A Theoretical Understanding of Chain-of-Thought: Coherent Reasoning and Error-Aware Demonstration (2410.16540v1)

Abstract: Few-shot Chain-of-Thought (CoT) prompting has demonstrated strong performance in improving the reasoning capabilities of LLMs. While theoretical investigations have been conducted to understand CoT, the underlying transformer used in these studies isolates the CoT reasoning process into separated in-context learning steps (Stepwise ICL). In this work, we theoretically show that, compared to Stepwise ICL, the transformer gains better error correction ability and more accurate predictions if the reasoning from earlier steps (Coherent CoT) is integrated. Given that this coherent reasoning changes the behavior of the transformer, we further investigate the sensitivity of the transformer with Coherent CoT when the demonstration examples are corrupted at the inference stage. Our theoretical results indicate that the transformer is more sensitive to errors in intermediate reasoning steps than the final outcome. Building upon this observation, we propose an improvement on CoT by incorporating both correct and incorrect reasoning paths in the demonstration. Our experiments validate the effectiveness of the proposed approach.

A Theoretical Understanding of Chain-of-Thought: Coherent Reasoning and Error-Aware Demonstration

This paper explores the theoretical underpinnings of Chain-of-Thought (CoT) prompting—an approach known for enhancing the reasoning capabilities of LLMs in few-shot learning scenarios. The authors challenge the conventional understanding by establishing a framework comparing Coherent CoT with Stepwise In-Context Learning (ICL), elucidating the performance benefits of incorporating comprehensive reasoning processes.

In Chain-of-Thought prompting, LLMs are presented with reasoning examples leading to more effective problem-solving tasks, specifically in mathematical and commonsense inference. However, prior analyses often treated CoT as isolated steps without interdependencies between multi-step processes. The authors address this limitation by investigating the holistic integration in Coherent CoT against Stepwise ICL.

Theoretical Insights

Model Setup and Assumptions:

The research is grounded in a structured data format using a Gaussian distribution, exploring both intermediate and final responses in a regression framework where $\beta$ is uniformly sampled. The model applies a single-head attention mechanism characterized by specific key, query, and value matrices, aligning with prior methodologies.

Comparison Between Coherent CoT and Stepwise ICL:

The authors demonstrate that Coherent CoT, which synthesizes information from all preceding reasoning steps, surpasses Stepwise ICL. The theoretical evaluation suggests that Coherent CoT allows LLMs to better adjust for errors in intermediate predictions, thereby enhancing accuracy—a significant finding supported by both theoretical predictions and simulation results indicating lower convergence error rates.

Sensitivity Analysis

The authors explore the sensitivity of Coherent CoT models to perturbations in reasoning steps during inference, identifying a pronounced sensitivity in intermediate steps compared to final responses. This sensitivity analysis distinguishes Coherent CoT as particularly robust in adjusting to errors within the reasoning chain, illuminating the model's error correction capabilities.

Proposed Methodology and Experimental Validation

Inspired by the sensitivity findings, the paper introduces a novel demonstration method combining correct and incorrect reasoning paths. This approach strengthens the model’s robustness by teaching it to recognize and rectify potential logical errors. Experiments on multiple datasets confirm the efficacy of this technique, indicating substantial performance improvements across LLMs like GPT-3.5-Turbo and Gemini Pro.

Implications and Future Directions

The research offers significant theoretical and practical implications for advancing LLMs' reasoning capabilities. By redefining how intermediate steps inform final predictions in CoT, the authors propose a robust blueprint for improving reasoning performance in AI. Future research may delve into expanding these findings across various datasets and model architectures, potentially exploring extensions to non-linear transformations or alternative data distributions.

In summary, this paper emphasizes the importance of treating Chain-of-Thought reasoning as an interconnected process. Such understanding not only clarifies existing models’ capabilities but also paves the way for developing more sophisticated error-aware AI systems.