# A Simple Model of Inference Scaling Laws

arXiv: https://arxiv.org/abs/2410.16377 · HTML: https://mlsanity.com/paper/2410.16377/

## Key facts

- Cited by 36 papers in the map (data through 14 Oct 2026).
- Builds on 18 of its 23 references that are in the map.
- First posted on arXiv 21 Oct 2024.

## Builds on (most cited first, 18 of 18)

- [Language Models are Few-Shot Learners](https://mlsanity.com/paper/2005.14165/) (2020)
- [Scaling Laws for Neural Language Models](https://mlsanity.com/paper/2001.08361/) (2020)
- [Fashion-MNIST: a Novel Image Dataset for Benchmarking Machine Learning Algorithms](https://mlsanity.com/paper/1708.07747/) (2017)
- [Scaling LLM Test-Time Compute Optimally can be More Effective than Scaling Model Parameters](https://mlsanity.com/paper/2408.03314/) (2024)
- [Large Language Monkeys: Scaling Inference Compute with Repeated Sampling](https://mlsanity.com/paper/2407.21787/) (2024)
- [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)
- [A Constructive Prediction of the Generalization Error Across Scales](https://mlsanity.com/paper/1909.12673/) (2019)
- [Spectrum Dependent Learning Curves in Kernel Regression and Wide Neural Networks](https://mlsanity.com/paper/2002.02561/) (2020)
- [A Solvable Model of Neural Scaling Laws](https://mlsanity.com/paper/2210.16859/) (2022)
- [Sobolev Norm Learning Rates for Regularized Least-Squares Algorithm](https://mlsanity.com/paper/1702.07254/) (2017)
- [The Efficiency Misnomer](https://mlsanity.com/paper/2110.12894/) (2021)
- [Generalization Error Rates in Kernel Regression: The Crossover from the Noiseless to Noisy Regime](https://mlsanity.com/paper/2105.15004/) (2021)
- [Learning Curve Theory](https://mlsanity.com/paper/2102.04074/) (2021)
- [Hydragen: High-Throughput LLM Inference with Shared Prefixes](https://mlsanity.com/paper/2402.05099/) (2024)
- [Smaller, Weaker, Yet Better: Training LLM Reasoners via Compute-Optimal Sampling](https://mlsanity.com/paper/2408.16737/) (2024)
- [Scaling Laws for Associative Memories](https://mlsanity.com/paper/2310.02984/) (2023)
- [Critical Percolation as a Framework to Analyze the Training of Deep Networks](https://mlsanity.com/paper/1802.02154/) (2018)

## Built on by (most cited first, 30 of 36)

- [s1: Simple test-time scaling](https://mlsanity.com/paper/2501.19393/) (2025)
- [Missing Premise exacerbates Overthinking: Are Reasoning Models losing Critical Thinking Skill?](https://mlsanity.com/paper/2504.06514/) (2025)
- [Think Twice: Enhancing LLM Reasoning by Scaling Multi-round Test-time Thinking](https://mlsanity.com/paper/2503.19855/) (2025)
- [Crosslingual Reasoning through Test-Time Scaling](https://mlsanity.com/paper/2505.05408/) (2025)
- [Rethinking Fine-Tuning when Scaling Test-Time Compute: Limiting Confidence Improves Mathematical Reasoning](https://mlsanity.com/paper/2502.07154/) (2025)
- [To Backtrack or Not to Backtrack: When Sequential Search Limits Model Reasoning](https://mlsanity.com/paper/2504.07052/) (2025)
- [Revisiting Multi-Agent Debate as Test-Time Scaling: A Systematic Study of Conditional Effectiveness](https://mlsanity.com/paper/2505.22960/) (2025)
- [Efficient Prediction of Pass@k Scaling in Large Language Models](https://mlsanity.com/paper/2510.05197/) (2025)
- [Let Me Think! A Long Chain-of-Thought Can Be Worth Exponentially Many Short Ones](https://mlsanity.com/paper/2505.21825/) (2025)
- [AgentHER: Hindsight Experience Replay for LLM Agent Trajectory Relabeling](https://mlsanity.com/paper/2603.21357/) (2026)
- [PiCSAR: Probabilistic Confidence Selection And Ranking for Reasoning Chains](https://mlsanity.com/paper/2508.21787/) (2025)
- [First Finish Search: Efficient Test-Time Scaling in Large Language Models](https://mlsanity.com/paper/2505.18149/) (2025)
- [CyberV: Cybernetics for Test-time Scaling in Video Understanding](https://mlsanity.com/paper/2506.07971/) (2025)
- [UniT: Unified Multimodal Chain-of-Thought Test-time Scaling](https://mlsanity.com/paper/2602.12279/) (2026)
- [Predictable Scale: Part II, Farseer: A Refined Scaling Law in Large Language Models](https://mlsanity.com/paper/2506.10972/) (2025)
- [Learning Shrinks the Hard Tail: Training-Dependent Inference Scaling in a Solvable Linear Model](https://mlsanity.com/paper/2601.03764/) (2026)
- [When More Sampling Hurts: The Modal Ceiling and Correlation Ceiling of Test-Time Scaling](https://mlsanity.com/paper/2606.28661/) (2026)
- [A Theory of Inference Compute Scaling: Reasoning through Directed Stochastic Skill Search](https://mlsanity.com/paper/2507.00004/) (2025)
- [Pretraining Scaling Laws for Generative Evaluations of Language Models](https://mlsanity.com/paper/2509.24012/) (2025)
- [Demystifying LLM-as-a-Judge: Analytically Tractable Model for Inference-Time Scaling](https://mlsanity.com/paper/2512.19905/) (2025)
- [MOOSE-Star: Unlocking Tractable Training for Scientific Discovery by Breaking the Complexity Barrier](https://mlsanity.com/paper/2603.03756/) (2026)
- [Jailbreak Scaling Laws for Large Language Models: Polynomial-Exponential Crossover](https://mlsanity.com/paper/2603.11331/) (2026)
- [X-Factor: Quality Is a Dataset-Intrinsic Property](https://mlsanity.com/paper/2505.22813/) (2025)
- [Evaluating the Robustness of Chinchilla Compute-Optimal Scaling](https://mlsanity.com/paper/2509.23963/) (2025)
- [Guided by the Plan: Enhancing Faithful Autoregressive Text-to-Audio Generation with Guided Decoding](https://mlsanity.com/paper/2601.14304/) (2026)
- [More Bang for the Buck: Improving the Inference of Large Language Models at a Fixed Budget using Reset and Discard (ReD)](https://mlsanity.com/paper/2601.21522/) (2026)
- [Predicting and improving test-time scaling laws via reward tail-guided search](https://mlsanity.com/paper/2602.01485/) (2026)
- [Item Response Scaling Laws: A Measurement Theory Approach for Efficient and Generalizable Neural Scaling Estimation](https://mlsanity.com/paper/2606.07616/) (2026)
- [Follow the Path: Reasoning over Knowledge Graph Paths to Improve Large Language Model Factuality](https://mlsanity.com/paper/2505.11140/) (2025)
- [Hard or Just Unreached? Diagnosing the Sampling Blind Spot in Math-Reasoning Difficulty Estimation](https://mlsanity.com/paper/2606.19636/) (2026)

## Abstract (arXiv)

Neural scaling laws have garnered significant interest due to their ability to predict model performance as a function of increasing parameters, data, and compute. In this work, we propose a simple statistical ansatz based on memorization to study scaling laws in the context of inference, specifically how performance improves with multiple inference attempts. We explore the coverage, or pass@k metric, which measures the chance of success over repeated attempts and provide a motivation for the observed functional form of the inference scaling behavior of the coverage in large language models (LLMs) on reasoning tasks. We then define an "inference loss", which exhibits a power law decay as the number of trials increases, and connect this result with prompting costs. We further test our construction by conducting experiments on a simple generative model, and find that our predictions are in agreement with the empirical coverage curves in a controlled setting. Our simple framework sets the ground for incorporating inference scaling with other known scaling laws.