Learning to Think from Multiple Thinkers
For machine learning researchers, this work reveals fundamental limitations of passive CoT supervision from multiple thinkers and proposes an active learning solution, though the results are theoretical and incremental.
This paper studies learning with Chain-of-Thought (CoT) supervision from multiple thinkers, showing that under cryptographic assumptions, learning can be hard from a few thinkers in passive settings, but provides an efficient active learning algorithm requiring only a small amount of CoT data per thinker and moderate number of thinkers.
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 $\frac{1}{\varepsilon}\cdot poly\log\frac{1}{\varepsilon}$.