# Scaling Scaling Laws with Board Games

arXiv: https://arxiv.org/abs/2104.03113 · HTML: https://mlsanity.com/paper/2104.03113/

## Key facts

- Cited by 63 papers in the map (data through 14 Oct 2026).
- Builds on 24 of its 52 references that are in the map.
- First posted on arXiv 7 Apr 2021.

## Builds on (most cited first, 24 of 24)

- [PyTorch: An Imperative Style, High-Performance Deep Learning Library](https://mlsanity.com/paper/1912.01703/) (2019)
- [Scaling Laws for Neural Language Models](https://mlsanity.com/paper/2001.08361/) (2020)
- [Energy and Policy Considerations for Deep Learning in NLP](https://mlsanity.com/paper/1906.02243/) (2019)
- [Dota 2 with Large Scale Deep Reinforcement Learning](https://mlsanity.com/paper/1912.06680/) (2019)
- [Deep Learning Scaling is Predictable, Empirically](https://mlsanity.com/paper/1712.00409/) (2017)
- [Scaling Laws for Autoregressive Generative Modeling](https://mlsanity.com/paper/2010.14701/) (2020)
- [An Empirical Model of Large-Batch Training](https://mlsanity.com/paper/1812.06162/) (2018)
- [Scaling Laws for Transfer](https://mlsanity.com/paper/2102.01293/) (2021)
- [Thinking Fast and Slow with Deep Learning and Tree Search](https://mlsanity.com/paper/1705.08439/) (2017)
- [A Constructive Prediction of the Generalization Error Across Scales](https://mlsanity.com/paper/1909.12673/) (2019)
- [OpenSpiel: A Framework for Reinforcement Learning in Games](https://mlsanity.com/paper/1908.09453/) (2019)
- [Phasic Policy Gradient](https://mlsanity.com/paper/2009.04416/) (2020)
- [Learning Curve Theory](https://mlsanity.com/paper/2102.04074/) (2021)
- [Trivializations for Gradient-Based Optimization on Manifolds](https://mlsanity.com/paper/1909.09501/) (2019)
- [Real World Games Look Like Spinning Tops](https://mlsanity.com/paper/2004.09468/) (2020)
- [Accelerating Self-Play Learning in Go](https://mlsanity.com/paper/1902.10565/) (2019)
- [Monte-Carlo Tree Search as Regularized Policy Optimization](https://mlsanity.com/paper/2007.12509/) (2020)
- [Re-evaluating Evaluation](https://mlsanity.com/paper/1806.02643/) (2018)
- [Accelerated Methods for Deep Reinforcement Learning](https://mlsanity.com/paper/1803.02811/) (2018)
- [ELF OpenGo: An Analysis and Open Reimplementation of AlphaZero](https://mlsanity.com/paper/1902.04522/) (2019)
- [On the Predictability of Pruning Across Scales](https://mlsanity.com/paper/2006.10621/) (2020)
- [Multiagent Evaluation under Incomplete Information](https://mlsanity.com/paper/1909.09849/) (2019)
- [Polygames: Improved Zero Learning](https://mlsanity.com/paper/2001.09832/) (2020)
- [Neurohex: A Deep Q-learning Hex Agent](https://mlsanity.com/paper/1604.07097/) (2016)

## Built on by (most cited first, 30 of 63)

- [Scaling LLM Test-Time Compute Optimally can be More Effective than Scaling Model Parameters](https://mlsanity.com/paper/2408.03314/) (2024)
- [A General Language Assistant as a Laboratory for Alignment](https://mlsanity.com/paper/2112.00861/) (2021)
- [Large Language Monkeys: Scaling Inference Compute with Repeated Sampling](https://mlsanity.com/paper/2407.21787/) (2024)
- [Are Emergent Abilities of Large Language Models a Mirage?](https://mlsanity.com/paper/2304.15004/) (2023)
- [Towards Reasoning Era: A Survey of Long Chain-of-Thought for Reasoning Large Language Models](https://mlsanity.com/paper/2503.09567/) (2025)
- [Cognitive Behaviors that Enable Self-Improving Reasoners, or, Four Habits of Highly Effective STaRs](https://mlsanity.com/paper/2503.01307/) (2025)
- [Inference Scaling Laws: An Empirical Analysis of Compute-Optimal Inference for Problem-Solving with Language Models](https://mlsanity.com/paper/2408.00724/) (2024)
- [Compute Trends Across Three Eras of Machine Learning](https://mlsanity.com/paper/2202.05924/) (2022)
- [Optimizing Test-Time Compute via Meta Reinforcement Fine-Tuning](https://mlsanity.com/paper/2503.07572/) (2025)
- [Broken Neural Scaling Laws](https://mlsanity.com/paper/2210.14891/) (2022)
- [Towards System 2 Reasoning in LLMs: Learning How to Think With Meta Chain-of-Thought](https://mlsanity.com/paper/2501.04682/) (2025)
- [Scaling Test-Time Compute Without Verification or RL is Suboptimal](https://mlsanity.com/paper/2502.12118/) (2025)
- [Beyond Autoregression: Fast LLMs via Self-Distillation Through Time](https://mlsanity.com/paper/2410.21035/) (2024)
- [ZeroBench: An Impossible Visual Benchmark for Contemporary Large Multimodal Models](https://mlsanity.com/paper/2502.09696/) (2025)
- [Does equivariance matter at scale?](https://mlsanity.com/paper/2410.23179/) (2024)
- [How Do Large Language Monkeys Get Their Power (Laws)?](https://mlsanity.com/paper/2502.17578/) (2025)
- [Scaling laws for single-agent reinforcement learning](https://mlsanity.com/paper/2301.13442/) (2023)
- [Machine Learning Model Sizes and the Parameter Gap](https://mlsanity.com/paper/2207.02852/) (2022)
- [Scaling Laws for a Multi-Agent Reinforcement Learning Model](https://mlsanity.com/paper/2210.00849/) (2022)
- [Linguistic Generalizability of Test-Time Scaling in Mathematical Reasoning](https://mlsanity.com/paper/2502.17407/) (2025)
- [Pretraining on the Test Set Is All You Need](https://mlsanity.com/paper/2309.08632/) (2023)
- [MINERVA: Evaluating Complex Video Reasoning](https://mlsanity.com/paper/2505.00681/) (2025)
- [Rethinking Fine-Tuning when Scaling Test-Time Compute: Limiting Confidence Improves Mathematical Reasoning](https://mlsanity.com/paper/2502.07154/) (2025)
- [(Mis)Fitting: A Survey of Scaling Laws](https://mlsanity.com/paper/2502.18969/) (2025)
- [When AI Co-Scientists Fail: SPOT-a Benchmark for Automated Verification of Scientific Research](https://mlsanity.com/paper/2505.11855/) (2025)
- [Metastable Dynamics of Chain-of-Thought Reasoning: Provable Benefits of Search, RL and Distillation](https://mlsanity.com/paper/2502.01694/) (2025)
- [Scaling Laws Under the Microscope: Predicting Transformer Performance from Small Scale Experiments](https://mlsanity.com/paper/2202.06387/) (2022)
- [Compositional Capabilities of Autoregressive Transformers: A Study on Synthetic, Interpretable Tasks](https://mlsanity.com/paper/2311.12997/) (2023)
- [ATLAS: Adaptive Transfer Scaling Laws for Multilingual Pretraining, Finetuning, and Decoding the Curse of Multilinguality](https://mlsanity.com/paper/2510.22037/) (2025)
- [Scaling Embedding Layers in Language Models](https://mlsanity.com/paper/2502.01637/) (2025)

## Abstract (arXiv)

The largest experiments in machine learning now require resources far beyond the budget of all but a few institutions. Fortunately, it has recently been shown that the results of these huge experiments can often be extrapolated from the results of a sequence of far smaller, cheaper experiments. In this work, we show that not only can the extrapolation be done based on the size of the model, but on the size of the problem as well. By conducting a sequence of experiments using AlphaZero and Hex, we show that the performance achievable with a fixed amount of compute degrades predictably as the game gets larger and harder. Along with our main result, we further show that the test-time and train-time compute available to an agent can be traded off while maintaining performance.