Sharp Spectral Thresholds for Logit Fixed Points
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Computer Science > Machine Learning
Title:Sharp Spectral Thresholds for Logit Fixed Points
Abstract:Softmax feedback systems are a common mathematical core of entropy-regularized reinforcement learning, logit game dynamics, population choice, and mean-field variational updates. Their central stability question is simple: when does a self-reinforcing softmax system produce a unique and globally predictable outcome? Classical theory gives a conservative answer. By treating softmax as a unit-scale response, it certifies stability only in a strongly randomized regime.
We prove that the classical approach misses an entire stable regime and does not identify the point at which the qualitative change truly occurs. For finite-dimensional affine logit systems, the sharp dimension-free Euclidean threshold is $$\beta\|\Pi W\Pi\|_{\mathcal T\to\mathcal T}<2,$$ rather than the previously used condition, which certifies stability only while the softmax system remains safely over-regularized. Our theorem fills the previously missing pre-bifurcation regime, extending stability guarantees for affine softmax feedback systems to reward-responsive yet globally predictable systems. It enlarges the certified stability boundary for these systems and identifies where the model genuinely undergoes a phase transition.
| Subjects: | Machine Learning (cs.LG); Artificial Intelligence (cs.AI); Computer Science and Game Theory (cs.GT) |
| Cite as: | arXiv:2605.15651 [cs.LG] |
| (or arXiv:2605.15651v1 [cs.LG] for this version) | |
| https://doi.org/10.48550/arXiv.2605.15651
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