Manifold Trajectories in Next-Token Prediction: From Replicator Dynamics to Softmax Equilibrium
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arXiv
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| Formato: | Preprint |
| Publicado: |
2025
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| _version_ | 1866912558711046144 |
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| author | Lee-Jenkins, Christopher R. |
| author_facet | Lee-Jenkins, Christopher R. |
| contents | Decoding in large language models is often described as scoring tokens and normalizing with softmax. We give a minimal, self-contained account of this step as a constrained variational principle on the probability simplex. The discrete, normalization-respecting ascent is the classical multiplicative-weights (entropic mirror) update; its continuous-time limit is the replicator flow. From these ingredients we prove that, for a fixed context and temperature, the next-token distribution follows a smooth trajectory inside the simplex and converges to the softmax equilibrium. This formalizes the common ``manifold traversal'' intuition at the output-distribution level. The analysis yields precise, practice-facing consequences: temperature acts as an exact rescaling of time along the same trajectory, while top-k and nucleus sampling restrict the flow to a face with identical guarantees. We also outline a controlled account of path-dependent score adjustments and their connection to loop-like, hallucination-style behavior. We make no claims about training dynamics or internal representations; those are deferred to future work. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2508_21186 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Manifold Trajectories in Next-Token Prediction: From Replicator Dynamics to Softmax Equilibrium Lee-Jenkins, Christopher R. Machine Learning Artificial Intelligence Dynamical Systems Decoding in large language models is often described as scoring tokens and normalizing with softmax. We give a minimal, self-contained account of this step as a constrained variational principle on the probability simplex. The discrete, normalization-respecting ascent is the classical multiplicative-weights (entropic mirror) update; its continuous-time limit is the replicator flow. From these ingredients we prove that, for a fixed context and temperature, the next-token distribution follows a smooth trajectory inside the simplex and converges to the softmax equilibrium. This formalizes the common ``manifold traversal'' intuition at the output-distribution level. The analysis yields precise, practice-facing consequences: temperature acts as an exact rescaling of time along the same trajectory, while top-k and nucleus sampling restrict the flow to a face with identical guarantees. We also outline a controlled account of path-dependent score adjustments and their connection to loop-like, hallucination-style behavior. We make no claims about training dynamics or internal representations; those are deferred to future work. |
| title | Manifold Trajectories in Next-Token Prediction: From Replicator Dynamics to Softmax Equilibrium |
| topic | Machine Learning Artificial Intelligence Dynamical Systems |
| url | https://arxiv.org/abs/2508.21186 |