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The Potential of CoT for Reasoning: A Closer Look at Trace Dynamics

Gregor Bachmann, Yichen Jiang, Seyed Mohsen Moosavi Dezfooli, Moin Nabi

TL;DR

The paper investigates chain-of-thought (CoT) reasoning by introducing the potential, a metric that quantifies how conditioning on partial CoT tokens affects the probability of a correct answer $y^{*}$, i.e., $\operatorname{pot}(\bm{c}_{<t};\bm{x})$. Using sampling with $N=128$ trajectories, it analyzes CoT traces from competition-level mathematics (notably AIME-2024) to reveal that potential curves are often non-monotonic and feature sharp reasoning insights, tangents, and occasional late guessing; it also defines an estimator and proves a monotonicity property in expectation for correct runs. The study then demonstrates that CoT can be optimized to produce more monotonic progress and that a weaker model can be substantially aided by partial CoT from a stronger model, with as little as $20\%$ partial CoT delivering meaningful improvements across model families. Together, these findings advance understanding of CoT dynamics, offer a principled way to assess and improve reasoning traces, and suggest practical implications for transfer learning and reinforcement learning reward shaping in large language models.$

Abstract

Chain-of-thought (CoT) prompting is a de-facto standard technique to elicit reasoning-like responses from large language models (LLMs), allowing them to spell out individual steps before giving a final answer. While the resemblance to human-like reasoning is undeniable, the driving forces underpinning the success of CoT reasoning still remain largely unclear. In this work, we perform an in-depth analysis of CoT traces originating from competition-level mathematics questions, with the aim of better understanding how, and which parts of CoT actually contribute to the final answer. To this end, we introduce the notion of a potential, quantifying how much a given part of CoT increases the likelihood of a correct completion. Upon examination of reasoning traces through the lens of the potential, we identify surprising patterns including (1) its often strong non-monotonicity (due to reasoning tangents), (2) very sharp but sometimes tough to interpret spikes (reasoning insights and jumps) as well as (3) at times lucky guesses, where the model arrives at the correct answer without providing any relevant justifications before. While some of the behaviours of the potential are readily interpretable and align with human intuition (such as insights and tangents), others remain difficult to understand from a human perspective. To further quantify the reliance of LLMs on reasoning insights, we investigate the notion of CoT transferability, where we measure the potential of a weaker model under the partial CoT from another, stronger model. Indeed aligning with our previous results, we find that as little as 20% of partial CoT can ``unlock'' the performance of the weaker model on problems that were previously unsolvable for it, highlighting that a large part of the mechanics underpinning CoT are transferable.

The Potential of CoT for Reasoning: A Closer Look at Trace Dynamics

TL;DR

The paper investigates chain-of-thought (CoT) reasoning by introducing the potential, a metric that quantifies how conditioning on partial CoT tokens affects the probability of a correct answer , i.e., . Using sampling with trajectories, it analyzes CoT traces from competition-level mathematics (notably AIME-2024) to reveal that potential curves are often non-monotonic and feature sharp reasoning insights, tangents, and occasional late guessing; it also defines an estimator and proves a monotonicity property in expectation for correct runs. The study then demonstrates that CoT can be optimized to produce more monotonic progress and that a weaker model can be substantially aided by partial CoT from a stronger model, with as little as partial CoT delivering meaningful improvements across model families. Together, these findings advance understanding of CoT dynamics, offer a principled way to assess and improve reasoning traces, and suggest practical implications for transfer learning and reinforcement learning reward shaping in large language models.$

Abstract

Chain-of-thought (CoT) prompting is a de-facto standard technique to elicit reasoning-like responses from large language models (LLMs), allowing them to spell out individual steps before giving a final answer. While the resemblance to human-like reasoning is undeniable, the driving forces underpinning the success of CoT reasoning still remain largely unclear. In this work, we perform an in-depth analysis of CoT traces originating from competition-level mathematics questions, with the aim of better understanding how, and which parts of CoT actually contribute to the final answer. To this end, we introduce the notion of a potential, quantifying how much a given part of CoT increases the likelihood of a correct completion. Upon examination of reasoning traces through the lens of the potential, we identify surprising patterns including (1) its often strong non-monotonicity (due to reasoning tangents), (2) very sharp but sometimes tough to interpret spikes (reasoning insights and jumps) as well as (3) at times lucky guesses, where the model arrives at the correct answer without providing any relevant justifications before. While some of the behaviours of the potential are readily interpretable and align with human intuition (such as insights and tangents), others remain difficult to understand from a human perspective. To further quantify the reliance of LLMs on reasoning insights, we investigate the notion of CoT transferability, where we measure the potential of a weaker model under the partial CoT from another, stronger model. Indeed aligning with our previous results, we find that as little as 20% of partial CoT can ``unlock'' the performance of the weaker model on problems that were previously unsolvable for it, highlighting that a large part of the mechanics underpinning CoT are transferable.
Paper Structure (20 sections, 1 theorem, 16 equations, 16 figures, 4 tables, 1 algorithm)

This paper contains 20 sections, 1 theorem, 16 equations, 16 figures, 4 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Related Work
  3. Potential of CoT
  4. Setup.
  5. Potential.
  6. Shape of Potential Curves
  7. Quantifying the shape.
  8. Amount of guessing.
  9. Optimizing the potential.
  10. A Closer Look at Chain-of-Thought Reasoning
  11. Transferability of CoT Through the Lens of Potential
  12. Conclusion
  13. Appendix
  14. Experimental Details
  15. Proof of Proposition \ref{['prop:potential']}
  16. ...and 5 more sections

Key Result

Proposition 1

Conditional on the event that the full CoT $\bm{c}_{1:T}$ yields the correct final answer $y^{*}$, it holds for every $t\leq T$ that

Figures (16)

  • Figure 1: Left: Illustration of the calculation of the potential. Right: An example prompt and partial CoT, which in this case should intuitively raise the probability of success (i.e. the potential) significantly once discovered or provided by another model.
  • Figure 2: Potential curves. Potential of correct and wrong CoTs for Qwen2.5-7B on AIME-2024, Question 5 and 11. Strongly non-monotonic behaviour for both correct and incorrect CoTs.
  • Figure 3: The potential of optimal and standard CoT for AIME-2025-I, question 1 and 5. While standard CoT eventually arrives at the right answer, optimal CoT does so in a more robust way.
  • Figure 4: Inflated$\operatorname{pass@}\space k$. We show $\operatorname{pass@}\space k$ accuracies and the corresponding corrected values for Qwen2.5-1.5B (left) and Qwen2.5-7B (right).
  • Figure 5: Reasoning tangents and insights. Qwen2.5-7B's potential $\operatorname{pot}_{256}(\mathbin{\vcenter{\hbox{$\m@th\bullet$}}};\bm{x})$ behaving strongly non-monotonic. The reasoning tangent ② hurts the potential, while the reasoning insights ③ (observing the symmetry $x=y$ of the problem) and ④ (finding the root of the cubic equation) push the potential back on track. Finally, the model performs a reasoning jump ⑤ (for some non-obvious reason, this particular calculation is difficult for the model).
  • ...and 11 more figures

Theorems & Definitions (2)

  • Proposition 1
  • proof