MathFusion: Enhancing Mathematical Problem-solving of LLM through Instruction Fusion

Abstract: LLMs have shown impressive progress in mathematical reasoning. While data augmentation is promising to enhance mathematical problem-solving ability, current approaches are predominantly limited to instance-level modifications-such as rephrasing or generating syntactic variations-which fail to capture and leverage the intrinsic relational structures inherent in mathematical knowledge. Inspired by human learning processes, where mathematical proficiency develops through systematic exposure to interconnected concepts, we introduce MathFusion, a novel framework that enhances mathematical reasoning through cross-problem instruction synthesis. MathFusion implements this through three fusion strategies: (1) sequential fusion, which chains related problems to model solution dependencies; (2) parallel fusion, which combines analogous problems to reinforce conceptual understanding; and (3) conditional fusion, which creates context-aware selective problems to enhance reasoning flexibility. By applying these strategies, we generate a new dataset, \textbf{MathFusionQA}, followed by fine-tuning models (DeepSeekMath-7B, Mistral-7B, Llama3-8B) on it. Experimental results demonstrate that MathFusion achieves substantial improvements in mathematical reasoning while maintaining high data efficiency, boosting performance by 18.0 points in accuracy across diverse benchmarks while requiring only 45K additional synthetic instructions, representing a substantial improvement over traditional single-instruction approaches. Our datasets, models, and code are publicly available at https://github.com/QizhiPei/mathfusion.

• The paper introduces MathFusion, a novel training framework that improves LLM mathematical reasoning by creating and using interconnected problems based on sequential, parallel, and conditional fusion strategies.
• MathFusion training data enhances existing LLMs like DeepSeekMath, Mistral, and Llama3, achieving an average 18.0 percentage point accuracy increase on math benchmarks with just 45,000 synthetic examples.
• This data-efficient approach suggests improved AI math tutors and scientific assistants are possible and demonstrates a method potentially applicable to enhancing broader LLM reasoning abilities.

Here is a summary of the paper "MathFusion: Enhancing Mathematic Problem-solving of LLM through Instruction Fusion" (2503.16212):

Rationale and Problem Solved

• Problem: Existing methods to improve the math skills of LLMs often modify individual problems (like rephrasing). This doesn't effectively teach the model how different mathematical concepts and problems relate to each other.
• Goal: The paper introduces MathFusion, a new approach inspired by how humans learn math. It aims to enhance LLM mathematical reasoning by teaching them using interconnected problems, rather than isolated ones.

Data and Methods

• Fusion Strategies: MathFusion creates new training data by combining existing math problems in three ways:
  • Sequential fusion: Links problems where solving one helps solve the next, mimicking dependencies.
  • Parallel fusion: Presents similar problems together to strengthen understanding of core concepts.
  • Conditional fusion: Creates problems that require choosing the right approach based on context, improving flexibility.
• New Dataset: Using these strategies, the researchers generated a new dataset called MathFusionQA.
• Scale: This new dataset contains 45,000 synthetic instructions (combined problems).

Models Used

• The researchers didn't create a new model from scratch. Instead, they applied the MathFusion training data to improve existing LLMs:
  • DeepSeekMath-7B
  • Mistral-7B
  • Llama3-8B

Performance Highlights

• Models trained with MathFusion showed significant improvements on various math problem-solving tests (benchmarks).
• Accuracy increased by an average of 18.0 percentage points.
• This improvement was achieved efficiently, requiring only 45,000 additional training examples, demonstrating better data efficiency compared to traditional methods.

Implications and Applications

• Better AI Math Tutors: This technique could lead to AI that better understands and explains mathematical concepts by recognizing relationships between problems.
• Improved Scientific Assistants: LLMs enhanced with MathFusion could be more effective tools for researchers and engineers tackling complex calculations and modeling.
• Efficient AI Training: The method shows that significant performance gains can be achieved with relatively small amounts of carefully structured synthetic data, making AI training more efficient.
• Broader Reasoning Skills: While focused on math, the idea of fusing instructions based on relationships might be applicable to improving LLM reasoning in other complex domains.

In conclusion, MathFusion offers a novel and data-efficient framework for enhancing the mathematical reasoning capabilities of LLMs by training them on strategically combined problems, better reflecting how interconnected mathematical knowledge is learned.