Towards Revealing the Mystery behind Chain of Thought: A Theoretical Perspective (2305.15408v5)

Abstract: Recent studies have discovered that Chain-of-Thought prompting (CoT) can dramatically improve the performance of LLMs, particularly when dealing with complex tasks involving mathematics or reasoning. Despite the enormous empirical success, the underlying mechanisms behind CoT and how it unlocks the potential of LLMs remain elusive. In this paper, we take a first step towards theoretically answering these questions. Specifically, we examine the expressivity of LLMs with CoT in solving fundamental mathematical and decision-making problems. By using circuit complexity theory, we first give impossibility results showing that bounded-depth Transformers are unable to directly produce correct answers for basic arithmetic/equation tasks unless the model size grows super-polynomially with respect to the input length. In contrast, we then prove by construction that autoregressive Transformers of constant size suffice to solve both tasks by generating CoT derivations using a commonly used math language format. Moreover, we show LLMs with CoT can handle a general class of decision-making problems known as Dynamic Programming, thus justifying its power in tackling complex real-world tasks. Finally, an extensive set of experiments show that, while Transformers always fail to directly predict the answers, they can consistently learn to generate correct solutions step-by-step given sufficient CoT demonstrations.

Theoretical Analysis of Chain-of-Thought in Decoder-Based Transformers

The paper "Revealing the Secret Behind CoT: A Theoretical Perspective" provides a detailed examination of the underlying capabilities of decoder-based transformer models when equipped with a Chain-of-Thought (CoT) mechanism. With an analytical lens, the authors dissect the computational prowess of transformer networks, focusing on their proficiency, particularly in reasoning tasks, and the pivotal role played by the CoT approach.

Decoder-based transformer models, notable for their deployment in systems like GPT, have shown remarkable success across various NLP applications. However, their inherent limitations when tackling complex reasoning tasks, without the CoT framework, necessitate further scrutiny. The paper takes a deep dive into how these transformers can effectively move beyond their constraints by leveraging CoT, which facilitates a stepwise reasoning process that decomposes complex tasks into manageable units. This not only results in solving intricate problems but also, as demonstrated theoretically, reveals their Turing-complete nature with sufficient output length.

Key Theoretical Insights

The authors undertake a rigorous theoretical exploration, utilizing classical circuit complexity theories to delineate the computational boundaries of transformer models. They identify that, without CoT, transformers are inherently restricted by their computational depth, which aligns with $TC^0$ circuit complexity. This reveals that they cannot solve problems necessitating non-constant computational depth efficiently.

Key findings include:

• Hierarchical Reasoning: The paper articulates how CoT enhances hierarchical reasoning capabilities within transformers, enabling task decomposition which is crucial for processing complex problems.
• Transformers' Limitations: Detailed analysis demonstrates that tasks such as arithmetic formula computation, Hidden Markov Models (HMM), and Circuit Value Problems exceed the capacity of transformer models when not utilizing CoT.
• Chain-of-Thought Impact: By employing CoT, decoder-based transformers can effectively simulate dynamic programming and finite state automata equipped with stacks, manifesting their ability to handle inherently sequential problems successfully.

Empirical and Theoretical Validation

The paper presents both empirical evidence and formalized theoretical proofs to substantiate the claims regarding the capacities of CoT-augmented transformer models. This dual approach strengthens the argument that CoT is instrumental in transcending the typical constraints of transformer architectures. The notion that transformers can simulate Turing machines when applying CoT is particularly compelling, offering a theoretical validation of their ability to potentially execute any algorithmic computation.

Implications and Future Directions

The implications of these findings are substantial for the field of machine learning, specifically in advancing the design of models capable of sophisticated reasoning tasks. Practically, this suggests pathways to harness the full potential of transformers in complex problem-solving scenarios by embedding chain-of-thought methodologies. The established Turing-completeness of CoT-enhanced models opens possibilities for their application across diverse computationally demanding fields.

Future research could explore optimization strategies for implementing CoT in practice, potentially improving efficiency and accuracy. Additionally, the exploration of how these theoretical insights translate into real-world applications, particularly in domains requiring intricate reasoning processes, warrants continued investigation.

In conclusion, this paper contributes a critical theoretical framework for understanding and leveraging the capabilities of decoder-based transformers, underscoring the transformative role of Chain-of-Thought approaches in expanding computational horizons within artificial intelligence.