# Rethinking External Slow-Thinking: From Snowball Errors to Probability of Correct Reasoning

arXiv: https://arxiv.org/abs/2501.15602 · HTML: https://mlsanity.com/paper/2501.15602/

## Key facts

- Cited by 36 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 22 of its 42 references that are in the map.
- First posted on arXiv 26 Jan 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, 22 of 22)

- [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)
- [Scaling Laws for Neural Language Models](https://mlsanity.com/paper/2001.08361/) (2020)
- [DeepSeek-R1: Incentivizing Reasoning Capability in LLMs via Reinforcement Learning](https://mlsanity.com/paper/2501.12948/) (2025)
- [Self-Consistency Improves Chain of Thought Reasoning in Language Models](https://mlsanity.com/paper/2203.11171/) (2022)
- [Let's Verify Step by Step](https://mlsanity.com/paper/2305.20050/) (2023)
- [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)
- [Scaling Relationship on Learning Mathematical Reasoning with Large Language Models](https://mlsanity.com/paper/2308.01825/) (2023)
- [ReST-MCTS*: LLM Self-Training via Process Reward Guided Tree Search](https://mlsanity.com/paper/2406.03816/) (2024)
- [Language Models Are Greedy Reasoners: A Systematic Formal Analysis of Chain-of-Thought](https://mlsanity.com/paper/2210.01240/) (2022)
- [Skywork-Reward: Bag of Tricks for Reward Modeling in LLMs](https://mlsanity.com/paper/2410.18451/) (2024)
- [Alphazero-like Tree-Search can Guide Large Language Model Decoding and Training](https://mlsanity.com/paper/2309.17179/) (2023)
- [Inference Scaling Laws: An Empirical Analysis of Compute-Optimal Inference for Problem-Solving with Language Models](https://mlsanity.com/paper/2408.00724/) (2024)
- [Imitate, Explore, and Self-Improve: A Reproduction Report on Slow-thinking Reasoning Systems](https://mlsanity.com/paper/2412.09413/) (2024)
- [The pitfalls of next-token prediction](https://mlsanity.com/paper/2403.06963/) (2024)
- [HelpSteer2-Preference: Complementing Ratings with Preferences](https://mlsanity.com/paper/2410.01257/) (2024)
- [Enhancing LLM Reasoning with Reward-guided Tree Search](https://mlsanity.com/paper/2411.11694/) (2024)
- [MindStar: Enhancing Math Reasoning in Pre-trained LLMs at Inference Time](https://mlsanity.com/paper/2405.16265/) (2024)
- [BottleSum: Unsupervised and Self-supervised Sentence Summarization using the Information Bottleneck Principle](https://mlsanity.com/paper/1909.07405/) (2019)
- [Understanding Chain-of-Thought in LLMs through Information Theory](https://mlsanity.com/paper/2411.11984/) (2024)
- [Towards a Theoretical Understanding of Synthetic Data in LLM Post-Training: A Reverse-Bottleneck Perspective](https://mlsanity.com/paper/2410.01720/) (2024)

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

- [Towards Reasoning Era: A Survey of Long Chain-of-Thought for Reasoning Large Language Models](https://mlsanity.com/paper/2503.09567/) (2025)
- [Multimodal Chain-of-Thought Reasoning: A Comprehensive Survey](https://mlsanity.com/paper/2503.12605/) (2025)
- [Demystifying Reasoning Dynamics with Mutual Information: Thinking Tokens are Information Peaks in LLM Reasoning](https://mlsanity.com/paper/2506.02867/) (2025)
- [Information Gain-based Policy Optimization: A Simple and Effective Approach for Multi-Turn Search Agents](https://mlsanity.com/paper/2510.14967/) (2025)
- [Think or Not? Exploring Thinking Efficiency in Large Reasoning Models via an Information-Theoretic Lens](https://mlsanity.com/paper/2505.18237/) (2025)
- [Understanding Agent Scaling in LLM-Based Multi-Agent Systems via Diversity](https://mlsanity.com/paper/2602.03794/) (2026)
- [A Survey of Slow Thinking-based Reasoning LLMs using Reinforced Learning and Inference-time Scaling Law](https://mlsanity.com/paper/2505.02665/) (2025)
- [Explainable Chain-of-Thought Reasoning: An Empirical Analysis on State-Aware Reasoning Dynamics](https://mlsanity.com/paper/2509.00190/) (2025)
- [CTRLS: Chain-of-Thought Reasoning via Latent State-Transition](https://mlsanity.com/paper/2507.08182/) (2025)
- [InT: Self-Proposed Interventions Enable Credit Assignment in LLM Reasoning](https://mlsanity.com/paper/2601.14209/) (2026)
- [CoAT: Chain-of-Associated-Thoughts Framework for Enhancing Large Language Models Reasoning](https://mlsanity.com/paper/2502.02390/) (2025)
- [Compositional Generalization from Learned Skills via CoT Training: A Theoretical and Structural Analysis for Reasoning](https://mlsanity.com/paper/2502.04667/) (2025)
- [Mitigating Overthinking in Large Reasoning Language Models via Reasoning Path Deviation Monitoring](https://mlsanity.com/paper/2603.14251/) (2026)
- [From Chaos to Order: The Atomic Reasoner Framework for Fine-grained Reasoning in Large Language Models](https://mlsanity.com/paper/2503.15944/) (2025)
- [Decoupling Knowledge and Reasoning in LLMs: An Exploration Using Cognitive Dual-System Theory](https://mlsanity.com/paper/2507.18178/) (2025)
- [Streaming Hallucination Detection in Long Chain-of-Thought Reasoning](https://mlsanity.com/paper/2601.02170/) (2026)
- [Token Signature: Predicting Chain-of-Thought Gains with Token Decoding Feature in Large Language Models](https://mlsanity.com/paper/2506.06008/) (2025)
- [A Stitch in Time Saves Nine: Proactive Self-Refinement for Language Models](https://mlsanity.com/paper/2508.12903/) (2025)
- [Why Does Reasoning Length Converge? Unveiling the Underfitting-Overfitting Trade-off in Chain-of-Thought](https://mlsanity.com/paper/2509.04027/) (2025)
- [ThinkPilot: Steering Reasoning Models via Automated Think-prefixes Optimization](https://mlsanity.com/paper/2510.12063/) (2025)
- [Diffuse Thinking: Exploring Diffusion Language Models as Efficient Thought Proposers for Reasoning](https://mlsanity.com/paper/2510.27469/) (2025)
- [Reliable Chain-of-Thought via Prefix Consistency](https://mlsanity.com/paper/2605.07654/) (2026)
- [From Passive Metric to Active Signal: The Evolving Role of Uncertainty Quantification in Large Language Models](https://mlsanity.com/paper/2601.15690/) (2026)
- [When Do LLMs Reason? A Dynamical Systems View via Entropy Phase Transitions](https://mlsanity.com/paper/2605.22873/) (2026)
- [The Diminishing Returns of Early-Exit Decoding in Modern LLMs](https://mlsanity.com/paper/2603.23701/) (2026)
- [From Inertia to Objectivity: Improving Deep Research Agents with Noise Isolation](https://mlsanity.com/paper/2608.23045/) (2026)
- [It's the Problem, Not the Path: Budget and Difficulty Confounds in LLM Reasoning Trajectories](https://mlsanity.com/paper/2609.03436/) (2026)
- [Stepwise Reasoning Checkpoint Analysis: A Test Time Scaling Method to Enhance LLMs' Reasoning](https://mlsanity.com/paper/2505.17829/) (2025)
- [Latent Reward Steering: An Adaptive Inference-Time Framework that Implicitly Promotes Cognitive Behaviors in Reasoning LLMs](https://mlsanity.com/paper/2606.00726/) (2026)
- [Test-Time Scaling in Multimodal Foundation Models: A Comprehensive Survey of Generation and Reasoning](https://mlsanity.com/paper/2606.08231/) (2026)

## Abstract (arXiv)

Test-time scaling, which is also often referred to as slow-thinking, has been demonstrated to enhance multi-step reasoning in large language models (LLMs). However, despite its widespread utilization, the mechanisms underlying slow-thinking methods remain poorly understood. This paper explores the mechanisms of external slow-thinking from a theoretical standpoint. We begin by examining the snowball error effect within the LLM reasoning process and connect it to the likelihood of correct reasoning using information theory. Building on this, we show that external slow-thinking methods can be interpreted as strategies to mitigate the error probability. We further provide a comparative analysis of popular external slow-thinking approaches, ranging from simple to complex, highlighting their differences and interrelationships. Our findings suggest that the efficacy of these methods is not primarily determined by the specific framework employed, and that expanding the search scope or the model's internal reasoning capacity may yield more sustained improvements in the long term. We open-source our code at https://github.com/ZyGan1999/Snowball-Errors-and-Probability.