Free Lunch Uncertainty: Sequence-Level R\'enyi Entropy from Particle Decoding for Adaptive LLM Reasoning

arXiv:2611.13414v1 Announce Type: new
Abstract: Quantifying uncertainty over an entire LLM completion is critical for selective prediction and adaptive compute. Zero-cost signals such as mean token entropy and log-probability are local to tokens or to one completion, and self-consistency needs many extra completions. We show that Sequential Monte Carlo (SMC) decoding for power sampling already estimates the normalizing constant of the power distribution, which gives the sequence-level R’enyi entropy (SRE) of the model’s completion distribution at zero marginal cost, with a guarantee at every particle budget. A R’enyi chain rule extends this readout to a family of stepwise entropies of any order. We prove that orders above one are tied to the most likely token of each step, which explains why they track log-probability, while the order-1/2 member accumulates the log of per-step search effort. This member is the best zero-cost selective-prediction signal in 10 of 13 settings. On MATH500 its AUROC is 0.022 to 0.044 above log-probability with all four models and within length quartiles, and the same order is selected when each setting is held out. An 8-particle probe identifies the problems that benefit from more particles, and a two-pass policy matches 64-particle accuracy on MATH500 with 34.2% fewer particles over three seeds.