We study learning with Chain-of-Thought (CoT) supervision from multiple thinkers, all of whom provide correct but possibly systematically different solutions, e.g., step-by-step solutions to math problems written by different thinkers, or step-by-step execution traces of different programs solving the same problem. We consider classes that are computationally easy to learn using CoT supervision from a single thinker, but hard to learn with only end-result supervision, i.e., without CoT (Joshi et al. 2025). We establish that, under cryptographic assumptions, learning can be hard from CoT supervision provided by two or a few different thinkers, in passive data-collection settings. On the other hand, we provide a generic computationally efficient active learning algorithm that learns with a small amount of CoT data per thinker that is completely independent of the target accuracy $\varepsilon$, a moderate number of thinkers that scales as $\log \frac{1}{\varepsilon}\log \log \frac{1}{\varepsilon}$, and sufficient passive end-result data that scales as $\frac{1}{\varepsilon}\cdot poly\log\frac{1}{\varepsilon}$.

翻译：我们研究了利用来自多位思考者的思维链（CoT）监督进行学习的问题，其中所有思考者都提供正确但可能系统性不同的解决方案，例如不同思考者编写的数学题分步解答，或解决同一问题的不同程序的逐步执行轨迹。我们考虑那些使用单一思考者的CoT监督在计算上易于学习，但仅依赖最终结果监督（即无CoT，参见Joshi et al., 2025）则难以学习的类别。研究表明，在密码学假设下，若采用被动数据收集设置，从两位或少数不同思考者的CoT监督中学习可能是困难的。另一方面，我们提出了一种通用的计算高效主动学习算法，该算法仅需每个思考者提供少量CoT数据（该数据量完全独立于目标精度$\varepsilon$）、适量思考者数量（按$\log \frac{1}{\varepsilon}\log \log \frac{1}{\varepsilon}$缩放），以及充足的被动最终结果数据（按$\frac{1}{\varepsilon}\cdot poly\log\frac{1}{\varepsilon}$缩放）。