Gradient Flow Polarizes Softmax Outputs towards Low-Entropy Solutions

Sparse attention patterns are prevalent in trained transformers, yet the mechanisms that favor their emergence during training remain poorly understood. In this work, we investigate the role of softmax in shaping these patterns by studying the gradient flow dynamics of an attention head trained to produce a fixed bias vector for all inputs, a task that dense and sparse attention solve equally well. Leveraging symmetries of this task, we show that the dynamics of the head simplify to a *value–softmax* model $V\sigma (a)$, where $V$ and $a$ are a learnable value matrix and attention vector. We prove that gradient flow on this model drives the attention scores toward low entropy: under logistic loss they converge to a one-hot vector, and under square loss we obtain quantitative polarization bounds. Finally, we use experiments in trained transformers to examine how the choice of attention nonlinearity affects the influence of individual tokens and sparse attention patterns.