LILO: Learning to Reason at the Frontier of Learnability

Reinforcement learning is a widely adopted component of large language model post-training, especially for reasoning-style tasks such as maths questions. However, as we show, most existing methods will provably fail to learn from questions that are too hard, where the model always fails, or too easy, where the model always succeeds. Much human effort is therefore spent producing datasets of questions of a suitable difficulty for state-of-the-art models. Given this, we consider how to algorithmically identify questions that allow for maximally efficient training. We introduce a method, LILO (Learnability Improves LLMs Optimally), that prioritises training on questions with high variance of success, known as learnability, and we provide theory which shows that LILO enables the expected improvement of the model to be large. We run a wide range of experiments over multiple base models, algorithms and reasoning datasets to demonstrate that LILO consistently reaches a higher final test accuracy, and can do so in 3× fewer training steps. We explore how questions with high learnability can be efficiently identified, and discuss how learnability can be scaled to produce LLM agents that autonomously and open-endedly expand the frontier of human knowledge.