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 clear 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 theory 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 quanta scaling theory, 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.

Scaling Laws for a Multi-Agent Reinforcement Learning Model

The recent observation of neural power-law scaling relations has made a significant impact in the field of deep learning. A substantial amount of attention has been dedicated as a consequence to the description of scaling laws, although mostly for supervised learning and only to a reduced extent for reinforcement learning frameworks. In this paper we present an extensive study of performance scaling for a cornerstone reinforcement learning algorithm, AlphaZero. On the basis of a relationship between Elo rating, playing strength and power-law scaling, we train AlphaZero agents on the games Connect Four and Pentago and analyze their performance. We find that player strength scales as a power law in neural network parameter count when not bottlenecked by available compute, and as a power of compute when training optimally sized agents. We observe nearly identical scaling exponents for both games. Combining the two observed scaling laws we obtain a power law relating optimal size to compute similar to the ones observed for language models. We find that the predicted scaling of optimal neural network size fits our data for both games. This scaling law implies that previously published state-of-the-art game-playing models are significantly smaller than their optimal size, given the respective compute budgets. We also show that large AlphaZero models are more sample efficient, performing better than smaller models with the same amount of training data.

Investment vs. reward in a competitive knapsack problem

Natural selection drives species to develop brains, with sizes that increase with the complexity of the tasks to be tackled. Our goal is to investigate the balance between the metabolic costs of larger brains compared to the advantage they provide in solving general and combinatorial problems. Defining advantage as the performance relative to competitors, a two-player game based on the knapsack problem is used. Within this framework, two opponents compete over shared resources, with the goal of collecting more resources than the opponent. Neural nets of varying sizes are trained using a variant of the AlphaGo Zero algorithm. A surprisingly simple relation, $N_A/(N_A+N_B)$, is found for the relative win rate of a net with $N_A$ neurons against one with $N_B$. Success increases linearly with investments in additional resources when the networks sizes are very different, i.e. when $N_A \ll N_B$, with returns diminishing when both networks become comparable in size.