One Sample to Rule Them All: Extreme Data Efficiency in RL Scaling

TL;DR

Polymath learning addresses data efficiency in RL for large language models by designing a single math reasoning sample with multidisciplinary coverage. It introduces Synthetic Prime and natural polymath samples, and a regimen of SALIENT math skills to guide sample construction. The key findings show that one carefully engineered sample can boost reasoning across mathematics, physics, chemistry, and biology and that synthetic, multidisciplinary samples outperform large natural datasets in many benchmarks. This work suggests a shift toward sample engineering as a scalable strategy for enhancing reasoning in LLMs without large data volumes.

Abstract

The reasoning ability of large language models (LLMs) can be unleashed with reinforcement learning (RL) (OpenAI, 2024; DeepSeek-AI et al., 2025a; Zeng et al., 2025). The success of existing RL attempts in LLMs usually relies on high-quality samples of thousands or beyond. In this paper, we challenge fundamental assumptions about data requirements in RL for LLMs by demonstrating the remarkable effectiveness of one-shot learning. Specifically, we introduce polymath learning, a framework for designing one training sample that elicits multidisciplinary impact. We present three key findings: (1) A single, strategically selected math reasoning sample can produce significant performance improvements across multiple domains, including physics, chemistry, and biology with RL; (2) The math skills salient to reasoning suggest the characteristics of the optimal polymath sample; and (3) An engineered synthetic sample that integrates multidiscipline elements outperforms training with individual samples that naturally occur. Our approach achieves superior performance to training with larger datasets across various reasoning benchmarks, demonstrating that sample quality and design, rather than quantity, may be the key to unlock enhanced reasoning capabilities in language models. Our results suggest a shift, dubbed as sample engineering, toward precision engineering of training samples rather than simply increasing data volume.

Figures (10)

• Figure 1: The subject-level performance of different learning strategies. OE stands for subjects with open-ended problems. The subjects are sorted by subject embedding distance to MATH500 (the grey dotted line), from low to high. The blue line represents pass ratio from 64 independent attempts of the base model. The stars and triangles represent best performance of in-context learning and polymath learning. Note that we only display the best polymath learning and in-context polymath learning results for demonstration.
• Figure 2: Skill spectrum between natural and synthetic polymath samples. The polygon represents number of salient skills identified in each math domain (Geo. and Precal. represents Geometry and Precalculus respectively). The real and dashed areas represent the natural and synthetic specialist samples except the last one, which represents the Synthetic Prime sample, and the synthetic samples include more comprehensive salient skill sets than the natural polymath samples.
• Figure 3: Average number of mathematical skills employed per problem in different subject domains. Algebra and Precalculus skills are the most prevalent.
• Figure 4: Self-verification patterns under different comprehensive and polymath samples across all subjects. Verification patterns like 're-evaluate' and 'recheck' appear most frequently in polymath learning with the 'number theory' sample, and the 'intermediate algebra' sample elicits the most code blocks in reasoning.
• Figure 5: Average domain performance over natural samples with different LIMR scores. The performance is reported the same way as in Table \ref{['table: main results by subject domains']}. The samples with LIMR score being 0.6 perform best.