Bidirectional Curriculum Generation: A Multi-Agent Framework for Data-Efficient Mathematical Reasoning

2026-03-05

AI Summary

This work addresses the inefficiency of current large language models in mathematical reasoning, which often rely on massive datasets, and the suboptimal sample utilization caused by traditional unidirectional curriculum learning that blindly escalates problem difficulty. To overcome these limitations, the authors propose a multi-agent framework featuring a bidirectional curriculum generation mechanism grounded in the optimal pacing theorem. This mechanism dynamically modulates problem difficulty, adaptively alternating between challenging the model’s capabilities and repairing its reasoning flaws. A closed-loop feedback system is integrated through diagnostic analysis of reasoning failures and targeted simplification strategies. The approach substantially reduces the required training data while significantly outperforming existing baselines in reasoning performance, thereby demonstrating the efficacy of data-efficient mathematical reasoning.

Technology Category

• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Multiagent Systems: Multiagent Learning
• Machine Learning: Large Multimodal Models (LMMs)

Application Category

• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking
• Semantics and Knowledge: Data modeling to support human-machine intelligence, including LLMs agents, intelligent system behavior, explanations, and user-friendly interactions
• User Modeling, Personalization and Recommendation: Large Language Models (LLM) for user modeling and recommendation

Abstract

Enhancing mathematical reasoning in Large Language Models typically demands massive datasets, yet data efficiency remains a critical bottleneck. While Curriculum Learning attempts to structure this process, standard unidirectional approaches (simple-to-complex) suffer from inefficient sample utilization: they blindly escalate complexity even when foundational gaps persist, leading to wasted computation on unsolvable problems. To maximize the instructional value of every training sample, we introduce a novel Bidirectional Curriculum Generation framework. Unlike rigid trajectories, our multi-agent ecosystem mimics adaptive pedagogy to establish a closed feedback loop. It dynamically generates data by either complicating problems to challenge the model or, crucially, simplying them to repair specific reasoning failures. This mechanism ensures that the model consumes only the most effective data at any given stage. Grounded in the Optimal Pacing Theorem, our approach optimizes the learning trajectory, significantly outperforming baselines while achieving superior reasoning performance with substantially fewer instruction samples.

Problem

• mathematical reasoning
• data efficiency
• curriculum learning
• sample utilization
• large language models

Innovation

• Bidirectional Curriculum Generation
• Multi-Agent Framework
• Data-Efficient Learning
• Adaptive Pedagogy
• Mathematical Reasoning