  This work focuses on the training dynamics of one associative memory module
storing outer products of token embeddings. We reduce this problem to the study
of a system of particles, which interact according to properties of the data
distribution and correlations between embeddings. Through theory and
experiments, we provide several insights. In overparameterized regimes, we
obtain logarithmic growth of the ``classification margins.'' Yet, we show that
imbalance in token frequencies and memory interferences due to correlated
embeddings lead to oscillatory transitory regimes. The oscillations are more
pronounced with large step sizes, which can create benign loss spikes, although
these learning rates speed up the dynamics and accelerate the asymptotic
convergence. In underparameterized regimes, we illustrate how the cross-entropy
loss can lead to suboptimal memorization schemes. Finally, we assess the
validity of our findings on small Transformer models.
