# OptScale: Probabilistic Optimality for Inference-time Scaling

arXiv: https://arxiv.org/abs/2506.22376 · HTML: https://mlsanity.com/paper/2506.22376/

## Key facts

- Cited by 3 papers in the map (data through 14 Oct 2026).
- Placed in Test-time compute and efficient thinking, one of 17 directions in LLM reasoning.
- Builds on 23 of its 32 references that are in the map.
- First posted on arXiv 27 Jun 2025.

## Placed in

- LLM reasoning / [Test-time compute and efficient thinking](https://mlsanity.com/lr/test-time-compute-and-efficient-thinking/)

## Builds on (most cited first, 23 of 23)

- [The Llama 3 Herd of Models](https://mlsanity.com/paper/2407.21783/) (2024)
- [Training Verifiers to Solve Math Word Problems](https://mlsanity.com/paper/2110.14168/) (2021)
- [Qwen3 Technical Report](https://mlsanity.com/paper/2505.09388/) (2025)
- [DeepSeek-R1: Incentivizing Reasoning Capability in LLMs via Reinforcement Learning](https://mlsanity.com/paper/2501.12948/) (2025)
- [Training a Helpful and Harmless Assistant with Reinforcement Learning from Human Feedback](https://mlsanity.com/paper/2204.05862/) (2022)
- [Scaling LLM Test-Time Compute Optimally can be More Effective than Scaling Model Parameters](https://mlsanity.com/paper/2408.03314/) (2024)
- [The Lessons of Developing Process Reward Models in Mathematical Reasoning](https://mlsanity.com/paper/2501.07301/) (2025)
- [Towards Large Reasoning Models: A Survey of Reinforced Reasoning with Large Language Models](https://mlsanity.com/paper/2501.09686/) (2025)
- [Large Language Models for Mathematical Reasoning: Progresses and Challenges](https://mlsanity.com/paper/2402.00157/) (2024)
- [Rewarding Progress: Scaling Automated Process Verifiers for LLM Reasoning](https://mlsanity.com/paper/2410.08146/) (2024)
- [A Survey on Test-Time Scaling in Large Language Models: What, How, Where, and How Well?](https://mlsanity.com/paper/2503.24235/) (2025)
- [Large Language Model Guided Tree-of-Thought](https://mlsanity.com/paper/2305.08291/) (2023)
- [Optimizing Test-Time Compute via Meta Reinforcement Fine-Tuning](https://mlsanity.com/paper/2503.07572/) (2025)
- [Towards Thinking-Optimal Scaling of Test-Time Compute for LLM Reasoning](https://mlsanity.com/paper/2502.18080/) (2025)
- [Let's Sample Step by Step: Adaptive-Consistency for Efficient Reasoning and Coding with LLMs](https://mlsanity.com/paper/2305.11860/) (2023)
- [A Survey of Frontiers in LLM Reasoning: Inference Scaling, Learning to Reason, and Agentic Systems](https://mlsanity.com/paper/2504.09037/) (2025)
- [Escape Sky-high Cost: Early-stopping Self-Consistency for Multi-step Reasoning](https://mlsanity.com/paper/2401.10480/) (2024)
- [SEAL: Steerable Reasoning Calibration of Large Language Models for Free](https://mlsanity.com/paper/2504.07986/) (2025)
- [Make Every Penny Count: Difficulty-Adaptive Self-Consistency for Cost-Efficient Reasoning](https://mlsanity.com/paper/2408.13457/) (2024)
- [Reasoning Aware Self-Consistency: Leveraging Reasoning Paths for Efficient LLM Sampling](https://mlsanity.com/paper/2408.17017/) (2024)
- [Think Twice: Enhancing LLM Reasoning by Scaling Multi-round Test-time Thinking](https://mlsanity.com/paper/2503.19855/) (2025)
- [Dynamic Parallel Tree Search for Efficient LLM Reasoning](https://mlsanity.com/paper/2502.16235/) (2025)
- [Improving LLM Reasoning through Scaling Inference Computation with Collaborative Verification](https://mlsanity.com/paper/2410.05318/) (2024)

## Built on by (most cited first, 3 of 3)

- [More Correct Mass, Worse Answers: Why Power Sampling Can Fail and How to Fix It](https://mlsanity.com/paper/2608.14420/) (2026)
- [MarkovScale: Towards Optimal Sequential Scaling at Inference Time](https://mlsanity.com/paper/2602.01120/) (2026)
- [HyPER: Bridging Exploration and Exploitation for Scalable LLM Reasoning with Hypothesis Path Expansion and Reduction](https://mlsanity.com/paper/2602.06527/) (2026)

## Abstract (arXiv)

Inference-time scaling has emerged as a powerful technique for enhancing the reasoning performance of Large Language Models (LLMs). However, existing approaches often rely on heuristic strategies for parallel sampling, lacking a principled foundation. To address this gap, we propose a probabilistic framework that formalizes the optimality of inference-time scaling under the assumption that parallel samples are independently and identically distributed (i.i.d.), and where the Best-of-N selection strategy follows a probability distribution that can be estimated. Within this framework, we derive a theoretical lower bound on the required number of samples to achieve a target performance level, providing the first principled guidance for compute-efficient scaling. Leveraging this insight, we develop textscOptScale, a practical algorithm that dynamically determines the optimal number of sampled responses. textscOptScale employs a language model-based predictor to estimate probabilistic prior parameters, enabling the decision of the minimal number of samples needed that satisfy predefined performance thresholds and confidence levels. Extensive experiments on representative reasoning benchmarks (including MATH-500, GSM8K, AIME, and AMC) demonstrate that textscOptScale significantly reduces sampling overhead while remaining better or on par with state-of-the-art reasoning performance. Our work offers both a theoretical foundation and a practical solution for principled inference-time scaling, addressing a critical gap in the efficient deployment of LLMs for complex reasoning.