Ghosts of Softmax: Complex Singularities That Limit Safe Step Sizes in Cross-Entropy
Optimization analyses for cross-entropy training rely on local Taylor models of the loss to predict whether a proposed step will decrease the objective. These surrogates are reliable only inside the Taylor convergence radius of the true loss along the update direction. That radius is set not by real-line curvature alone but by the nearest complex singularity. For cross-entropy, the softmax partition function $F=\sum_j \exp(z_j)$ has complex zeros -- ``ghosts of softmax'' -- that induce logarithmic singularities in the loss and cap this radius. To make this geometry usable, we derive closed-form expressions under logit linearization along the proposed update direction. In the binary case, ...
| Authors | Piyush Sao |
|---|---|
| Primary category | cs.LG |
| Published | 13 March 2026 |
| Last updated at source | 13 March 2026 |
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