mlsanityData to 7 Oct 2026

Scaling Scaling Laws with Board Games

arXiv 2104.03113 ↗First posted 7 Apr 2021cs.LG cs.MAcited by 62

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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.

Key facts. Cited by 62 papers in the map (data through 7 Oct 2026). Builds on 24 of its 52 references that are in the map. First posted on arXiv 7 Apr 2021.

Builds on

24 of 52 references.

References · 24

  1. 2019PyTorch: An Imperative Style, High-Performance Deep Learning Librarycited by 20,160
  2. 2020Scaling Laws for Neural Language Modelscited by 6,411
  3. 2019Energy and Policy Considerations for Deep Learning in NLPcited by 1,242
  4. 2019Dota 2 with Large Scale Deep Reinforcement Learningcited by 1,084
  5. 2017Deep Learning Scaling is Predictable, Empiricallycited by 556
  6. 2020Scaling Laws for Autoregressive Generative Modelingcited by 510
  7. 2018An Empirical Model of Large-Batch Trainingcited by 268
  8. 2021Scaling Laws for Transfercited by 265
  9. 2017Thinking Fast and Slow with Deep Learning and Tree Searchcited by 245
  10. 2019A Constructive Prediction of the Generalization Error Across Scalescited by 202
  11. 2019OpenSpiel: A Framework for Reinforcement Learning in Gamescited by 180
  12. 2020Phasic Policy Gradientcited by 121
12 more
  1. 2021Learning Curve Theorycited by 79
  2. 2019Trivializations for Gradient-Based Optimization on Manifoldscited by 78
  3. 2020Real World Games Look Like Spinning Topscited by 66
  4. 2019Accelerating Self-Play Learning in Gocited by 60
  5. 2020Monte-Carlo Tree Search as Regularized Policy Optimizationcited by 59
  6. 2018Re-evaluating Evaluationcited by 47
  7. 2018Accelerated Methods for Deep Reinforcement Learningcited by 40
  8. 2019ELF OpenGo: An Analysis and Open Reimplementation of AlphaZerocited by 37
  9. 2020On the Predictability of Pruning Across Scalescited by 33
  10. 2019Multiagent Evaluation under Incomplete Informationcited by 13
  11. 2020Polygames: Improved Zero Learningcited by 11
  12. 2016Neurohex: A Deep Q-learning Hex Agentcited by 5

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Cited by year (2026 to 7 Oct)

Citing papers per year: 2021: 2; 2022: 6; 2023: 5; 2024: 12; 2025: 30; 2026: 7.

62 papers.

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  1. 2024Scaling LLM Test-Time Compute Optimally can be More Effective than Scaling Model Parameterscited by 1,928
  2. 2021A General Language Assistant as a Laboratory for Alignmentcited by 1,023
  3. 2024Large Language Monkeys: Scaling Inference Compute with Repeated Samplingcited by 946
  4. 2023Are Emergent Abilities of Large Language Models a Mirage?cited by 515
  5. 2025Towards Reasoning Era: A Survey of Long Chain-of-Thought for Reasoning Large Language Modelscited by 381
  6. 2025Cognitive Behaviors that Enable Self-Improving Reasoners, or, Four Habits of Highly Effective STaRscited by 373
  7. 2024Inference Scaling Laws: An Empirical Analysis of Compute-Optimal Inference for Problem-Solving with Language Modelscited by 282
  8. 2022Compute Trends Across Three Eras of Machine Learningcited by 161
  9. 2025Optimizing Test-Time Compute via Meta Reinforcement Fine-Tuningcited by 121
  10. 2025Towards System 2 Reasoning in LLMs: Learning How to Think With Meta Chain-of-Thoughtcited by 108
  11. 2022Broken Neural Scaling Lawscited by 102
  12. 2025Scaling Test-Time Compute Without Verification or RL is Suboptimalcited by 82
38 more
  1. 2024Beyond Autoregression: Fast LLMs via Self-Distillation Through Timecited by 73
  2. 2025ZeroBench: An Impossible Visual Benchmark for Contemporary Large Multimodal Modelscited by 56
  3. 2024Does equivariance matter at scale?cited by 49
  4. 2025How Do Large Language Monkeys Get Their Power (Laws)?cited by 40
  5. 2022Machine Learning Model Sizes and the Parameter Gapcited by 39
  6. 2023Scaling laws for single-agent reinforcement learningcited by 39
  7. 2022Scaling Laws for a Multi-Agent Reinforcement Learning Modelcited by 37
  8. 2023Pretraining on the Test Set Is All You Needcited by 33
  9. 2025Linguistic Generalizability of Test-Time Scaling in Mathematical Reasoningcited by 33
  10. 2025MINERVA: Evaluating Complex Video Reasoningcited by 31
  11. 2025Rethinking Fine-Tuning when Scaling Test-Time Compute: Limiting Confidence Improves Mathematical Reasoningcited by 28
  12. 2025Metastable Dynamics of Chain-of-Thought Reasoning: Provable Benefits of Search, RL and Distillationcited by 25
  13. 2025When AI Co-Scientists Fail: SPOT-a Benchmark for Automated Verification of Scientific Researchcited by 25
  14. 2025(Mis)Fitting: A Survey of Scaling Lawscited by 24
  15. 2022Scaling Laws Under the Microscope: Predicting Transformer Performance from Small Scale Experimentscited by 23
  16. 2023Compositional Capabilities of Autoregressive Transformers: A Study on Synthetic, Interpretable Taskscited by 21
  17. 2023Beyond Scale: The Diversity Coefficient as a Data Quality Metric for Variability in Natural Language Datacited by 19
  18. 2025Scaling Embedding Layers in Language Modelscited by 19
  19. 2025ATLAS: Adaptive Transfer Scaling Laws for Multilingual Pretraining, Finetuning, and Decoding the Curse of Multilingualitycited by 19
  20. 2024AutoElicit: Using Large Language Models for Expert Prior Elicitation in Predictive Modellingcited by 15
  21. 2024gzip Predicts Data-dependent Scaling Lawscited by 14
  22. 2024Neural Scaling Laws Rooted in the Data Distributioncited by 14
  23. 2025Value-Based Deep RL Scales Predictablycited by 14
  24. 2025Let Me Think! A Long Chain-of-Thought Can Be Worth Exponentially Many Short Onescited by 12
  25. 2025Pushing on Multilingual Reasoning Models with Language-Mixed Chain-of-Thoughtcited by 10
  26. 2024Human-aligned Chess with a Bit of Searchcited by 9
  27. 2022Evaluation Beyond Task Performance: Analyzing Concepts in AlphaZero in Hexcited by 8
  28. 2024Towards an Improved Understanding and Utilization of Maximum Manifold Capacity Representationscited by 8
  29. 2025Best-of-Majority: Minimax-Optimal Strategy for Pass@k Inference Scalingcited by 8
  30. 2026Adaptive Test-Time Compute Allocation for Reasoning LLMs via Constrained Policy Optimizationcited by 8
  31. 2024AlphaZero Neural Scaling and Zipf's Law: a Tale of Board Games and Power Lawscited by 7
  32. 2025Diverse Inference and Verification for Advanced Reasoningcited by 6
  33. 2025Confucius3-Math: A Lightweight High-Performance Reasoning LLM for Chinese K-12 Mathematics Learningcited by 5
  34. 2025A Theory of Inference Compute Scaling: Reasoning through Directed Stochastic Skill Searchcited by 5
  35. 2026IsoCompute Playbook: Optimally Scaling Sampling Compute for LLM RLcited by 5
  36. 2025How to Upscale Neural Networks with Scaling Law? A Survey and Practical Guidelinescited by 4
  37. 2024Towards a Science Exocortexcited by 3
  38. 2025Meek Models Shall Inherit the Earthcited by 2

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