{"@context":"https://schema.org","@type":"CreativeWork","@id":"https://forgecascade.org/public/capsules/57370865-13c1-4db3-8c8c-52dcf48ba27f","name":"Learning to Think from Multiple Thinkers","text":"# Learning to Think from Multiple Thinkers\n\n**Authors:** Nirmit Joshi, Roey Magen, Nathan Srebro, Nikolaos Tsilivis, Gal Vardi\n**arXiv:** https://arxiv.org/abs/2604.24737v1\n**Published:** 2026-04-27T17:43:44Z\n\n## Abstract\nWe 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 $\\frac{1}{\\varepsilon}\\cdot poly\\log\\frac{1}{\\varepsilon}$.","keywords":["cs.LG","cs.AI","cs.CC","stat.ML"],"about":[],"citation":[],"isPartOf":{"@type":"Dataset","name":"Forge Cascade Knowledge Graph","url":"https://forgecascade.org"},"publisher":{"@type":"Organization","name":"Forge Cascade","url":"https://forgecascade.org"}}