Information-Theoretic Foundations of Online Temperature Sampling

Abstract

Repetitive, degenerate generation is a persistent failure of LLM decoding, and the standard remedies (temperature, nucleus, and top-$k$ sampling) are static heuristics. By Kac's lemma and the AEP, a stationary source takes time exponential in its entropy to repeat a block. We therefore maximize the entropy of the sampling distributions under one cross-entropy budget that keeps the output plausible under the model. Offline, the solution is temperature sampling, with the inverse temperature as the budget's Lagrange multiplier. \emph{Our contribution is the online problem}: the conditionals $p_1, p_2, \dots$ are revealed one at a time, and the sampling distribution $q_t$ must be chosen causally from $p_1, \dots, p_t$ while the budget couples the entire horizon. The optimal temperature is thus a global shadow price that must be estimated from a prefix. We propose Entropic Online Sampling (EOS), which re-solves for the temperature on each prefix. We prove it loses $O(R\log n)$ entropy against the offline optimum while overshooting the budget by $O(R\log n)$, with $R$ an instance-dependent drift parameter. A budget-tightened variant is exactly feasible at the same rate. Conversely, every exactly feasible online algorithm must lose $\Omega(R)$, so EOS is optimal up to a $\log n$ factor. When the conditionals are drawn from a finite mixture, $R=\tilde{O}(\sqrt{n})$. Experiments confirm that EOS tracks the offline temperature and beats fixed-temperature sampling in entropy.