A Mathematical Framework for Transformer Circuits

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### Contents

[Summary of Results](#summary-of-results)
[Transformer Overview](#transformer-overview)
* [Model Simplifications](#model-simplifications)
* [High-Level Architecture](#high-level-architecture)
* [Virtual Weights and the Residual Stream as a Communication Channel](#residual-comms)
* [Attention Heads are Independent and Additive](#architecture-attn-independent)
* [Attention Heads as Information Movement](#architecture-attn-as-movement)

[Zero-Layer Transformers](#zero-layer-transformers)
[One-Layer Attention-Only Transformers](#one-layer-attention-only-transformers)
* [The Path Expansion Trick](#onel-path-expansion)
* [Splitting Attention Head terms into Query-Key and Output-Value Circuits](#splitting-attention-head-terms-into-circuits)
* [Interpretation as Skip-Trigrams](#interpretation-as-skip-trigrams)
* [Summarizing OV/QK Matrices](#summarizing-ovqk-matrices)
* [Do We "Fully Understand" One-Layer Models?](#do-we-fully-understand-one-layer-models)

[Two-Layer Attention-Only Transformers](#two-layer-attention-only-transformers)
* [Three Kinds of Composition](#three-kinds-of-composition)
* [Path Expansion of Logits](#path-expansion-of-logits)
* [Path Expansion of Attention Scores QK Circuit](#path-expansion-of-attention-scores-qk-circuit)
* [Analyzing a Two-Layer Model](#analyzing-a-two-layer-model)
* [Induction Heads](#induction-heads)
* [Term Importance Analysis](#term-importance-analysis)
* [Virtual Attention Heads](#virtual-attention-heads)

[Where Does This Leave Us?](#where-does-this-leave-us)
[Related Work](#related-work)

[Comments](#comments)
[Acknowledgments](#acknowledgments)
[Author Contributions](#author-contributions)
[Citation Information](#citation)
[Additional Intuition and Observations](#additional-intuition)
[Notation](#notation)
[Technical Details](#technical-details)

Transformer language models are an emerging technology that is gaining increasingly broad real-world use, for example in systems like GPT-3 , LaMDA , Codex , Meena , Gopher , and similar models.  However, as these models scale, their open-endedness and high capacity creates an increasing scope for unexpected and sometimes harmful behaviors.  Even years after a large model is trained, both creators and users routinely discover model capabilities – including problematic behaviors – they were previously unaware of.

One avenue for addressing these issues is mechanistic interpretability, attempting to reverse engineer the detailed computations performed by transformers, similar to how a programmer might try to reverse engineer complicated binaries into human-readable source code.  If this were possible, it could potentially provide a more systematic approach to explaining current safety problems, identifying new ones, and perhaps even anticipating the safety problems of powerful future models that have not yet been built.  A previous project, the [Distill Circuits thread](https://distill.pub/2020/circuits/) , has attempted to reverse engineer vision models, but so far there hasn’t been a comparable project for transformers or language models.

In this paper, we attempt to take initial, very preliminary steps towards reverse-engineering transformers.  Given the incredible complexity and size of modern language models, we have found it most fruitful to start with the simplest possible models and work our way up from there.  Our aim is to discover simple algorithmic patterns, motifs, or frameworks that can subsequently be applied to larger and more complex models.  Specifically, in this paper we will study transformers with two layers or less which have only attention blocks – this is in contrast to a large, modern transformer like GPT-3, which has 96 layers and alternates attention blocks with MLP blocks.

We find that by conceptualizing the operation of transformers in a new but mathematically equivalent way, we are able to make sense of these small models and gain significant understanding of how they operate internally.  Of particular note, we find that specific attention heads that we term “induction heads” can explain in-context learning in these small models, and that these heads only develop in models with at least two attention layers.  We also go through some examples of these heads operating in action on specific data.

We don’t attempt to apply to our insights to larger models in this first paper, but in a [forthcoming paper](https://transformer-circuits.pub/2022/in-context-learning-and-induction-heads/index.html), we will show that both our mathematical framework for understanding transformers, and the concept of induction heads, continues to be at least partially relevant for much larger and more realistic models – though we remain a very long way from being able to fully reverse engineer such models.

---

Summary of Results
------------------

#### Reverse Engineering Results

To explore the challenge of reverse engineering transformers, we reverse engineer several toy, attention-only models. In doing so we find:

* Zero layer transformers model bigram statistics. The bigram table can be accessed directly from the weights.
* One layer attention-only transformers are an ensemble of bigram and “skip-trigram” (sequences of the form "A… B C") models. The bigram and skip-trigram tables can be accessed directly from the weights, without running the model. These skip-trigrams can be surprisingly expressive. This includes implementing a kind of very simple in-context learning.
* Two layer attention-only transformers can implement much more complex algorithms using compositions of attention heads. These compositional algorithms can also be detected directly from the weights. Notably, two layer models use attention head composition to create “induction heads”, a very general in-context learning algorithm.We’ll explore induction heads in much more detail in a forthcoming paper.
* One layer and two layer attention-only transformers use very different algorithms to perform in-context learning. Two layer attention heads use qualitatively more sophisticated inference-time algorithms — in particular, a special type of attention head we call an induction head — to perform in-context-learning, forming an important transition point that will be relevant for larger models.

#### Conceptual Take-Aways

We’ve found that many subtle details of the transformer architecture require us to approach reverse engineering it in a pretty different way from how the InceptionV1 Circuits work . We’ll unpack each of these points in the sections below, but for now we briefly summarize. We’ll also expand on a lot of the terminology we introduce here once we get to the appropriate sections. (To be clear, we don't intend to claim that any of these points are necessarily novel; many are implicitly or explicitly present in other papers.)

* Attention heads can be understood as independent operations, each outputting a result which is added into the residual stream. Attention heads are often described in an alternate “concatenate and multiply” formulation for computational efficiency, but this is mathematically equivalent.
* Attention-only models can be written as a sum of interpretable end-to-end functions mapping tokens to changes in logits. These functions correspond to “paths” through the model, and are linear if one freezes the attention patterns.
* Transformers have an enormous amount of linear structure. One can learn a lot simply by breaking apart sums and multiplying together chains of matrices.
* Attention heads can be understood as having two largely independent computations: a QK (“query-key”) circuit which computes the attention pattern, and an OV (“output-value”) circuit which computes how each token affects the output if attended to.
* Key, query, and value vectors can be thought of as intermediate results in the computation of the low-rank matrices W\_Q^TW\_K and W\_OW\_V. It can be useful to describe transformers without reference to them.
* Composition of attention heads greatly increases the expressivity of transformers. There are three different ways attention heads can compose, corresponding to keys, queries, and values. Key and query composition are very different from value composition.
* All components of a transformer (the token embedding, attention heads, MLP layers, and unembedding) communicate with each other by reading and writing to different subspaces of the residual stream. Rather than analyze the residual stream vectors, it can be helpful to decompose the residual stream into all these different communication channels, corresponding to paths through the model.

---

Transformer Overview
--------------------

Before we attempt to reverse engineer transformers, it's helpful to briefly review the high-level structure of transformers and describe how we think about them.

In many cases, we've found it helpful to reframe transformers in equivalent, but non-standard ways. Mechanistic interpretability requires us to break models down into human-interpretable pieces. An important first step is finding the representation which makes it easiest to reason about the model. In modern deep learning, there is — for good reason! — a lot of emphasis on computational efficiency, and our mathematical descriptions of models often mirror decisions in how one would write efficient code to run the model. But when there are many equivalent ways to represent the same computation, it is likely that the most human-interpretable representation and the most computationally efficient representation will be different.

Reviewing transformers will also let us align on terminology, which can sometimes vary. We'll also introduce some notation in the process, but since this notation is used across many sections, we provide a detailed description of all notation in the [notation appendix](#notation) as a concise reference for readers.

### Model Simplifications

To demonstrate the ideas in this paper in their cleanest form, we focus on "toy transformers" with some simplifications.

In most parts of this paper, we will make a very substantive change: we focus on “attention-only” transformers, which don't have MLP layers. This is a very dramatic simplification of the transformer architecture. We're partly motivated by the fact that circuits with attention heads present new challenges not faced by the Distill circuits work, and considering them in isolation allows us to give an especially elegant treatment of those issues. But we've also simply had much less success in understanding MLP layers so far; in normal transformers with both attention and MLP layers there are many circuits mediated primarily by attention heads which we can study, some of which seem very important, but the MLP portions have been much harder to get traction on. This is a major weakness of our work that we plan to focus on addressing in the future. Despite this, we will have some discussion of transformers with MLP layers in later sections.

We also make several changes that we consider to be more superficial and are mostly made for clarity and simplicity. We do not consider biases, but a model with biases can always be simulated without them by folding them into the weights and creating a dimension that is always one. Additionally, biases in attention-only transformers mostly multiply out to functionally be biases on the logits. We also ignore layer normalization. It adds a fair amount of complexity to consider explicitly, and up to a variable scaling, layer norm can be merged into adjacent weights. We also expect that, modulo some implementational annoyances, layer norm could be substituted for batch normalization (which can fully be folded into adjacent parameters).

### High-Level Architecture

There are several variants of transformer language models. We focus on autoregressive, decoder-only transformer language models, such as GPT-3. (The original transformer paper had a special encoder-decoder structure to support translation, but many modern language models don't include this.)

A transformer starts with a token embedding, followed by a series of “residual blocks”, and finally a token unembedding. Each residual block consists of an attention layer, followed by an MLP layer. Both the attention and MLP layers each “read” their input from the residual stream (by performing a linear projection), and then “write” their result to the residual stream by adding a linear projection back in. Each attention layer consists of multiple heads, which operate in parallel.

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### Virtual Weights and the Residual Stream as a Communication Channel

One of the main features of the high level architecture of a transformer is that each layer adds its results into what we call the “residual stream.”Constructing models with a residual stream traces back to early work by the Schmidhuber group, such as highway networks  and LSTMs, which have found significant modern success in the more recent residual network architecture . In transformers, the residual stream vectors are often called the “embedding.” We prefer the residual stream terminology, both because it emphasizes the residual nature (which we believe to be important) and also because we believe the residual stream often dedicates subspaces to tokens other than the present token, breaking the intuitions the embedding terminology suggests. The residual stream is simply the sum of the output of all the previous layers and the original embedding. We generally think of the residual stream as a communication channel, since it doesn't do any processing itself and all layers communicate through it.

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The residual stream has a deeply linear structure.It's worth noting that the completely linear residual stream is very unusual among neural network architectures: even ResNets , the most similar architecture in widespread use, have non-linear activation functions on their residual stream, or applied whenever the residual stream is accessed! Every layer performs an arbitrary linear transformation to "read in" information from the residual stream at the start,This ignores the layer normalization at the start of each layer, but up to a constant scalar, the layer normalization is a constant affine transformation and can be folded into the linear transformation. See discussion of how we handle layer normalization in the appendix. and performs another arbitrary linear transformation before adding to "write" its output back into the residual stream. This linear, additive structure of the residual stream has a lot of important implications. One basic consequence is that the residual stream doesn't have a ["privileged basis"](#def-privileged-basis); we could rotate it by rotating all the matrices interacting with it, without changing model behavior.

#### Virtual Weights

An especially useful consequence of the residual stream being linear is that one can think of implicit "virtual weights" directly connecting any pair of layers (even those separated by many other layers), by multiplying out their interactions through the residual stream. These virtual weights are the product of the output weights of one layer with the input weightsNote that for attention layers, there are three different kinds of input weights: W\_Q,  W\_K,  and W\_V. For simplicity and generality, we think of layers as just having input and output weights here. of another (ie. W\_{I}^2W\_{O}^1), and describe the extent to which a later layer reads in the information written by a previous layer.

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#### Subspaces and Residual Stream Bandwidth

The residual stream is a high-dimensional vector space. In small models, it may be hundreds of dimensions; in large models it can go into the tens of thousands. This means that layers can send different information to different layers by storing it in different subspaces. This is especially important in the case of attention heads, since every individual head operates on comparatively small subspaces (often 64 or 128 dimensions), and can very easily write to completely disjoint subspaces and not interact.

Once added, information persists in a subspace unless another layer actively deletes it. From this perspective, dimensions of the residual stream become something like "memory" or "bandwidth". The original token embeddings, as well as the unembeddings, mostly interact with a relatively small fraction of the dimensions.We performed PCA analysis of token embeddings and unembeddings. For models with large d\_\text{model}, the spectrum quickly decayed, with the embeddings/unembeddings being concentrated in a relatively small fraction of the overall dimensions. To get a sense for whether they occupied the same or different subspaces, we concatenated the normalized embedding and unembedding matrices and applied PCA. This joint PCA process showed a combination of both "mixed" dimensions and dimensions used only by one; the existence of dimensions which are used by only one might be seen as a kind of upper bound on the extent to which they use the same subspace. This leaves most dimensions "free" for other layers to store information in.

It seems like we should expect residual stream bandwidth to be in very high demand! There are generally far more "computational dimensions" (such as neurons and attention head result dimensions) than the residual stream has dimensions to move information. Just a single MLP layer typically has four times more neurons than the residual stream has dimensions. So, for example, at layer 25 of a 50 layer transformer, the residual stream has 100 times more neurons as it has dimensions before it, trying to communicate with 100 times as many neurons as it has dimensions after it, somehow communicating in superposition! We call tensors like this ["bottleneck activations"](#def-bottleneck-activation) and expect them to be unusually challenging to interpret. (This is a major reason why we will try to pull apart the different streams of communication happening through the residual stream apart in terms of virtual weights, rather than studying it directly.)

Perhaps because of this high demand on residual stream bandwidth, we've seen hints that some MLP neurons and attention heads may perform a kind of "memory management" role, clearing residual stream dimensions set by other layers by reading in information and writing out the negative version.Some MLP neurons have very negative cosine similarity between their input and output weights, which may indicate deleting information from the residual stream. Similarly, some attention heads have large negative eigenvalues in their W\_OW\_V matrix and primarily attend to the present token, potentially serving as a mechanism to delete information. It's worth noticing that while these may be generic mechanisms for "memory management" deletion of information, they may also be mechanisms for conditionally deleting information, operating only in some cases.

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### Attention Heads are Independent and Additive

As seen above, we think of transformer attention layers as several completely independent attention heads h\in H which operate completely in parallel and each add their output back into the residual stream. But this isn't how transformer layers are typically presented, and it may not be obvious they're equivalent.

In the original Vaswani et al. paper on transformers , the output of an attention layer is described by stacking the result vectors r^{h\_1}, r^{h\_2},..., and then multiplying by an output matrix W\_O^H. Let's split W\_O^H into equal size blocks for each head [W\_O^{h\_1}, W\_O^{h\_2}...]. Then we observe that:

W\_O^H \left[\begin{matrix}r^{h\_1}\\r^{h\_2}\\... \end{matrix}\right] ~~=~~ \left[W\_O^{h\_1},~ W\_O^{h\_2},~ ... \right]\cdot\left[\begin{matrix}r^{h\_1}\\r^{h\_2}\\...\end{matrix}\right] ~~=~~ \sum\_i W\_O^{h\_i} r^{h\_i}

Revealing it to be equivalent to running heads independently, multiplying each by its own output matrix, and adding them into the residual stream. The concatenate definition is often preferred because it produces a larger and more compute efficient matrix multiply. But for understanding transformers theoretically, we prefer to think of them as independently additive.

### Attention Heads as Information Movement

But if attention heads act independently, what do they do? The fundamental action of attention heads is moving information. They read information from the residual stream of one token, and write it to the residual stream of another token. The main observation to take away from this section is that which tokens to move information from is completely separable from what information is “read” to be moved and how it is “written” to the destination.

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)
To see this, it’s helpful to write attention in a non-standard way. Given an attention pattern, computing the output of an attention head is typically described in three steps:

1. Compute the value vector for each token from the residual stream (v\_i = W\_V x\_i).
2. Compute the “result vector” by linearly combining value vectors according to the attention pattern (r\_i = \sum\_j A\_{i,j} v\_j).
3. Finally, compute the output vector of the head for each token (h(x)\_i = W\_O r\_i).As discussed above, often multiplication by the output matrix is written as one matrix multiply applied to the concatenated results of all heads; however this version is equivalent.

Each of these steps can be written as matrix multiply: why don’t we collapse them into a single step? If you think of x as a 2d matrix (consisting of a vector for each token), we’re multiplying it on different sides. W\_V and W\_O multiply the “vector per token” side, while A multiplies the “position” side. Tensors can offer us a much more natural language for describing this kind of map between matrices (if tensor product notation isn't familiar, we've included a [short introduction](#notation-tensor-product) in the notation appendix).  One piece of motivation that may be helpful is to note that we want to express linear maps from matrices to matrices: [n\_\text{context},~ d\_\text{model}] ~\to~ [n\_\text{context},~ d\_\text{model}]. Mathematicians call such linear maps "(2,2)-tensors" (they map two input dimensions to two output dimensions). And so tensors are the natural language for expressing this transformation.

Using tensor products, we can describe the process of applying attention as:

h(x) ~=~ (\text{Id} \otimes W\_O)~~\cdot~~ Project result vectors out for each token (h(x)\_i = W\_O r\_i) ~ (A \otimes \text{Id})~~\cdot~~~Mix value vectors *across* tokens to compute result vectors (r\_i = \sum\_j A\_{i,j} v\_j) ~ (\text{Id} \otimes W\_V)~~\cdot~~~ Compute value vector for each token (v\_i=W\_V x\_i)~~ x
Applying the mixed product property and collapsing identities yields:

h(x) ~=~ (A ~~\otimes~~ W\_O W\_V) ~~~\cdot~~~~~~A mixes across tokens while W\_OW\_V acts on each vector independently. x
What about the attention pattern? Typically, one computes the keys k\_i = W\_K x\_i, computes the queries q\_i = W\_Q x\_i and then computes the attention pattern from the dot products of each key and query vector A = \text{softmax}(q^T k). But we can do it all in one step without referring to keys and queries: A = \text{softmax}(x^T W\_Q^T W\_K x).

It's worth noting that although this formulation is mathematically equivalent, actually implementing attention this way (ie. multiplying by W\_O W\_V and W\_Q^T W\_K) would be horribly inefficient!

#### Observations about Attention Heads

A major benefit of rewriting attention heads in this form is that it surfaces a lot of structure which may have previously been harder to observe:

* Attention heads move information from the residual stream of one token to another.

* A corollary of this is that the residual stream vector space — which is often interpreted as a “contextual word embedding” — will generally have linear subspaces corresponding to information copied from other tokens and not directly about the present token.

* An attention head is really applying two linear operations, A and W\_OW\_V, which operate on different dimensions and act independently.

* A governs which token's information is moved from and to.
* W\_O W\_V governs which information is read from the source token and how it is written to the destination token.What do we mean when we say that W\_{OV}=W\_O W\_V governs which subspace of the residual stream the attention head reads and writes to when it moves information? It can be helpful to consider the singular value decomposition USV = W\_{OV}. Since d\_{head} < d\_{model}, W\_{OV} is low-rank and only a subset of the diagonal entries in S are non-zero. The right singular vectors V describe which subspace of the residual stream being attended to is “read in” (somehow stored as a value vector), while the left singular vectors U describe what subspace of the destination residual stream they are written to.

* A is the only non-linear part of this equation (being computed from a softmax). This means that if we fix the attention pattern, attention heads perform a linear operation. This also means that, without fixing A, attention heads are “half-linear” in some sense, since the per-token linear operation is constant.
* W\_Q and W\_K always operate together. They’re never independent. Similarly, W\_O and W\_V always operate together as well.

* Although they’re parameterized as separate matrices, W\_O W\_V and W\_Q^T W\_K can always be thought of as individual, low-rank matrices.
* Keys, queries and value vectors are, in some sense, superficial. They’re intermediary by-products of computing these low-rank matrices. One could easily reparametrize both factors of the low-rank matrices to create different vectors, but still function identically.
* Because W\_O W\_V and W\_Q W\_K always operate together, we like to define variables representing these combined matrices, W\_{OV} = W\_O W\_V and W\_{QK} = W\_Q^T W\_K.

* Products of attention heads behave much like attention heads themselves. By the distributive property, (A^{h\_2}\otimes W\_{OV}^{h\_2}) \cdot (A^{h\_1}\otimes W\_{OV}^{h\_1}) = (A^{h\_2}A^{h\_1})\otimes(W\_{OV}^{h\_2}W\_{OV}^{h\_1}). The result of this product can be seen as functionally equivalent to an attention head, with an attention pattern which is the composition of the two heads A^{h\_2}A^{h\_1} and an output-value matrix W\_{OV}^{h\_2}W\_{OV}^{h\_1}. We call these “virtual attention heads”, discussed in more depth later.

---

Zero-Layer Transformers
-----------------------

Watch videos covering similar content to this section: [0 layer theory](https://www.youtube.com/watch?v=V3NQaDR3xI4&list=PLoyGOS2WIonajhAVqKUgEMNmeq3nEeM51&index=1)

Before moving on to more complex models, it’s useful to briefly consider a “zero-layer” transformer. Such a model takes a token, embeds it, unembeds it to produce logits predicting the next token:

T ~=~ W\_U W\_E

Because the model cannot move information from other tokens, we are simply predicting the next token from the present token. This means that the optimal behavior of W\_U W\_E is to approximate the bigram log-likelihood.This parallels an observation by Levy & Goldberg, 2014 that many early word embeddings can be seen as matrix factorizations of a log-likelihood matrix.

This is relevant to transformers more generally. Terms of the form W\_U W\_E will occur in the expanded form of equations for every transformer, corresponding to the “direct path” where a token embedding flows directly down the residual stream to the unembedding, without going through any layers. The only thing it can affect is the bigram log-likelihoods. Since other aspects of the model will predict parts of the bigram log-likelihood, it won’t exactly represent bigram statistics in larger models, but it does represent a kind of “residual”. In particular, the W\_U W\_E term seems to often help represent bigram statistics which aren’t described by more general grammatical rules, such as the fact that “Barack” is often followed by “Obama”. An interesting corollary of this is to note is that, though W\_U is often referred to as the “un-embedding” matrix, we should not expect this to be the inverse of embedding with W\_E.

---

One-Layer Attention-Only Transformers
-------------------------------------

Watch videos covering similar content to this section: [1 layer theory](https://www.youtube.com/watch?v=7crsHGsh3p8&list=PLoyGOS2WIonajhAVqKUgEMNmeq3nEeM51&index=3), [1 layer results](https://www.youtube.com/watch?v=ZBlHFFE-ng8&list=PLoyGOS2WIonajhAVqKUgEMNmeq3nEeM51&index=4).

We claim that one-layer attention-only transformers can be understood as an ensemble of a bigram model and several "skip-trigram" models (affecting the probabilities of sequences "A… BC").Our use of the term "skip-trigram" to describe sequences of the form "A… BC" is inspired by Mikolov et al. 's use of the term "skip-gram" in their classic paper on word embeddings. Intuitively, this is because each attention head can selectively attend from the present token ("B") to a previous token ("A") and copy information to adjust the probability of possible next tokens ("C").

The goal of this section is to rigorously show this correspondence, and demonstrate how to convert the raw weights of a transformer into interpretable tables of skip-trigram probability adjustments.

### The Path Expansion Trick

Recall that a one-layer attention-only transformer consists of a token embedding, followed by an attention layer (which [independently applies](#architecture-attn-independent) attention heads), and finally an unembedding:

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QmCC
)
Using [tensor notation](#notation-tensor-product) and the [alternative representation of attention heads](#architecture-attn-as-movement) we previously derived, we can represent the transformer as a product of three terms.

![](data:image/PNG;base64,iVBORw0KGgoAAAANSUhEUgAABCEAAAJACAYAAACzCnw7AAAMbWlDQ1BJQ0MgUHJvZmlsZQAASImV
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Our key trick is to simply expand the product. This transforms the product (where every term corresponds to a layer), into a sum where every term corresponds to an end-to-end path.

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We claim each of these end-to-end path terms is tractable to understand, can be reasoned about independently, and additively combine to create model behavior.

The direct path term, \text{Id} \otimes W\_U W\_E, also occurred when we looked at the zero-layer transformer. Because it doesn’t move information between positions (that's what \text{Id} \otimes … denotes!), the only thing it can contribute to is the bigram statistics, and it will fill in missing gaps that other terms don’t handle there.

The more interesting terms are the attention head terms.

### Splitting Attention Head terms into Query-Key and Output-Value Circuits

For each attention head h we have a term A^h \otimes (W\_UW\_{OV}^hW\_E) where A^h= \text{softmax}\left( t^T \cdot W\_E^T W\_{QK}^h W\_E \cdot t \right). How can we map these terms to model behavior? And while we’re at it, why do we get these particular products of matrices on our equations?

The key thing to notice is that these terms consist of two separable operations, which are at their heart two [n\_\text{vocab},~ n\_\text{vocab}] matrices:

* W\_E^T W\_{QK}^h W\_E — We call this matrix the "query-key (QK) circuit." It provides the attention score for every query and key token. That is, each entry describes how much a given query token "wants" to attend to a given key token.
* W\_UW\_{OV}^hW\_E — We call this matrix the “Output-Value (OV) circuit.” It describes how a given token will affect the output logits if attended to.

To intuitively understand these products, it can be helpful to think of them as paths through the model, starting and ending at tokens. The QK circuit is formed by tracing the computation of a query and key vector up to their attention head, where they dot product to create a bilinear form. The OV circuit is created by tracing the path computing a value vector and continuing it through up to the logits.

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The attention pattern is a function of both the source and destination tokenTechnically, it is a function of all possible source tokens from the start to the destination token, as the softmax calculates the score for each via the QK circuit, exponentiates and then normalises, but once a destination token has decided how much to attend to a source token, the effect on the output is solely a function of that source token. That is, if multiple destination tokens attend to the same source token the same amount, then the source token will have the same effect on the logits for the predicted output token.

#### OV and QK Independence (The Freezing Attention Patterns Trick)

Thinking of the OV and QK circuits separately can be very useful, since they're both individually functions we can understand (linear or bilinear functions operating on matrices we understand).

But is it really principled to think about them independently? One thought experiment which might be helpful is to imagine running the model twice. The first time you collect the attention patterns of each head. This only depends on the QK circuit.In models with more than one layer, we'll see that the QK circuit can be more complicated than W\_E^T W\_{QK}^h W\_E. The second time, you replace the attention patterns with the "frozen" attention patterns you collected the first time. This gives you a function where the logits are a linear function of the tokens! We find this a very powerful way to think about transformers.

### Interpretation as Skip-Trigrams

One of the core challenges of mechanistic interpretability is to make neural network parameters meaningful by contextualizing them (see discussion by Voss et al. in [Visualizing Weights](https://distill.pub/2020/circuits/visualizing-weights/) ). By multiplying out the OV and QK circuits, we've succeeded in doing this: the neural network parameters are now simple linear or bilinear functions on tokens. The QK circuit determines which "source" token the present "destination" token attends back to and copies information from, while the OV circuit describes what the resulting effect on the "out" predictions for the next token is. Together, the three tokens involved form a "skip-trigram" of the form `[source]... [destination][out]`, and the "out" is modified.

It's important to note that this doesn't mean that interpretation is trivial. For one thing, the resulting matrices are enormous (our vocabulary is ~50,000 tokens, so a single expanded OV matrix has ~2.5 billion entries); we revealed the one-layer attention-only model to be a compressed Chinese room, and we're left with a giant pile of cards. There's also all the usual issues that come with understanding the weights of generalized linear models acting on correlated variables and fungibility between variables. For example, an attention head might have a weight of zero because another attention head will attend to the same token and perform the same role it would have.  Finally, there's a technical issue where QK weights aren't comparable between different query vectors, and there isn't a clear right answer as to how to normalize them.

Despite this, we do have transformers in a form where all parameters are contextualized and understandable. And despite these subtleties, we can simply read off skip-trigrams from the joint OV and QK matrices. In particular, searching for large entries in these matrices reveals lots of interesting behavior.

In the following subsections, we give a curated tour of some interesting skip-trigrams and how they're embedded in the QK/OV circuits. But full, non-cherrypicked examples of the largest entries in several models are available by following the links:

* [**Large QK/OV entries - 12 heads, d\_head=64**](https://transformer-circuits.pub/2021/framework/head_dump/small_a.html)
* [**Large QK/OV entries - 32 heads, d\_head=128**](https://transformer-circuits.pub/2021/framework/head_dump/larger.html)

#### Copying / Primitive In-Context Learning

One of the most striking things about looking at these matrices is that most attention heads in one layer models dedicate an enormous fraction of their capacity to copying. The OV circuit sets things up so that tokens, if attended to by the head, increase the probability of that token, and to a lesser extent, similar tokens. The QK circuit then only attends back to tokens which could plausibly be the next token. Thus, tokens are copied, but only to places where bigram-ish statistics make them seem plausible.

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In the above example, we fix a given source token and look at the largest corresponding QK entries (the destination token) and largest corresponding OV entries (the out token). The source token is selected to show interesting behavior, but the destination and out token are the top entries unless entries are explicitly skipped with an ellipsis; they are colored by the intensity of their value in the matrix.

Most of the examples are straightforward, but two deserve explanation: the fourth example (with skip-trigrams like `lambda… $\lambda`) appears to be the model learning LaTeX, while the fifth example (with the skip-trigram `nbsp… >&nbsp`) appears to be the model learning HTML escape sequences.

Note that most of these examples are copying; this appears to be very common.

We also see more subtle kinds of copying. One particularly interesting one is related to how tokenization for transformers typically works. Tokenizers typically merge spaces onto the start of words. But occasionally a word will appear in a context where there isn't a space in front of it, such as at the start of a new paragraph or after a dialogue open quote. These cases are rare, and as such, the tokenization isn't optimized for them. So for less common words, it's quite common for them to map to a single token when a space is in front of them (`" Ralph" → [" Ralph"]`) but split when there isn't a space (`"Ralph" → ["R", "alph"]`).

It's quite common to see skip-trigram entries dealing with copying in this case. In fact, we sometimes observe attention heads which appear to partially specialize in handling copying for words that split into two tokens without a space. When these attention heads observe a fragmented token (e.g. `"R"`) they attend back to tokens which might be the complete word with a space (`" Ralph"`) and then predict the continuation (`"alph"`). (It's interesting to note that this could be thought of as a very special case where a one-layer model can kind of mimic the [induction heads](#induction-heads) we'll see in two layer models.)

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VcMMsYhJZJQAAAAASUVORK5CYII=
)
We can summarize this copying behavior into a few abstract patterns that we've observed:

![](data:image/PNG;base64,iVBORw0KGgoAAAANSUhEUgAABDUAAAD5CAIAAAC4d5lqAAAMbWlDQ1BJQ0MgUHJvZmlsZQAASImV
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nqM953u2eH7x8vaSetV5dXlbeWd6r/O+ydRlxjEXMc/5EHyCfWb5HPb56OvlW+i73/dPPye/PL+d
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All of these can be seen as a kind of very primitive in-context learning. The ability of transformers to adapt to their context is one of their most interesting properties, and this kind of simple copying is a very basic form of it. However, we'll see when we look at a two-layer transformer that a much more interesting and powerful algorithm for in-context learning is available to deeper transformers.

#### Other Interesting Skip-Trigrams

Of course, copying isn't the only behavior these attention heads encode.

Skip-trigrams seem trivial, but can actually produce more complex behavior than one might expect. Below are some particularly striking skip-trigram examples we found in looking through the largest entries in the expanded OV/QK matrices of our models.

* [Python] Predicting that the python keywords `else`, `elif` and `except` are more likely after an indentation is reduced using skip-trigrams of the form: `\n\t\t\t … \n\t\t → else/elif/except` where the first part is indented N times, and the second part N-1, for various values of N, and where the whitespace can be tabs or spaces.
* [Python] Predicting that `open()` will have a file mode string argument: `open … "," → [rb / wb / r / w]` (for example `open("abc.txt","r")`)
* [Python] The first argument to a function is often `self`: `def … ( → self` (for example `def method_name(self):`)
* [Python] In Python 2, `super` is often used to call `.__init__()` after being invoked on `self`: `super … self → ).__` (for example `super(Parent, self).__init__()`)
* [Python] increasing probability of method/variables/properties associated with a library: `upper … . → upper/lower/capitalize/isdigit`, `tf … . → dtype/shape/initializer`, `datetime… → date / time / strftime / isoformat`, `QtWidgets … . → QtCore / setGeometry / QtGui`, `pygame … . → display / rect / tick`
* [Python] common patterns `for... in [range/enumerate/sorted/zip/tqdm]`
* [HTML] `tbody` is often followed by `<td>` tags: `tbody … < → td`
* [Many] Matching of open and closing brackets/quotes/punctuation: `(** … X → **)`, `(' … X → ')` , `"% … X → %"` , `'</ … X → >'` (see [32 head model, head 0:27](https://transformer-circuits.pub/2021/framework/head_dump/larger.html#head-0-27))
* [LaTeX] In LaTeX, every `\left` command must have a corresponding `\right` command; conversely `\right` can only happen after a `\left`. As a result, the model predicts that future LaTex commands are more likely to be `\right` after `\left`: `left … \ → right`
* [English] Common phrases and constructions (e.g. `keep … [in → mind / at → bay / under → wraps]`,  `difficult … not → impossible`)

* For a single head, here are some trigrams associated with the query `" and"`:   `back and → forth`,  `eat and → drink`, `trying and → failing`, `day and → night`, `far and → away`, `created and → maintained`, `forward and → backward`, `past and → present`, `happy and → satisfied`, `walking and → talking`, `sick and → tired`, … (see [12 head model, head 0:0](https://transformer-circuits.pub/2021/framework/head_dump/small_a.html#head-0-0))

* [URLs] Common URL schemes: `twitter … / → status`, `github … / → [issues / blob / pull / master]`, `gmail … . → com`, `http … / → [www / google / localhost / youtube / amazon]`, `http … : → [8080 / 8000]`, `www … . → [org / com / net]`

One thing to note is that the learned skip-trigrams are often related to idiosyncrasies of one's tokenization. For example collapsing whitespace together allows individual tokens to reveal indentation. Not merging backslash into text tokens means that when the model is predicting LaTeX, there's a token after backslash that must be an escape sequence. And so on.

Many skip tri-grams can be difficult to interpret without specific knowledge (e.g. `Israel … K → nes` only makes sense if you know Israel's legislative body is called the "Knesset"). A useful tactic can be to try typing potential skip tri-grams into Google search (or similar tools) and look at autocompletions.

#### Primarily Positional Attention Heads

Our treatment of attention heads hasn't discussed how attention heads handle position, largely because there are now several competing methods (e.g. ) and they would complicate our equations. (In the case of standard positional embeddings, the one-layer math works out to multiplying W\_{QK} by the positional embeddings.)

In practice, the one-layer models tend to have a small number of attention heads that are primarily positional, strongly preferring certain relative positions. Below, we present one attention head which either attends to the present token or the previous token.How can a one layer model learn an attention head that attends to a relative position? For a position mechanism that explicitly encodes relative position like rotary  the answer is straightforward. However, we use a mechanism similar to  (and, for the purposes of this point, ) where each token index has a position embedding that affects keys and queries. Let's assume that the embeddings are either fixed to be sinusoidal, or the model learns to make them sinusoidal. Observe that, in such an embedding, translation is equivalent to multiplication by a rotation matrix. Then W\_{QK} can select for any relative positional offset by appropriately rotating the dimensions containing sinusoidal information.

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#### Skip-Trigram "Bugs"

One of the most interesting things about looking at the expanded QK and OV matrices of one layer transformers is that they can shed light on transformer behavior that seems incomprehensible from the outside.

Our one-layer models represent skip-trigrams in a "factored form" split between the OV and QK matrices. It's kind of like representing a function f(a,b,c) = f\_1(a,b) f\_2(a,c). They can't really capture the three way interactions flexibly. For example, if a single head increases the probability of both `keep… in mind` and `keep… at bay`, it must also increase the probability of `keep… in bay` and `keep… at mind`. This is likely a good trade for the model on balance, but is also, in some sense, a bug. We frequently observe these in attention heads.

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)
Highlighted text denotes skip-trigram continuations that the model presumably ideally wouldn't increase the probability of. Note that `QCanvas` [is a class](https://doc.qt.io/archives/3.3/qcanvas.html) involving pixmaps in the popular Qt library. `Lloyd... Catherine` likely refers to Catherine Lloyd Burns. These examples are slightly cherry-picked to be interesting, but very common if you look at the expanded weights for models linked above.

Even though these particular bugs seem in some sense trivial, we’re excited about this result as an early demonstration of using interpretability to understand model failures. We have not further explored this phenomenon, but we’d be curious to do so in more detail. For instance, could we characterize how much performance (in points of loss or otherwise) these “bugs” are costing the model? Does this particular class continue to some extent in larger models (presumably partially, but not completely, masked by other effects)?

### Summarizing OV/QK Matrices

We've turned the problem of understanding one-layer attention-only transformers into the problem of understanding their expanded OV and QK matrices. But as mentioned above, the expanded OV and QK matrices are enormous, with easily billions of entries. While searching for the largest entries is interesting, are there better ways to understand them? There are at least three reasons to expect there are:

* The OV and QK matrices are extremely low-rank. They are 50,000 x 50,000 matrices, but only rank d\_\text{head} (64 or 128). In some sense, they're quite small even though they appear large in their expanded form.
* Looking at individual entries often reveals hints of much simpler structure. For example, we observe one head where names of people all have the top queries like `" by"` (e.g. `"Anne… by → Anne"`) while location names have top queries like `" from"` (e.g. `"Canada… from → Canada"`). This hints at something like cluster structure in the matrix.
* Copying behavior is widespread in OV matrices and arguably one of the most interesting behaviors. (We'll see in the next section that there's analogous QK matrix structure in two layer models that's used to search for similar tokens to a query.) It seems like it should be possible to formalize this.

We don't yet feel like we have a clear right answer, but we're optimistic that the right kind of matrix decomposition or dimensionality reduction could be highly informative. (See the technical details appendix for notes on how to efficiently work with these large matrices.)

#### Detecting Copying Behavior

The type of behavior we're most excited to detect in an automated way is copying. Since copying is fundamentally about mapping the same vector to itself (for example, having a token increase its own probability) it seems unusually amenable to being captured in some kind of summary statistic.

However, we've found it hard to pin down exactly what the right notion is; this is likely because there are lots of slightly different ways one could draw the boundaries of whether something is a "copying matrix" and we're not yet sure what the most useful one is. For example, we don't observe this in the models discussed in this paper, but in slightly larger models we often observe attention heads which "copy" some mixture of gender, plurality, and tense from nearby words, helping the model use the correct pronouns and conjugate verbs. The matrices for these attention heads aren't exactly copying individual tokens, but it seems like they are copying in some very meaningful sense. So copying is actually a more complex concept than it might first appear.

One natural approach might be to use eigenvectors and eigenvalues. Recall that v\_i is an eigenvector of the matrix M with an eigenvalue \lambda\_i if Mv\_i = \lambda\_i v\_i. Let's consider what that means for an OV circuit M=W\_UW^h\_{OV}W\_E if \lambda\_i is a positive real number. Then we're saying that there's a linear combination of tokensBefore token embedding, we think of tokens as being one-hot vectors in a very high-dimensional space. Logits are also vectors. As a result, we can think about linear combinations of tokens in both spaces. which increases the linear combination of logits of those same tokens. Very roughly you could think of this as a set of tokens (perhaps all tokens representing plural words for a very broad one, or all tokens starting with a given first letter, or all tokens representing different capitalizations and inclusions of space for a single word for a narrow one) which mutually increase their own probability. Of course, in general we expect the eigenvectors have both positive and negative entries, so it's more like there are two sets of tokens (e.g. tokens representing male and female words, or tokens representing singular and plural words) which increase the probability of other tokens in the same set and decrease those in others.

The eigendecomposition expresses the matrix as a set of such eigenvectors and eigenvalues. For a random matrix, we expect to have an equal number of positive and negative eigenvalues, and for many to be complex.The most similar class of random matrix for which eigenvalues are well characterized is likely Ginibre matrices, which have Gaussian-distributed entries similar to our neural network matrices at initialization. Real valued Ginibre matrices are known to have positive-negative symmetric eigenvalues, with extra probability mass on the real numbers, and "repulsion" near them . Of course, in practice we are dealing with products of matrices, but empirically the distribution of eigenvalues for the OV circuit with our randomly initialized weights appears to mirror the Ginibre distribution.  But copying requires positive eigenvalues, and indeed we observe that many attention heads have positive eigenvalues, apparently mirroring the copying structure:

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2L/uWyzGAAAAAElFTkSuQmCC
)
One can even collapse that down further and get a histogram of how many of the attention heads are copying (if one trusts the eigenvalues as a summary statistic):

![](data:image/PNG;base64,iVBORw0KGgoAAAANSUhEUgAABSsAAAMXCAYAAADSWgDJAAAMbWlDQ1BJQ0MgUHJvZmlsZQAASImV
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)
It appears that 10 out of 12 heads are significantly copying! (This agrees with qualitative inspection of the expanded weights.)

But while copying matrices must have positive eigenvalues, it isn't clear that all matrices with positive eigenvalues are things we necessarily want to consider to be copying. A matrix's eigenvectors aren't necessarily orthogonal, and this allows for pathological examples;Non-orthogonal eigenvectors can have unintuitive properties. If one tries to express a matrix in terms of eigenvectors, one needs to multiply by the inverse of the eigenvector matrix, which can behave quite differently than naively projecting onto the eigenvectors in the non-orthogonal case. for example, there can be matrices with all positive eigenvalues that actually map some tokens to decreasing the logits of that same token. Positive eigenvalues still mean that the matrix is, in some sense, "copying on average", and they're still quite strong evidence of copying in that they seem improbable by default and empirically seem to align with copying. But they shouldn't be considered a dispositive proof that a matrix is copying in all senses one might reasonably mean.

One might try to formalize "copying matrices" in other ways. One possibility is to look at the diagonal of a matrix, which describes how each token affects its own probability. As expected, entries on the diagonal are very positive-leaning. We can also ask how often a random token increases its own probability more than any other token (or is one of the k-most increased tokens, to allow for tokens which are the same with a different capitalization or with a space). All of these seem to point in the direction of these attention heads being copying matrices, but it's not clear that any of them is a fully robust formalization of "the primary behavior of this matrix is copying". It's worth noting that all of these potential notions of copying are linked by the fact that the sum of the eigenvalues is equal to the trace is equal to the sum of the diagonal.

For the purposes of this paper, we'll continue to use the eigenvalue-based summary statistic. We don't think it's perfect, but it seems like quite strong evidence of copying, and empirically aligns with manual inspection and other definitions.

### Do We "Fully Understand" One-Layer Models?

There's often skepticism that it's even possible or worth trying to truly reverse engineer neural networks. That being the case, it's tempting to point at one-layer attention-only transformers and say "look, if we take the most simplified, toy version of a transformer, at least that minimal version can be fully understood."

But that claim really depends on what one means by fully understood. It seems to us that we now understand this simplified model in the same sense that one might look at the weights of a giant linear regression and understand it, or look at a large database and understand what it means to query it. That is a kind of understanding. There's no longer any algorithmic mystery. The contextualization problem of neural network parameters has been stripped away. But without further work on summarizing it, there's far too much there for one to hold the model in their head.

Given that regular one layer neural networks are just generalized linear models and can be interpreted as such, perhaps it isn't surprising that a single attention layer is mostly one as well.

---

Two-Layer Attention-Only Transformers
-------------------------------------

Videos covering similar content to this section: [2 layer theory](https://www.youtube.com/watch?v=UM-eJbx_YDk&list=PLoyGOS2WIonajhAVqKUgEMNmeq3nEeM51&index=5), [2 layer term importance](https://www.youtube.com/watch?v=qom0nxou4f4&list=PLoyGOS2WIonajhAVqKUgEMNmeq3nEeM51&index=6), [2 layer results](https://www.youtube.com/watch?v=VuxANJDXnIY&list=PLoyGOS2WIonajhAVqKUgEMNmeq3nEeM51&index=7)

Deep learning studies models that are deep, which is to say they have many layers. Empirically, such models are very powerful. Where does that power come from? One intuition might be that depth allows composition, which creates powerful expressiveness.

Composition of attention heads is the key difference between one-layer and two-layer attention-only transformers. Without composition, a two-layer model would simply have more attention heads to implement skip-trigrams with. But we'll see that in practice, two-layer models discover ways to exploit attention head composition to express a much more powerful mechanism for accomplishing in-context learning. In doing so, they become something much more like a computer program running an algorithm, rather than look-up tables of skip-trigrams we saw in one-layer models.

### Three Kinds of Composition

Recall that we think of the residual stream [as a communication channel](#residual-comms). Every attention head reads in subspaces of the residual stream determined by W\_Q, W\_K, and W\_V, and then writes to some subspace determined by W\_O. Since the attention head vectors are much smaller than the size of the residual stream (typical values of d\_\text{head} / d\_\text{model} might vary from around 1/10 to 1/100), attention heads operate on small subspaces and can easily avoid significant interaction.

When attention heads do compose, there are three options:

* Q-Composition: W\_Q reads in a subspace affected by a previous head.
* K-Composition: W\_K reads in a subspace affected by a previous head.
* V-Composition: W\_V reads in a subspace affected by a previous head.

Q- and K-Composition are quite different from V-Composition. Q- and K-Composition both affect the attention pattern, allowing attention heads to express much more complex patterns. V-Composition, on the other hand, affects what information an attention head moves when it attends to a given position; the result is that V-composed heads really act more like a single unit and can be thought of as creating an additional "virtual attention heads". Composing movement of information with movement of information gives movement of information, whereas attention heads affecting attention patterns is not reducible in this way.

To really understand these three kinds of composition, we'll need to study the OV and QK circuits again.

### Path Expansion of Logits

The most basic question we can ask of a transformer is "how are the logits computed?" Following [our approach](#onel-path-expansion) to the one-layer model, we write out a product where every term is a layer in the model, and expand to create a sum where every term is an end-to-end path through the model.

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Nelson Elhage
Neel Nanda
Catherine Olsson
Tom Henighan
Nicholas Joseph
Ben Mann
Amanda Askell
Yuntao Bai
Anna Chen
Tom Conerly
Nova DasSarma
Dawn Drain
Deep Ganguli
Zac Hatfield-Dodds
Danny Hernandez
Andy Jones
Jackson Kernion
Liane Lovitt
Kamal Ndousse
Dario Amodei
Tom Brown
Jack Clark
Jared Kaplan
Sam McCandlish
Chris Olah