Yusuf Kalayci

Shaddin Dughmi, Yusuf Kalayci, Grayson York

Much of learning theory is concerned with the design and analysis of probably approximately correct (PAC) learners. The closely related transductive model of learning has recently seen more scrutiny, with its learners often used as precursors to PAC learners. Our goal in this work is to understand and quantify the exact relationship between these two models. First, we observe that modest extensions of existing results show the models to be essentially equivalent for realizable learning for most natural loss functions, up to low order terms in the error and sample complexity. The situation for agnostic learning appears less straightforward, with sample complexities potentially separated by a $\frac{1}ε$ factor. This is therefore where our main contributions lie. Our results are two-fold: 1. For agnostic learning with bounded losses (including, for example, multiclass classification), we show that PAC learning reduces to transductive learning at the cost of low-order terms in the error and sample complexity via an adaptation of the reduction of arXiv:2304.09167 to the agnostic setting. 2. For agnostic binary classification, we show the converse: transductive learning is essentially no more difficult than PAC learning. Together with our first result this implies that the PAC and transductive models are essentially equivalent for agnostic binary classification. This is our most technical result, and involves two steps: A symmetrization argument on the agnostic one-inclusion graph (OIG) of arXiv:2309.13692 to derive the worst-case agnostic transductive instance, and expressing the error of the agnostic OIG algorithm for this instance in terms of the empirical Rademacher complexity of the class. We leave as an intriguing open question whether our second result can be extended beyond binary classification to show the transductive and PAC models equivalent more broadly.

Ömer Faruk Akgül, Yusuf Hakan Kalaycı, Rajgopal Kannan et al.

Large reasoning models achieve strong performance on complex tasks by generating extended chains of thought, but they often "overthink": continuing to reason long after they have enough information to answer correctly. This wastes inference-time compute and can hurt accuracy. Existing attempts to stop early either manipulate decoding with extra sampling and heuristics, rely on auxiliary verifier models, or operate only as post-hoc analysis pipelines without formal guarantees. We introduce LYNX, an online early-exit mechanism that turns a model's own hidden-state awareness into confidence-controlled stopping decisions. LYNX attaches exit decisions to naturally occurring reasoning cues (e.g., "hmm", "wait") during generation, trains a lightweight probe on hidden states at those cue tokens using supervision from forced exits, and wraps the resulting scores in split conformal prediction to obtain distribution-free control over premature exits. Crucially, we train and calibrate this probe once on a generic mathematical corpus and reuse it unchanged across benchmarks, decoding temperatures, and even non-mathematical tasks. Across three model families spanning 1.5B to 32B parameters, a single mathematically trained probe per base model yields strong accuracy--efficiency tradeoffs. On GSM8K, LYNX matches or improves baseline accuracy while reducing tokens by 40--65\%; on MATH-500 it improves accuracy by up to 12 points with roughly 35--60\% fewer tokens; on AIME 2024 it recovers baseline accuracy with more than 50\% token savings; and on CommonsenseQA, a non-math benchmark, it transfers zero-shot with modest accuracy gains and up to 70\% fewer tokens. Compared to state-of-the-art early-exit methods, LYNX offers competitive or superior Pareto frontiers while remaining fully online, requiring no proxy models at inference, and providing explicit, user-tunable confidence guarantees.

Yusuf Kalayci, Vinod Raman, Shaddin Dughmi

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.