# On the Cost and Benefit of Chain of Thought: A Learning-Theoretic Perspective

arXiv: https://arxiv.org/abs/2605.21260 · HTML: https://mlsanity.com/paper/2605.21260/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Placed in Prompting and chain-of-thought reasoning, one of 17 directions in LLM reasoning.
- Builds on 17 of its 119 references that are in the map.
- First posted on arXiv 20 May 2026.

## Placed in

- LLM reasoning / [Prompting and chain-of-thought reasoning](https://mlsanity.com/lr/prompting-and-chain-of-thought-reasoning/)

## Builds on (most cited first, 17 of 17)

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- [DeepSeek-R1: Incentivizing Reasoning Capability in LLMs via Reinforcement Learning](https://mlsanity.com/paper/2501.12948/) (2025)
- [OpenAI GPT-5 System Card](https://mlsanity.com/paper/2601.03267/) (2025)
- [Show Your Work: Scratchpads for Intermediate Computation with Language Models](https://mlsanity.com/paper/2112.00114/) (2021)
- [Gemini: A Family of Highly Capable Multimodal Models](https://mlsanity.com/paper/2312.11805/) (2023)
- [Reasoning Models Don't Always Say What They Think](https://mlsanity.com/paper/2505.05410/) (2025)
- [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)
- [Scaling Latent Reasoning via Looped Language Models](https://mlsanity.com/paper/2510.25741/) (2025)
- [Implicit Reasoning in Large Language Models: A Comprehensive Survey](https://mlsanity.com/paper/2509.02350/) (2025)
- [A Survey of Slow Thinking-based Reasoning LLMs using Reinforced Learning and Inference-time Scaling Law](https://mlsanity.com/paper/2505.02665/) (2025)
- [Why Can Large Language Models Generate Correct Chain-of-Thoughts?](https://mlsanity.com/paper/2310.13571/) (2023)
- [Not All Thoughts are Generated Equal: Efficient LLM Reasoning via Multi-Turn Reinforcement Learning](https://mlsanity.com/paper/2505.11827/) (2025)
- [Why Can't Transformers Learn Multiplication? Reverse-Engineering Reveals Long-Range Dependency Pitfalls](https://mlsanity.com/paper/2510.00184/) (2025)
- [Beyond the Black Box: A Survey on the Theory and Mechanism of Large Language Models](https://mlsanity.com/paper/2601.02907/) (2026)
- [Evolution of meta's llama models and parameter-efficient fine-tuning of large language models: a survey](https://mlsanity.com/paper/2510.12178/) (2025)
- [Breaking the Capability Ceiling of LLM Post-Training by Reintroducing Markov States](https://mlsanity.com/paper/2603.19987/) (2026)

## Abstract (arXiv)

We develop a learning-theoretic framework for understanding Chain of Thought (CoT). We model CoT as the interaction between an answer map and a chain rule that generates intermediate questions autoregressively, and define the reasoning risk of a hypothesis under this interaction. Our first result is a tight canonical decomposition of this risk into two terms with opposing roles: an oracle-trajectory risk (OTR), which captures the benefit of CoT and reduces to a target-domain risk in a domain adaptation problem, and a trajectory-mismatch risk (TMR), which captures the cost of CoT through error accumulation along mismatched reasoning trajectories. We then show that this cost is unavoidable without structure: if any one of the loss, the hypothesis answer map, or the chain rule lacks stability, the TMR can be arbitrarily large even when the OTR is zero and the hypothesis is uniformly close to the ground truth. Conversely, under stability, we prove a tight upper bound on the TMR governed by an exact amplification factor that identifies bounded, linear, and exponential error-growth regimes. Together, these results give a precise theory of when CoT helps, when it hurts, and what controls the transition between the two.