Optimal Stopping vs Best-of-$N$ for Inference Time Optimization

Large language model (LLM) generation often requires balancing output quality against inference cost, especially when using multiple generations. We introduce a new framework for inference-time optimization based on the classical Pandora's Box problem. Viewing each generation as opening a costly "box" with random reward, we develop algorithms that decide when to stop generating without knowing the underlying reward distribution. Our first contribution is a UCB-style Pandora's Box algorithm, which achieves performance that is provably close to Weitzman's algorithm, the optimal strategy when the distribution is known. We further adapt this method to practical LLM settings by addressing reward scaling across prompts via a Bradley-Terry inspired transformation. This leads to an adaptive inference-time optimization method that normalizes rewards and learns stopping thresholds on the fly. Experiments on the AlpacaFarm and HH-RLHF datasets, using multiple LLM-reward model pairs, show that our adaptive strategy can obtain the same performance as non-adaptive Best-of-N sampling while requiring 15-35 percent fewer generations on average. Our results establish a principled bridge between optimal stopping theory and inference-time scaling, providing both theoretical performance bounds and practical efficiency gains for LLM deployment.

Is Transductive Learning Equivalent to PAC Learning?

Most work in the area of learning theory has focused on designing effective Probably Approximately Correct (PAC) learners. Recently, other models of learning such as transductive error have seen more scrutiny. We move toward showing that these problems are equivalent by reducing agnostic learning with a PAC guarantee to agnostic learning with a transductive guarantee by adding a small number of samples to the dataset. We first rederive the result of Aden-Ali et al. arXiv:2304.09167 reducing PAC learning to transductive learning in the realizable setting using simpler techniques and at more generality as background for our main positive result. Our agnostic transductive to PAC conversion technique extends the aforementioned argument to the agnostic case, showing that an agnostic transductive learner can be efficiently converted to an agnostic PAC learner. Finally, we characterize the performance of the agnostic one inclusion graph algorithm of Asilis et al. arXiv:2309.13692 for binary classification, and show that plugging it into our reduction leads to an agnostic PAC learner that is essentially optimal. Our results imply that transductive and PAC learning are essentially equivalent for supervised learning with pseudometric losses in the realizable setting, and for binary classification in the agnostic setting. We conjecture this is true more generally for the agnostic setting.