# Transformer电路的数学框架

- 来源：Anthropic：Transformer Circuits（可解释性研究）
- 发布时间：2021-12-22 00:00
- AIHOT 分数：58
- AIHOT 链接：https://aihot.virxact.com/items/cmoegbh74007gslxx4vwf1za6
- 原文链接：https://transformer-circuits.pub/2021/framework/index.html

## AI 摘要

本研究提出一个用于逆向工程Transformer语言模型的数学框架，通过分析仅含注意力机制的超简模型（两层或以下）来揭示其内部工作机制。研究发现：零层模型直接建模二元统计；单层模型是二元与“跳跃三元”模型的组合；两层模型则能通过注意力头组合形成“归纳头”，实现复杂的上下文学习算法。该框架将注意力头分解为独立的QK与OV电路，并将模型输出表述为可解释函数之和，为理解更大规模模型的计算原理提供了初步基础。

## 正文

A Mathematical Framework for Transformer Circuits

Transformer Circuits Thread

A Mathematical Framework for Transformer Circuits

Authors

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‡

Affiliation

Anthropic

Published

Dec 22, 2021

Core Research Contributor;† Core Infrastructure Contributor;‡ Correspondence to colah@anthropic.com;Author contributions statement below.

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, 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, 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 WQ^TWK and WOWV. 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 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.

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.

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"; 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 weights Note that for attention layers, there are three different kinds of input weights: WQ, WK, and WV. 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.

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" 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 WOWV 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.

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^{h1}, r^{h2},..., and then multiplying by an output matrix WO^H. Let's split WO^H into equal size blocks for each head [WO^{h1}, WO^{h2}...]. Then we observe that:

WO^H \left[\begin{matrix}r^{h1}\\r^{h2}\\... \end{matrix}\right] ~~=~~ \left[WO^{h1},~ WO^{h2},~ ... \right]\cdot\left[\begin{matrix}r^{h1}\\r^{h2}\\...\end{matrix}\right] ~~=~~ \sumi WO^{hi} r^{hi}

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.

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 (vi = WV xi). 2. Compute the “result vector” by linearly combining value vectors according to the attention pattern (ri = \sumj A{i,j} vj). 3. Finally, compute the output vector of the head for each token (h(x)i = WO ri).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. WV and WO 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 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 WO)~~\cdot~~

Project result vectors out for each token (h(x)i = WO ri)

~(A \otimes \text{Id})~~\cdot~~~

Mix value vectors across tokens to compute result vectors (ri = \sumj A{i,j} vj)

~(\text{Id} \otimes WV)~~\cdot~~~

Compute value vector for each token (vi=WV xi)~~

x

Applying the mixed product property and collapsing identities yields:

h(x) ~=~(A ~~\otimes~~ WO WV) ~~~\cdot~~~~~~

A mixes across tokens while WOWV acts on each vector independently.

x

What about the attention pattern? Typically, one computes the keys ki = WK xi, computes the queries qi = WQ xi 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 WQ^T WK x).

It's worth noting that although this formulation is mathematically equivalent, actually implementing attention this way (ie. multiplying by WO WV and WQ^T WK) 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 WOWV, which operate on different dimensions and act independently.

A governs which token's information is moved from and to. WO WV 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}=WO WV 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} &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 we'll see in two layer models.)

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

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 methodname(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 tags: tbody … ' (see 32 head model, 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)

[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.

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) = f1(a,b) f2(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.

Highlighted text denotes skip-trigram continuations that the model presumably ideally wouldn't increase the probability of. Note that QCanvasis a class 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 vi is an eigenvector of the matrix M with an eigenvalue \lambdai if Mvi = \lambdai vi. Let's consider what that means for an OV circuit M=WUW^h{OV}WE if \lambdai is a positive real number.Then we're saying that there's a linear combination of tokens Before 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:

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):

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, 2 layer term importance, 2 layer results

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. Every attention head reads in subspaces of the residual stream determined by WQ, WK, and WV, and then writes to some subspace determined by WO. 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: WQ reads in a subspace affected by a previous head. K-Composition: WK reads in a subspace affected by a previous head. V-Composition: WV 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 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.

Two of these terms, the direct path term and individual head terms, are identical to the one-layer model. The final "virtual attention head" term corresponds to V-Composition.Virtual attention heads are conceptually very interesting, and we'll discuss them more later. However, in practice, we'll find that they tend to not play a significant role in small two-layer models.

Path Expansion of Attention Scores QK Circuit

Just looking at the logit expansion misses what is probably the most radically different property of a two-layer attention-only transformer: Q-composition and K-composition cause them to have much more expressive second layer attention patterns.

To see this, we need to look at the QK circuits computing the attention patterns. Recall that the attention pattern for a head h is A^h~=~ \text{softmax}^\!\left( t^T \cdot C{QK}^h t \right), where C{QK}^h is the "QK-circuit" mapping tokens to attention scores. For first layer attention heads, the QK-circuit is just the same matrix we saw in the one-layer model: C^{\,h\in H1}{\,QK}~=~ WE^T W{QK}^h WE.

But for the second layer QK-circuit, both Q-composition and K-composition come into play, with previous layer attention heads potentially influencing the construction of the keys and queries. Ultimately, W{QK} acts on the residual stream. In the case of the first layer this reduced to just acting on the token embeddings: C^{\,h\in H1}{\,QK}~=~ x0^T W{QK}^h x0=~ WE^T W{QK}^h WE. But by the second layer, C^{\,h\in H2}{\,QK}~=~ x1^T W{QK}^h x1 is acting on x1, the residual stream after first layer attention heads. We can write this down as a product, with the first layer both on the "key side" and "query side."Then we apply our path expansion trick to the product.

One complicating factor is that we have to write it as a 6-dimensional tensor, using two tensor products on matrices. This is because we're trying to express a multilinear function of the form [n\text{context},~ d\text{model}] \times [n\text{context},~ d\text{model}] ~\to~ [n\text{context},~ n\text{context}]. In the one-layer case, we could side step this by implicitly doing an outer product, but that no longer works. A natural way to express this is as a (4,2)-tensor (one with 4 input dimensions and 2 output dimensions). Each term will be of the form Aq \otimes Ak \otimes W where x (Aq \otimes Ak \otimes W) y = Aq^T x W y Ak, meaning that Aq describes the movement of query-side information between tokens, Ak describes the movement of key-side information between tokens, and W describes how they product together to form an attention score.

Each of these terms corresponds to a way the model can implement more complex attention patterns. In the abstract, it can be hard to reason about them. But we'll return to them with a concrete case shortly, when we talk about induction heads.

Analyzing a Two-Layer Model

So far, we've developed a theoretical model for understanding two-layer attention-only models. We have an overall equation describing the logits (the OV circuit), and then an equation describing how each attention head's attention pattern is computed (the QK circuit). But how do we understand them in practice? In this section, we'll reverse engineer a single two-layer model.

Recall that the key difference between a two-layer model and a one-layer model is Q-, K-, and V-composition. Without composition, the model is just a one-layer model with extra heads.

Small two-layer models seem to often (though not always) have a very simple structure of composition, where the only type of composition is K-composition between a single first layer head and some of the second layer heads.There appears to be no significant V- or Q- composition in this particular model. The following diagram shows Q-, K-, and V-composition between first and second layer heads in the model we wish to analyze. We've colored the heads involved by our understanding of their behavior. The first layer head has a very simple attention pattern: it primarily attends to the previous token, and to a lesser extent the present token and the token two back. The second layer heads are what we call induction heads.

Correction

The following diagram has an error introduced by a bug in an underlying library we wrote to accelerate linear algebra on low-rank matrices. A detailed comment on this, along with a corrected figure, can be found below.

The above diagram shows Q-, K-, and V-Composition between attention heads in the first and second layer. That is, how much does the query, key or value vector of a second layer head read in information from a given first layer head? This is measured by looking at the Frobenius norm of the product of the relevant matrices, divided by the norms of the individual matrices. For Q-Composition, ||W{QK}^{h2~T}W{OV}^{h1}||F / (||W{QK}^{h2~T}||F ||W{OV}^{h1}||F), for K-Composition ||W{QK}^{h2}W{OV}^{h1}||F / (||W{QK}^{h2}||F ||W{OV}^{h1}||F), for V-Composition ||W{OV}^{h2}W{OV}^{h1}||F / (||W{OV}^{h2}||F ||W{OV}^{h1}||F). By default, we subtract off the empirical expected amount for random matrices of the same shapes (most attention heads have a much smaller composition than random matrices). In practice, for this model, there is only significant K-composition, and only with one layer 0 head.

One quick observation from this is that most attention heads are not involved in any substantive composition.We can think of them as, roughly, a larger collection of skip tri-grams. This two-layer model has a mystery for us to figure out, but it's a fairly narrowly scoped one.(We speculate this means that having a couple induction heads in some sense "outcompetes" a few potential skip-trigram heads, but no other type of composition did. That is, having more skip-trigram heads is a competitive use of second layer attention heads in a small model.)

In the next few sections, we'll develop a theory of what's going on, but before we do, we provide an opportunity to poke around at the attention heads using the interactive diagram below, which displays value-weighted attention patterns over the first paragraph of Harry Potter and the Philosopher's Stone. We've colored the attention heads involved in K-composition using the same scheme as above. (This makes it a bit hard to investigate the other heads; if you want to look at those, an interface for general exploration is available here).

We recommend isolating individual heads and both looking at the pattern and hovering over tokens. For induction heads, note especially the off-diagonal lines in the attention pattern, and the behavior on the tokens compositing Dursley and Potters.

The above diagram shows the value-weighted attention pattern for various attention heads; that is, the attention patterns with attention weights scaled by the norm of the value vector at the source position ||v{src}^h||. You can think of the value-weighted attention pattern as showing "how big a vector is moved from each position." (This approach was also recently introduced by Kobayashi et al..) This is especially useful because attention heads will sometimes use certain tokens as a kind of default or resting position when there isn't a token that matches what they're looking for; the value vector at these default positions will be small, and so the value weighted pattern is more informative.

The interface allows one to isolate attention heads, shows the overall attention pattern, and allows one to explore the attention for individual tokens. Attention heads involved in K-composition are colored using the same scheme as above. We suggest trying to isolate these heads.

If you look carefully, you'll notice that the aqua colored "induction heads" often attend back to previous instances of the token which will come next. We'll investigate this more in the next section. Of course, looking at attention patterns on a single

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