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A Fix for Softmax and Cross-Entropy

Jonathan T. Barron, Dan Gnanapragasam, Michael Tschannen · 2026 · arXiv (Cornell University)

DOI: 10.48550/arxiv.2610.31642 · cited by 0 works (OpenAlex) · open-access PDF

Spanish

Las implementaciones estándar de softmax, log-softmax y cross-entropy tienen dos errores numéricos. El primer error, Softmax Collapse, redondea el gradiente del logit objetivo a cero si ese logit es mucho más grande que los otros, resultando en gradientes que ya no suman a cero. El segundo error, Softmax Cancelación, ocurre cuando los logits de los estudiantes están cerca de los logits de los maestros, y produce gradientes no-zero incluso cuando los logits de los estudiantes y de los maestros son iguales. Presentamos implementaciones simples de log-softmax, softmax y cross-entropy que solucionan ambos errores. Nuestra solución elimina el fenómeno "Slingshot" previamente observado, acelera el grokking con decay peso, previene el colap

Machine translation (Meta M2M100); the English original follows.

Abstract

Standard implementations of softmax, log-softmax, and cross-entropy have two numerical bugs. The first bug, Softmax Collapse, rounds the target logit's gradient to zero if that one logit is much larger than the others, resulting in gradients that no longer sum to zero. The second bug, Softmax Cancellation, occurs when student logits are close to teacher logits, and produces non-zero gradients even when the student and teacher logits are equal. We present simple implementations of log-softmax, softmax, and cross-entropy that fix both bugs. Our fix eliminates the previously-observed "Slingshot" phenomenon, speeds up grokking with weight decay, prevents accuracy collapse without weight decay, stops student networks from drifting away from the teacher during distillation, and shows that after memorization, logit growth is bounded by an equilibrium between weight decay and Adam's $ε$ hyperparameter.

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Topics: Softmax function, Logistic regression, Logit, Simple (philosophy), Bounded function, Mathematics

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