Leaner transformers can easily learn to cluster

Transformers have in-context learning capabilities, where some known learning algorithms can be executed in the forward pass through the model. Recent work show that transformers can exactly perform Lloyd's algorithm for $k$-means clustering with $n$ points in $d$ dimensions with an embedding size $d_{\textsf{emb}} = d+k$ (thus, requiring attention projection matrices of size $(d+k)^2$). In this work, we build upon this result in the following ways: First, we present an equally expressive but smaller transformer that executes Lloyd's algorithm with embedding size $d_{\textsf{emb}} = (d + \lceil \log_2 k \rceil )$. Next, we train these transformers to learn the clustering algorithms given a distribution of clustering tasks, and theoretically characterize and empirically validate the factors affecting the convergence and in-distribution generalization of stochastic gradients based learning algorithms. Finally, we probe the general clustering abilities of these learned algorithms (in the form of transformers), and try to understand situations where they succeed and fail.