Toy Models of Superposition

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A new Anthropic interpretability paper—“Toy Models of Superpostion”—came out last week that I think is quite exciting and hasn’t been discussed here yet.

Twitter thread from Anthropic

Twitter thread from Chris

Introduction:

It would be very convenient if the individual neurons of artificial neural networks corresponded to cleanly interpretable features of the input. For example, in an “ideal” ImageNet classifier, each neuron would fire only in the presence of a specific visual feature, such as the color red, a left-facing curve, or a dog snout. Empirically, in models we have studied, some of the neurons do cleanly map to features. But it isn’t always the case that features correspond so cleanly to neurons, especially in large language models where it actually seems rare for neurons to correspond to clean features. This brings up many questions. Why is it that neurons sometimes align with features and sometimes don’t? Why do some models and tasks have many of these clean neurons, while they’re vanishingly rare in others?

In this paper, we use toy models — small ReLU networks trained on synthetic data with sparse input features — to investigate how and when models represent more features than they have dimensions. We call this phenomenon superposition. When features are sparse, superposition allows compression beyond what a linear model would do, at the cost of “interference” that requires nonlinear filtering.

Consider a toy model where we train an embedding of five features of varying importance in two dimensions, add a ReLU afterwards for filtering, and vary the sparsity of the features. With dense features, the model learns to represent an orthogonal basis of the most important two features (similar to what Principal Component Analysis might give us), and the other three features are not represented. But if we make the features sparse, this changes:

Not only can models store additional features in superposition by tolerating some interference, but we’ll show that, at least in certain limited cases, models can perform computation while in superposition. (In particular, we’ll show that models can put simple circuits computing the absolute value function in superposition.) This leads us to hypothesize that the neural networks we observe in practice are in some sense noisily simulating larger, highly sparse networks. In other words, it’s possible that models we train can be thought of as doing “the same thing as” an imagined much-larger model, representing the exact same features but with no interference.

Feature superposition isn’t a novel idea. A number of previous interpretability papers have speculated about it, and it’s very closely related to the long-studied topic of compressed sensing in mathematics, as well as the ideas of distributed, dense, and population codes in neuroscience and deep learning. What, then, is the contribution of this paper?

For interpretability researchers, our main contribution is providing a direct demonstration that superposition occurs in artificial neural networks given a relatively natural setup, suggesting this may also occur in practice. We offer a theory of when and why this occurs, revealing a phase diagram for superposition. We also discover that, at least in our toy model, superposition exhibits complex geometric structure.

But our results may also be of broader interest. We find preliminary evidence that superposition may be linked to adversarial examples and grokking, and might also suggest a theory for the performance of mixture of experts models. More broadly, the toy model we investigate has unexpectedly rich structure, exhibiting phase changes, a geometric structure based on uniform polytopes, “energy level”-like jumps during training, and a phenomenon which is qualitatively similar to the fractional quantum Hall effect in physics. We originally investigated the subject to gain understanding of cleanly-interpretable neurons in larger models, but we’ve found these toy models to be surprisingly interesting in their own right.

KEY RESULTS FROM OUR TOY MODELS

In our toy models, we are able to demonstrate that:

• Superposition is a real, observed phenomenon.

• Both monosemantic and polysemantic neurons can form.

• At least some kinds of computation can be performed in superposition.

• Whether features are stored in superposition is governed by a phase change.

• Superposition organizes features into geometric structures such as digons, triangles, pentagons, and tetrahedrons.

Our toy models are simple ReLU networks, so it seems fair to say that neural networks exhibit these properties in at least some regimes, but it’s very unclear what to generalize to real networks.

And the beginning of their section on the overall strategic picture, which echoes some of my sentiment from “A transparency and interpretability tech tree”, including the importance of universal quantification (what I called “worst-case transparency”) and unknown unknowns:

The Strategic Picture of Superposition

Although superposition is scientifically interesting, much of our interest comes from a pragmatic motivation: we believe that superposition is deeply connected to the challenge of using interpretability to make claims about the safety of AI systems. In particular, it is a clear challenge to the most promising path we see to be able to say that neural networks won’t perform certain harmful behaviors or to catch “unknown unknowns” safety problems. This is because superposition is deeply linked to the ability to identify and enumerate over all features in a model, and the ability to enumerate over all features would be a powerful primitive for making claims about model behavior.

We begin this section by describing how “solving superposition” in a certain sense is equivalent to many strong interpretability properties which might be useful for safety. Next, we’ll describe three high level strategies one might take to “solving superposition.” Finally, we’ll describe a few other additional strategic considerations.

Safety, Interpretability, & “Solving Superposition”

We’d like a way to have confidence that models will never do certain behaviors such as “deliberately deceive” or “manipulate.” Today, it’s unclear how one might show this, but we believe a promising tool would be the ability to identify and enumerate over all features. The ability to have a universal quantifier over the fundamental units of neural network computation is a significant step towards saying that certain types of circuits don’t exist. 18 It also seems like a powerful tool for addressing “unknown unknowns”, since it’s a way that one can fully cover network behavior, in a sense.

How does this relate to superposition? It turns out that the ability to enumerate over features is deeply intertwined with superposition. One way to see this is to imagine a neural network with a privileged basis and without superposition (like the monosemantic neurons found in early InceptionV1): features would simply correspond to neurons, and you could enumerate over features by enumerating over neurons. The connection also goes the other way: if one has the ability to enumerate over features, one can perform compressed sensing using the feature directions to (with high probability) “unfold” a superposition models activations into those of a larger, non-superposition model. For this reason, we’ll call any method that gives us the ability to enumerate over features – and equivalently, unfold activations – a “solution to superposition”. Any solution is on the table, from creating models that just don’t have superposition, to identifying what directions correspond to features after the fact. We’ll discuss the space of possibilities shortly.

We’ve motivated “solving superposition” in terms of feature enumeration, but it’s worth noting that it’s equivalent to (or necessary for) many other interpretability properties one might care about:

• Decomposing Activation Space. The most fundamental challenge of any interpretability agenda is to defeat the curse of dimensionality. For mechanistic interpretability, this ultimately reduces to whether we can decompose activation space into independently understandable components, analogous to how computer program memory can be decomposed into variables. Identifying features is what allows us to decompose the model in terms of them.

• Describing Activations in Terms of Pure Features. One of the most obvious casualties of superposition is that we can’t describe activations in terms of pure features. When features are relatively basis aligned, we can take an activation – say the activations for a dog head in a vision model – and decompose them into individual underlying features, like a floppy ear, short golden fur, and a snout. (See the “semantic dictionary” interface in Building Blocks). Solving superposition would allow us to do this for every model.

• Understanding Weights (ie. Circuit Analysis). Neural network weights can typically only be understood when they’re connecting together understandable features. All the circuit analysis seen in the original circuit thread (see especially), see specially was fundamentally only possible because the weights connected non-polysemantic neurons. We need to solve superposition for this to work in general.

• Even very basic approaches become perilous with superposition. It isn’t just sophisticated approaches to interpretability which are harmed by superposition. Even very basic methods one might consider become unreliable. For example, if one is concerned about language models exhibiting manipulative behavior, one might ask if an input has a significant cosine similarity to the representations of other examples of deceptive behavior. Unfortunately, superposition means that cosine similarity has the potential to be misleading, since unrelated features start to be embedded with positive dot products to each other. However, if we solve superposition, this won’t be an issue – either we’ll have a model where features align with neurons, or a way to use compressed sensing to lift features to a space where they no longer have positive dot products.

[Alex_Altair]

This paper, like others from Anthropic, is is exemplary science and exceptional science communication. The authors are clear, precise and thorough. It is evident that their research motivation is to solve a problem, and not to publish a paper, and that their communication motivation is to help others understand, and not to impress.

[RogerDearnaley]

I thought this was a very interesting paper — I particularly liked the relationship to phase transitions.

However, I think there’s a likely to be another ‘phase’ that they don’t discuss (possibly it didn’t crop up in their small models, since it’s only viable in a sufficiently large model): one where you pack a very large number of features (thousands or millions, say) into a fairly large number of dimensions (hundreds, say). In spaces with dimensionality >= O(100), the statistics of norm and dot product are such that even randomly chosen unit norm vectors are almost invariably nearly orthogonal. So gradient descent would have very little work to do to find a very large number of vectors (much larger than the number of dimensions) that are all mutually almost-orthogonal, so that show very little interference between them. This is basically the limiting case of the pattern observed in the paper of packing n features in superposition into d dimensions where n > d >= 1, taking this towards the limit where n >> d >> 1.

Intuitively this phase seems particularly likely in a context like the residual stream of an LLM, where (in theory, if not in practice) the embedding space is invariant under arbitrary rotations, so there’s no obvious reason to expect vectors to align with the coordinate axes. On the other hand, in a system where there was a preferred basis (such as a system using L1 regularization), you might get such vectors that were themselves sparse, with most components zero but a significant number of non-zero components, enough for the randomness to still give low interference.

More speculatively, in a neural net that was using at least some of its neurons in this high-dimensionality dense superposition phase, the model will presumably learn ways to manipulate these vectors to do computation in superposition. One possibility for this might be methods comparable to some of the possible Vector Symbolic Architectures (also known as hyperdimensional computing) outlined in e.g. https://​​arxiv.org/​​pdf/​​2106.05268.pdf. Of the primitives used in that, a fully connected layer can clearly be trained to implement both addition of vectors and permutations of their elements, I suspect something functionally comparable to the vector elementwise-multiplication (Hadamard product) operation could be produced by using the nonlinearity of a smooth activation function such as GELU or Swish, and I suspect their their clean-up memory operation could be implemented using attention. If it turned out to be the case that SGD actually often finds solutions of this form, then an understanding of vector symbolic architectures might be helpful for interpretability of models where portions of them used this phase. This seems most likely in models that need to pack vast numbers of features into large numbers of dimensions, such as modern large LLMs.

[Charlie Steiner]

I think this really incentivizes things like network dissection over “interpret this neuron” approaches.

[Slider]

I dislike you of the word “superposition” for the concept. It doesn’t seem very analogical to the physical superposition. And having two complex concepts next to each other will produce, ironically enough, a lto of semantic interference. Holography seems more fitting.