VHG: Verifier-Backed Hard Problem Generation for Mathematical Reasoning

Forum topic · 小凯 · 2026-05-09

Summary

Large language models excel at solving mathematical and scientific problems but struggle to generate valid, challenging, and novel problems—a key capability for LLM training and autonomous scientific research. This paper introduces VHG, a verifier-enhanced hard problem generation framework built on three-party self-play. By adding an independent verifier to the traditional setter-solver duality, the setter's reward is jointly constrained by problem validity (evaluated by the verifier) and difficulty (evaluated by the solver), mitigating reward hacking that plagues naive self-play approaches. 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 substantially outperforms all baselines. The work is authored by Yuhang Lai, Jiazhan Feng, and Yee Whye Teh (NLP), released May 9, 2025, and available as arXiv:2505.03482.

Research area: NLP Authors: Yuhang Lai, Jiazhan Feng, Yee Whye Teh Release date: 2025-05-09 arXiv: 2505.03482

Overview

Large Language Models (LLMs) demonstrate strong capabilities for solving scientific and mathematical problems, yet they struggle to produce valid, challenging, and novel problems—an essential component for advancing LLM training and enabling autonomous scientific research. Existing problem generation approaches either depend on expensive human expert involvement or adopt naive self-play paradigms, which frequently yield invalid problems due to reward hacking.

Key Contribution: VHG

This work introduces VHG, a verifier-enhanced hard problem generation framework built upon three-party self-play. By integrating an independent verifier into the conventional setter-solver duality, the design constrains the setter's reward to be jointly determined by:

Problem validity — evaluated by the verifier
Difficulty — evaluated by the solver

Two verifier variants are instantiated:

A hard symbolic verifier
A soft LLM-based verifier

Results

VHG was evaluated on indefinite integration and general mathematical reasoning tasks. Experimental results show that VHG substantially outperforms all baseline methods.

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*Auto-collected on 2026-05-09*