loss, citing ability of the models to fit noise. Breiman frank commentary back in 2001, “Statistical Modelling: The Two Cultures”, touch on this among other worries about ML models. The success of ML has turn this worry into the generalisation puzzle. Zhang et. al. 2017 being a call to arms when DL greatly exacerbated the scale and urgency of this problem.
Yeah it surprises me that Zhang et al. (2018) has had the impact it did when, like you point out, the ideas have been around for so long. Deep learning theorists like Telgarsky point to it as a clear turning point.
Naive optimism: hopefully progress towards a strong resolution to the generalisation puzzle give us understanding enough to gain control on what kind of solutions are learned. And one day we can ask for more than generalisation, like “generalise and be safe”.
This I can stand behind.
Thanks for raising that, it’s a good point. I’d appreciate it if you also cross-posted this to the approximation post here.
I think this mostly has to do with the fact that learning theory grew up in/next to computer science where the focus is usually worst-case performance (esp. in algorithmic complexity theory). This naturally led to the mindset of uniform bounds. That and there’s a bit of historical contingency: people started doing it this way, and early approaches have a habit of sticking.
This is probably true for neural networks in particular, but mathematically speaking, it completely depends on how you parameterise the functions. You can create a parameterisation in which this is not true.
Agreed. So maybe what I’m actually trying to get at it is a statement about what “universality” means in the context of neural networks. Just as the microscopic details of physical theories don’t matter much to their macroscopic properties in the vicinity of critical points (“universality” in statistical physics), just as the microscopic details of random matrices don’t seem to matter for their bulk and edge statistics (“universality” in random matrix theory), many of the particular choices of neural network architecture doesn’t seem to matter for learned representations (“universality” in DL).
What physics and random matrix theory tell us is that a given system’s universality class is determined by its symmetries. (This starts to get at why we SLT enthusiasts are so obsessed with neural network symmetries.) In the case of learning machines, those symmetries are fixed by the parameter-function map, so I totally agree that you need to understand the parameter-function map.
However, focusing on symmetries is already a pretty major restriction. If a universality statement like the above holds for neural networks, it would tell us that most of the details of the parameter-function map are irrelevant.
There’s another important observation, which is that neural network symmetries leave geometric traces. Even if the RLCT on its own does not “solve” generalization, the SLT-inspired geometric perspective might still hold the answer: it should be possible to distinguish neural networks from the polynomial example you provided by understanding the geometry of the loss landscape. The ambitious statement here might be that all the relevant information you might care about (in terms of understanding universality) are already contained in the loss landscape.
If that’s the case, my concern about focusing on the parameter-function map is that it would pose a distraction. It could miss the forest for the trees if you’re trying to understand the structure that develops and phenomena like generalization. I expect the more fruitful perspective to remain anchored in geometry.
Is this not satisfied trivially due to the fact that the RLCT has a certain maximum and minimum value within each model class? (If we stick to the assumption that is compact, etc.)
Hmm, maybe restrict so it has to range over .
The easiest way to explain why this is the case will probably be to provide an example. Suppose we have a Bayesian learning machine with 15 parameters, whose parameter-function map is given by
and whose loss function is the KL divergence. This learning machine will learn 4-degree polynomials.
I’m not sure, but I think this example is pathological. One possible reason for this to be the case is that the symmetries in this model are entirely “generic” or “global.” The more interesting kinds of symmetry are “nongeneric” or “local.”
What I mean by “global” is that each point in the parameter space has the same set of symmetries (specifically, the product of a bunch of hyperboloids ). In neural networks there are additional symmetries that are only present for a subset of the weights. My favorite example of this is the decision boundary annihilation (see below).For the sake of simplicity, consider a ReLU network learning a 1D function (which is just piecewise linear approximation). Consider what happens when you you rotate two adjacent pieces so they end up sitting on the same line, thereby “annihilating” the decision boundary between them, so this now-hidden decision boundary no longer contributes to your function. You can move this decision boundary along the composite linear piece without changing the learned function, but this only holds until you reach the next decision boundary over. I.e.: this symmetry is local. (Note that real-world networks actually seem to take advantage of this property.)
This is the more relevant and interesting kind of symmetry, and it’s easier to see what this kind of symmetry has to do with functional simplicity: simpler functions have more local degeneracies. We expect this to be true much more generally — that algorithmic primitives like conditional statements, for loops, composition, etc. have clear geometric traces in the loss landscape.
So what we’re really interested in is something more like the relative RLCT (to the model class’s maximum RLCT). This is also the relevant quantity from a dynamical perspective: it’s relative loss and complexity that dictate transitions, not absolute loss or complexity.
This gets at another point you raised:
2. It is a type error to describe a function as having low RLCT. A given function may have a high RLCT or a low RLCT, depending on the architecture of the learning machine.
You can make the same critique of Kolmogorov complexity. Kolmogorov complexity is defined relative to some base UTM. Fixing a UTM lets you set an arbitrary constant correction. What’s really interesting is the relative Kolmogorov complexity.
In the case of NNs, the model class is akin to your UTM, and, as you show, you can engineer the model class (by setting generic symmetries) to achieve any constant correction to the model complexity. But those constant corrections are not the interesting bit. The interesting bit is the question of relative complexities. I expect that you can make a statement similar to the equivalence-up-to-a-constant of Kolmogorov complexity for RLCTs. Wild conjecture: given two model classes and and some true distribution , their RLCTs obey:
where is some monotonic function.
I think there’s some chance of models executing treacherous turns in response to a particular input, and I’d rather not trigger those if the model hasn’t been sufficiently sandboxed.
One would really want to know if the complexity measure can predict ‘emergence’ of capabilities like inner-monologue, particularly if you can spot previously-unknown capabilities emerging which may not be covered in any of your existing benchmarks.
That’s our hope as well. Early ongoing work on toy transformers trained to perform linear regression seems to bear out that lambdahat can reveal transitions where the loss can’t.
But this type of ‘emergence’ tends to happen with such expensive models that the available checkpoints are too separated to be informative (if you get an emergence going from 1b vs 10b vs 100b, what does it mean to compute a complexity measure there? You’d really want to compare them at wherever the emergence actually really happens, like 73.5b vs 74b, or whatever.)
The kind of emergence we’re currently most interested in is emergence over training time, which makes studying these transitions much more tractable (the main cost you’re paying is storage for checkpoints, and storage is cheap). It’s still a hurdle in that we have to start training large models ourselves (or setting up collaborations with other labs).
But the induction bump happens at pretty small (ie. cheap) model sizes, so it could be replicated many times and in many ways within-training-run and across training-runs, and one see how the complexity metric reflects or predicts the induction bump. Is that one of the ‘hidden’ transitions you plan to test? And if not, why not?
The induction bump is one of the main things we’re looking into now.
Oops yes this is a typo. Thanks for pointing it out.
Should be fixed now, thank you!
To be clear, our policy is not publish-by-default. Our current assessment is that the projects we’re prioritizing do not pose a significant risk of capabilities externalities. We will continue to make these decisions on a per-project basis.
We don’t necessarily expect all dangerous capabilities to exhibit phase transitions. The ones that do are more dangerous because we can’t anticipate them, so this just seems like the most important place to start.
It’s an open question to what extent the lottery-ticket style story of a subnetwork being continually upweighted contradicts (or supports) the phase transition perspective. Just because a subnetwork’s strength is growing constantly doesn’t mean its effect on the overall computation is. Rather than grokking, which is a very specific kind of phase transition, it’s probably better to have in mind the emergence of in-context learning in tandem with induction heads, which seems to us more like the typical case we’re interested in when we speak about structure in neural networks developing across training.
We expect there to be a deeper relation between degeneracy and structure. As an intuition pump, think of a code base where you have two modules communicating across some API. Often, you can change the interface between these two modules without changing the information content being passed between them and without changing their internal structure. Degeneracy — the ways in which you can change your interfaces — tells you something about the structure of these circuits, the boundaries between them, and maybe more. We’ll have more to say about this in the future.
Now that the deadline has arrived, I wanted to share some general feedback for the applicants and some general impressions for everyone in the space about the job market:
My number one recommendation for everyone is to work on more legible projects and outputs. A super low-hanging fruit for >50% of the applications would be to clean up your GitHub profiles or to create a personal site. Make it really clear to us which projects you’re proud of, so we don’t have to navigate through a bunch of old and out-of-use repos from classes you took years ago. We don’t have much time to spend on every individual application, so you want to make it really easy for us to become interested in you. I realize most people don’t even know how to create a GitHub profile page, so check out this guide.
We got 70 responses and will send out 10 invitations for interviews.
We rejected a reasonable number of decent candidates outright because they were looking for part-time work. If this is you, don’t feel dissuaded.
There were quite a few really bad applications (...as always): poor punctuation/capitalization, much too informal, not answering the questions, totally unrelated background, etc. Two suggestions: (1) If you’re the kind of person who is trying to application-max, make sure you actually fill in the application. A shitty application is actually worse than no application, and I don’t know why I have to say that. (2) If English is not your first language, run your answers through ChatGPT. GPT-3.5 is free. (Actually, this advice is for everyone).
Between 5 and 10 people expressed interest in an internship option. We’re going to think about this some more. If this includes you, and you didn’t mention it in your application, please reach out.
Quite a few people came from a data science / analytics background. Using ML techniques is actually pretty different from researching ML techniques, so for many of these people I’d recommend you work on some kind of project in interpretability or related areas to demonstrate that you’re well-suited to this kind of research.
Remember that job applications are always noisy. We almost certainly made mistakes, so don’t feel discouraged!
Hey Thomas, I wrote about our reasoning for this in response to Winston:
All in all, we’re expecting most of our hires to come from outside the US where the cost of living is substantially lower. If lower wages are a deal-breaker for anyone but you’re still interested in this kind of work, please flag this in the form. The application should be low-effort enough that it’s still worth applying.
Hey Winston, thanks for writing this out. This is something we talked a lot about internally. Here are a few thoughts:
Comparisons: At 35k a year, it seems it might be considerably lower than industry equivalent even when compared to other programs
I think the more relevant comparison is academia, not industry. In academia, $35k is (unfortunately) well within in the normal range for RAs and PhD students. This is especially true outside the US, where wages are easily 2x − 4x lower.
Often academics justify this on the grounds that you’re receiving more than just monetary benefits: you’re receiving mentorship and training. We think the same will be true for these positions.
The actual reason is that you have to be somewhat crazy to even want to go into research. We’re looking for somewhat crazy.
If I were applying to this, I’d feel confused and slightly underappreciated if I had the right set of ML/Software Engineering skills but to be barely paid subsistence level for my full-time work (in NY).
If it helps, we’re paying ourselves even less. As much as we’d like to pay the RAs (and ourselves) more, we have to work with what we have.
Of course… money is tight: The grant constraint is well acknowledged here. But potentially the number of RAs expected to hire can be further down adjusted as while potentially increasing the submission rate of the candidates that truly fits the requirement of the research program.
For exceptional talent, we’re willing to pay higher wages.
The important thing is that both funding and open positions are exceptionally scarce. We expect there to be enough strong candidates who are willing to take the pay cut.
All in all, we’re expecting most of our hires to come from outside the US where the cost of living is substantially lower. If lower wages are a deal-breaker for anyone but you’re still interested in this kind of work, please flag this in the form. The application should be low-effort enough that it’s still worth applying.
If you’ll allow linguistics, Pāṇini was two and a half thousand years ahead of modern descriptive linguists.