Dean Foster

João Sedoc, Baotong Zhang, Dean Foster

Reliably knowing when a language model is correct is almost as important as being correct. We introduce prover-verifier deliberation (PVD), an inference-time protocol grounded in interactive proof theory, as a mechanism for selective prediction: the protocol produces both an answer and a structured confidence verdict, allowing a system to report high-confidence answers while abstaining on uncertain cases. In each dialogue, a prover defends a candidate answer through checkable sub-claims while a verifier issues targeted challenges and returns \textsc{Accept}, \textsc{Challenge}, or \textsc{Reject}. Because frozen language models are imperfect provers and verifiers operating over a noisy channel, formal soundness and completeness guarantees do not transfer; instead, we characterize the protocol empirically through its coverage-precision behavior. Our main experiment uses Claude Sonnet 4.6 as prover and Claude Haiku 4.5 as verifier on GPQA Diamond. Questions accepted with no answer revision, which we call Accept + No Change (ANC), are reported as the high-confidence subset; we evaluate this subset by its precision and coverage. ANC separates reliable from unreliable answers, yielding a $\sim$30pp HC-Prec gap over the non-ANC complement. Robustness experiments with GPT and Gemini pairings show that high HC-Prec can transfer across model families, while verifier strictness and domain competence largely determine the size of the selection gap. On Humanity's Last Exam, weaker prover-verifier pairings can collapse or invert the ANC signal, illustrating a practical failure mode when the verifier operates outside its effective region. Comparisons with self-consistency, universal self-consistency, multi-agent debate, and Reflexion suggest that prover-verifier deliberation supplies a distinct argument-defensibility signal for selective prediction.

Jing Wang, Jie Shen, Dean Foster et al.

The trade-off between labeled data availability and downstream accuracy remains a central challenge in fine-tuning large language models (LLMs). We propose a principled framework for \emph{budget-aware supervised fine-tuning} by casting LLM adaptation as a contextual Stackelberg game. In our formulation, the learner (leader) commits to a scoring policy and a label-querying strategy, while an adaptive environment (follower) selects challenging supervised alternatives in response. To explicitly address label efficiency, we incorporate a finite supervision budget directly into the learning objective. Our algorithm operates in the full-feedback regime and achieves $\tilde{O}(d\sqrt{T})$ regret under standard linear contextual assumptions. We extend the framework with a Largest-Latency-First (LLF) confidence gate that selectively queries labels, achieving a budget-aware regret bound of $\tilde{O}(\sqrt{dB} + c\sqrt{B})$ with $B=βT$.

Yuda Song, Hanlin Zhang, Carson Eisenach et al.

Self-improvement is a mechanism in Large Language Model (LLM) pre-training, post-training and test-time inference. We explore a framework where the model verifies its own outputs, filters or reweights data based on this verification, and distills the filtered data. Despite several empirical successes, a fundamental understanding is still lacking. In this work, we initiate a comprehensive, modular and controlled study on LLM self-improvement. We provide a mathematical formulation for self-improvement, which is largely governed by a quantity which we formalize as the generation-verification gap. Through experiments with various model families and tasks, we discover a scaling phenomenon of self-improvement -- a variant of the generation-verification gap scales monotonically with the model pre-training flops. We also examine when self-improvement is possible, an iterative self-improvement procedure, and ways to improve its performance. Our findings not only advance understanding of LLM self-improvement with practical implications, but also open numerous avenues for future research into its capabilities and boundaries.