Abstract
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}.
