A Mathematical Framework for Transformer Circuits

Model Simplifications
High-Level Architecture
Virtual Weights and the Residual Stream as a Communication Channel
Attention Heads are Independent and Additive
Attention Heads as Information Movement
The Path Expansion Trick
Splitting Attention Head terms into Query-Key and Output-Value Circuits
Interpretation as Skip-Trigrams
Summarizing OV/QK Matrices
Do We "Fully Understand" One-Layer Models?
[Three Kinds of Composition](#three-kin…

Model Simplifications
High-Level Architecture
Virtual Weights and the Residual Stream as a Communication Channel
Attention Heads are Independent and Additive
Attention Heads as Information Movement
The Path Expansion Trick
Splitting Attention Head terms into Query-Key and Output-Value Circuits
Interpretation as Skip-Trigrams
Summarizing OV/QK Matrices
Do We "Fully Understand" One-Layer Models?
Three Kinds of Composition
Path Expansion of Logits
Path Expansion of Attention Scores QK Circuit
Analyzing a Two-Layer Model
Induction Heads
Term Importance Analysis
Virtual Attention Heads

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 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 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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