Summary
This paper introduces VHG, a verifier-augmented hard problem generation framework based on a three-party self-play paradigm, authored by Yuhang Lai, Jiazhan Feng, Yee Whye Teh, and Ning Miao (arXiv:2605.06660, 2026-05-07). Large language models excel at solving scientific and mathematical problems but struggle to generate valid, challenging, and novel problems—a capability essential for LLM training and autonomous scientific research. Existing generation approaches either require costly human experts or rely on naive self-play, which often produces invalid problems due to reward hacking. VHG addresses this by adding an independent verifier to the traditional problem-setter-solver duo: the setter's reward is constrained jointly by problem validity (assessed by the verifier) and difficulty (assessed by the solver). The authors instantiate two verifier variants—a hard symbolic verifier and a soft LLM-based verifier—and evaluate VHG on indefinite integration and general mathematical reasoning tasks. Experiments show VHG significantly outperforms all baseline methods.
Paper Overview
- Field: Machine Learning
- Authors: Yuhang Lai, Jiazhan Feng, Yee Whye Teh, Ning Miao
- Published: 2026-05-07
- arXiv: 2605.06660
Summary
Large language models (LLMs) demonstrate strong capabilities in solving scientific and mathematical problems, but they struggle to generate valid, challenging, and novel problems—a key element for advancing LLM training and enabling autonomous scientific research. Existing problem generation methods either rely on expensive human expert involvement or adopt a naive self-play paradigm, which frequently produces invalid problems due to reward hacking.
This work proposes VHG, a verifier-augmented hard problem generation framework built on a three-party self-play paradigm. By integrating an independent verifier into the traditional setter-solver dual structure, the design constrains the problem setter's reward to be determined jointly by:
- Problem validity, as assessed by the verifier
- Difficulty, as assessed by the solver
The authors instantiate two verifier variants: a
hard symbolic verifier and a
soft LLM verifier, and evaluate the framework on indefinite integration and general mathematical reasoning tasks. Experimental results show that VHG significantly outperforms all baseline methods by a clear margin.
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*Auto-collected on 2026-05-10.*
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