Grokking Group Multiplication with Cosets
Dashiell Stander Qinan Yu Honglu Fan Stella Biderman
We use the group Fourier transform over the symmetric group $S_n$ to reverse
engineer a 1-layer feedforward network that has "grokked" the multiplication of
$S_5$ and $S_6$. Each model discovers the true subgroup structure of the full
group and converges on circuits that decompose the group multiplication into
the multiplication of the group's conjugate subgroups. We demonstrate the value
of using the symmetries of the data and models to understand their mechanisms
and hold up the ``coset circuit'' that the model uses as a fascinating example
of the way neural networks implement computations. We also draw attention to
current challenges in conducting mechanistic interpretability research by
comparing our work to Chughtai et al. [6] which alleges to find a different
algorithm for this same problem.