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.

Performance scaling laws with resources are heavily studied in supervised deep learning models but not in reinforcement learning.We examine the scaling of the AlphaZero [1] algorithm's performance with model size by training agents on three competitive two-player games, Connect Four, Oware and Pentago.We find that performance, in the form of the Elo rating, scales logarithmically with the number of free neural network parameters, a trend consistent across games and when using deeper neural networks.This leads to a universal expression for the average match outcome which depends only on the ratio of sizes between opponents, which is supported by an agnostic rating method.