Min-$k$ Sampling: Decoupling Truncation from Temperature Scaling via Relative Logit Dynamics

Large language models generate text by sampling each token from a predicted distribution, and a temperature parameter sets how far the draw strays from the most probable tokens. Truncation samplers such as top-p and min-p discard the least probable tokens before the draw, so that sampling at high temperature stays coherent. Their reported accuracy gains come from temperatures of 1.5 to 3, while the defaults of deployed systems cluster between 0.6 and 1.0. Whether they change accuracy at those defaults, and for which models, has not been measured. We test thirteen open-weight models on GSM8K and MMLU-Pro at temperatures 0.7, 1.0, and 1.3 in one controlled pipeline, ten of them under eight decoding configurations. Six of the thirteen models lose 17 to 38 accuracy points on MMLU-Pro between 0.7 and 1.3, and the other seven lose at most 10. The lost accuracy comes from generations that run to the token limit or never state an answer. These results suggest that truncation samplers improve accuracy primarily when higher temperatures substantially degrade model performance. Where accuracy remains stable across temperatures, none of the tested truncation samplers improves on plain temperature sampling.

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. 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 $Ω(R)$, 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.

Sampling plays an important role in long-form language-model reasoning. Over thousands of decoding steps, small changes in the candidate token set can compound into different reasoning trajectories, stability profiles, and final answers. Existing truncation methods such as top-$p$, min-$p$, and fixed top-$nσ$ sampling improve over unrestricted sampling, but they rely on fixed thresholds that cannot adapt to changes in entropy, task difficulty, training stage, or generation budget. We introduce Adaptive Nucleus Truncation Sampling (ANTS), which extends top-$nσ$ sampling from a fixed decoding rule into an adaptive rollout-control mechanism for long-form generation. ANTS selects standardized neighborhoods around the maximum logit before temperature scaling, adapts the truncation width using an entropy-conditioned controller, and retains a no-truncation fallback arm to stabilize training when truncation becomes unsafe. On a 33B-total / 4B-active sparse Mixture-of-Experts reasoning model, ANTS improves average performance over percentage-based benchmarks by +1.9, +3.8, and +5.2 points at 8K, 16K, and 32K generation budgets, respectively. The strongest gains appear on instruction following and mathematical reasoning, with IFBench improving by more than 10 points at 32K and AIME 2025 improving by 7 points. Code generation reveals an important budget interaction. On Codeforces, ANTS trails the baseline at 8K, but reverses this gap and substantially improves ELO at 16K and 32K. These results suggest that sampler design should be treated not just as a decoding hyperparameter, but as part of how we stabilize and scale long-budget reasoning.