Learning Decision-Stump Thresholds in Context: Dynamics of Softmax Attention

Estimating a decision threshold requires locating observations near an unknown boundary. We study how gradient-based pretraining learns this statistical rule in a two-parameter softmax-attention model with a fixed feature and inequality direction. Pretraining uses labeled contexts and their true thresholds; a fresh threshold must be inferred from context alone. Under a large-resolution initialization, constant-step gradient descent on $m$ tasks with $n$ examples each produces a frozen estimator with error $\tilde{O}((m\wedge n{)}^{-1}+N^{-1})$ for each fixed interior threshold and every fresh-context size $N$. The two terms separate finite-pretraining accuracy from fresh-context localization. The mechanism is coordinated parameter divergence: population training calibrates the relative label and feature scores, then increases the attention scale as $t^{1/4}$, giving population threshold error $O(t^{-1/4})$. To transfer this mechanism to a fixed finite corpus, we control gradient errors relative to the shrinking directions of progress at successive parameter scales. This certifies a growing training interval without requiring long-time tracking of the population trajectory. We also identify the boundary limitation of the one-head model and explain statistically what a reflected symmetrization could achieve.