Function-Structured Reinforcement Learning with Executable Verifiers for Mathematical Reasoning
FSG-RL connects subproblem graphs with Python code and multi-verifier feedback to improve math reasoning, raising final-answer accuracy from 43.25% to 67.50% over supervised fine-tuning.
Published Oct 1, 2026arXiv ↗
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FSG-RL delivers a rigorous verifier-guided framework with answer-gated span credit and structured memory that raises full solution success substantially, yet its 52% ceiling, opaque verifier specifications, missing split-level analysis, and unverified memory durability leave it an…
Abstract
Algorithmic mathematical reasoning requires reliable decomposition, computation, and aggregation. Final-answer rewards provide limited guidance on intermediate errors, while successful execution does not guarantee mathematical correctness. This work proposes Function-Structured Graph Reinforcement Learning (FSG-RL), connecting subproblem graphs and Python implementations with multi-verifier feedback. The policy first learns to generate code from function graphs through supervised fine-tuning (SFT). Group Relative Policy Optimization (GRPO) then optimizes the policy using answer-gated rewards and span-level credit assignment. The framework also supports teacher supervision and structured memory. A benchmark curated from Grade School Math 8K (GSM8K), MathQA, MATH, and Omni-MATH pairs public function graphs with private verification specifications. Under a unified evaluation protocol, GRPO improves final-answer accuracy from 43.25% to 67.50% and full solution success from 32.25% to 52.25% over SFT. Continued reinforcement learning (RL) with teacher supervision yields additional gains. The gains extend beyond producing correctly formatted code, supporting verifier-guided reinforcement learning for mathematical reasoning. Code is available at https://github.com/ZihanLiummyycc/FSG-RL.