AlphaZero Neural Scaling and Zipf's Law: a Tale of Board Games and Power Laws

Neural scaling laws are observed in a range of domains, to date with no universal understanding of why they occur. Recent theories suggest that loss power laws arise from Zipf's law, a power law observed in domains like natural language. One theory suggests that language scaling laws emerge when Zipf-distributed task quanta are learned in descending order of frequency. In this paper we examine power-law scaling in AlphaZero, a reinforcement learning algorithm, using a model of language-model scaling. We find that game states in training and inference data scale with Zipf's law, which is known to arise from the tree structure of the environment, and examine the correlation between scaling-law and Zipf'slaw exponents. In agreement with the quanta scaling model, we find that agents optimize state loss in descending order of frequency, even though this order scales inversely with modelling complexity. We also find that inverse scaling, the failure of models to improve with size, is correlated with unusual Zipf curves where end-game states are among the most frequent states. We show evidence that larger models shift their focus to these less-important states, sacrificing their understanding of important early-game states.

1. We find that AlphaZero agents that exhibit scaling laws, such as those trained in Neumann & Gros (2022) [5], produce train and test data that follows Zipf's law, suggesting that learned tasks also scale with Zipf's law. 2. We show how these smooth Zipf laws form when agent policies are combined with the tree-structure of board games, which is known to create a Zipf-like state distribution [23].

39th Conference on Neural Information Processing Systems (NeurIPS 2025).