# A Mathematical Framework for Transformer Circuits

2021 #Content/Paper by 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, and Chris Olah. There’s also an accompanying YouTube playlist here and walkthrough by Neel Nanda here

Since so much of the basic computation done in transformers is linear algebra, there are many equivalent factorisations giving rise to different intuitive stories of how they work. This paper advocates for a particular one.

The start of the story is the recognition that individual attention heads operate on low-dimensional projections of the residual stream rather than the full space, allowing them to write to disjoint subspaces with minimal interaction. In turn, it can be fruitful to think of a forward pass as decomposing into mostly-independent paths of computation (i.e. circuits) weaving in and out of the residual stream. From an interpretability perspective, we may hope that only a few of these are critical to model behaviour in any given situation.

- #Comment - This view seems to de-emphasise the residual stream itself as a target for interpretability. With Dictionary Learning, this is now firmly back on the table.

This perspective is applied to small decoder-only transformer models containing up to two layers of attention heads and *no* fully-connected MLP layers. The complexity of the analysis is built up by adding one layer at a time.

**Zero-layer:** the only path is that defined by the product of the embedding and un-embedding matrices. The optimal behaviour of this path is to approximate a lookup table for the log-likelihood of bigrams […B] $\rightarrow$ C, and zero-layer transformers actually try to do this in practice.

**One-layer:** for an $n$-head attention layer, there are $n$ computation paths. The optimal behaviour of each path is to learn the log-likelihood of ‘skip-trigrams’ […A…B] $\rightarrow$ C. Empirical investigations suggest that simple copying behaviour […A…B] $\rightarrow$ A is often promoted.

**Two-layer:** multiple attention layers can be thought of as creating circuits connecting the output of one head to any/all of the query, key and value inputs of another head. The degree of composition between two heads can be quantified using a kind of cosine similarity between their weight matrices. Composition means that much more powerful mechanisms start to emerge at this stage. A prominent example is *induction heads*, which promote the repetition of bigram patterns […AB…A] $\rightarrow$ B.

- The authors go on to study induction heads in much larger models in a follow-up paper.