Prefix Likelihood-Ratio Control: Tail-Stable Training for Long-Horizon Language Generation

Token-level maximum likelihood and mean negative log-likelihood dominate the training and evaluation of large language models (LLMs), yet long-horizon generation often suffers from late-stage degradation despite improved mean loss. We identify two coupled causes of this mismatch: mean token loss is not scale-free with respect to sequence length, so small per-token divergences can amplify into large sequence-level distribution shifts, and long-horizon failures are spike-dominated, where rare but abrupt prefix deviations trigger errors that are largely invisible to mean objectives. To address these issues, we propose \textbf{Prefix Likelihood-Ratio Control} (PLRC), which centers training on the prefix log-likelihood ratio process and its increments to directly capture sequence-level deviation and spikes. As the data likelihood is unknown, PLRC learns a prefix ratio critic via noise-contrastive prefix discrimination and introduces tail-sensitive regularizers on the terminal ratio and its increments, yielding guarantees that bound long-horizon event distortion and the probability of the first spike. PLRC preserves standard teacher-forcing training without architectural changes while explicitly targeting failure modes that mean token loss cannot capture.