Ghosts of Softmax: Complex Singularities That Limit Safe Step Sizes in Cross-Entropy

TL;DR

This paper reveals that complex singularities in the softmax function impose fundamental limits on safe step sizes in cross-entropy optimization, and proposes a geometric method to improve training stability.

Contribution

The authors derive explicit formulas for the Taylor convergence radius in cross-entropy, linking complex analysis to optimization stability and proposing a practical normalization scheme.

Findings

Normalized step size $r<1$ ensures stable training across models.

Complex singularities in softmax limit the Taylor convergence radius.

A controller enforcing $ au extless ho_a$ improves robustness significantly.

Abstract

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, the exact radius is $\rho^*=\sqrt{\delta^2+ \pi^2}/\Delta_a$. In the multiclass case, we obtain the lower bound $\rho_a=\pi/\Delta_a$, where $\Delta_a=\max_k a_k-\min_k a_k$ is the spread of directional logit derivatives $a_k=\nabla z_k\cdot v$. This bound costs one Jacobian-vector product and reveals what makes a step fragile: samples that are both near a decision flip and highly sensitive to the proposed direction tighten the radius. The normalized step size $r=\tau/\rho_a$ separates safe from dangerous updates. Across six tested architectures and multiple step directions, no model fails for $r<1$, yet collapse appears once $r\ge 1$. Temperature scaling confirms the mechanism: normalizing by $\rho_a$ shrinks the onset-threshold spread from standard deviation $0.992$ to $0.164$. A controller that enforces $\tau\le\rho_a$ survives learning-rate spikes up to $10{,} 000\times$ in our tests, where gradient clipping still collapses. Together, these results identify a geometric constraint on cross-entropy optimization that operates through Taylor convergence rather than Hessian curvature.