Boosting Math Problem Solving in Small LLMs via Ensembles
摘要
Large Language Models (LLMs) have shown impressive capabilities in solving complex mathematical problems, making them valuable tools for education, research, and automated tutoring. However, top-performing models on benchmarks like MATH500, such as GPT-4 and DeepSeek-R1, are often large, proprietary, and costly to use, limiting their accessibility. In contrast, smaller open-source models are more affordable and easier to deploy locally but typically underperform in mathematical reasoning tasks. In this work, we explore the math-problem-solving potential of six small-scale, open-source LLMs (all under 10 billion parameters): Arithmo-Mistral-7B, MAmmoTH-7B, MAmmoTH-8B, MetaMath-7B, MetaMath-Llemma-7B, and MetaMath-Mistral-7B, on the MATH500 benchmark. To enhance their accuracy, we apply two “test-time” ensemble strategies: (1) Intra-model ensemble, where each model generates five independent outputs and the most frequent prediction is selected; and (2) Inter-model ensemble, where 2-level majority voting is performed: first at the intra-level, then across all the models in the ensemble. Our results show that the Intra-model ensemble consistently improves performance over individual runs, and combining outputs across models yields further gains. An ensemble of all six models achieves 38% accuracy on MATH500, outperforming each model’s standalone accuracy (typically 20–30%). Furthermore, a smaller ensemble of the top three models, MetaMath-Llemma-7B, MAmmoTH-7B, and MAmmoTH-8B, achieves a best result of 39.4%. These findings demonstrate that lightweight ensemble techniques can significantly boost math problem solving performance in small LLMs without additional training or expensive computational resources.