AlphaZero Neural Scaling and Zipf's Law: a Tale of Board Games and Power Laws
This work provides insights into scaling phenomena in reinforcement learning, potentially aiding algorithm design, but it is incremental as it applies existing theories to a new domain.
The paper investigates neural scaling laws in AlphaZero, finding that game states follow Zipf's law and that agents optimize state loss in descending frequency order, with inverse scaling linked to unusual Zipf curves where end-game states become overly frequent.
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's-law 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.