揭示思维链提示方法的统计基础
Unveiling the Statistical Foundations of Chain-of-Thought Prompting Methods
AI总结：
本研究从统计估计视角分析思维链提示，引入多步隐变量模型证明其等价于贝叶斯估计量，并将统计误差分解为提示误差与预训练LLM误差，证明提示误差随示例增加呈指数衰减，同时扩展至多种CoT变体并经数值实验验证。
摘要：
思维链（Chain-of-Thought, CoT）提示及其变体作为使用预训练大型语言模型（LLMs）解决多步推理问题的有效方法而广受欢迎。在本研究中，我们从统计估计的视角分析CoT提示，全面刻画了其样本复杂度。为此，我们引入了一个多步隐变量模型来概括推理过程，其中隐变量编码了任务信息。在此框架下，我们证明了当预训练数据集足够大时，由CoT提示形成的估计量等价于一个贝叶斯估计量。该估计量通过聚合从提示中的示例推断出的后验分布，有效地解决了多步推理问题。此外，我们证明了CoT估计量的统计误差可分解为两个主要部分：(i) 提示误差，源于使用CoT提示推断真实任务时产生；(ii) 预训练LLM的统计误差。我们证明，在适当假设下，随着示例数量的增加，提示误差呈指数级衰减至零。此外，我们明确刻画了预训练LLM的逼近误差和泛化误差。值得注意的是，我们构建了一个transformer模型，其逼近多步推理问题目标分布的误差随transformer块数量的增加呈指数级下降。我们的分析扩展到了CoT的其他变体，包括Self-Consistent CoT、Tree-of-Thought和Selection-Inference，为这些方法的有效性提供了广泛视角。我们还提供了数值实验以验证理论发现。
英文摘要：
Chain-of-Thought (CoT) prompting and its variants have gained popularity as effective methods for solving multi-step reasoning problems using pretrained large language models (LLMs). In this work, we analyze CoT prompting from a statistical estimation perspective, providing a comprehensive characterization of its sample complexity. To this end, we introduce a multi-step latent variable model that encapsulates the reasoning process, where the latent variable encodes the task information. Under this framework, we demonstrate that when the pretraining dataset is sufficiently large, the estimator formed by CoT prompting is equivalent to a Bayesian estimator. This estimator effectively solves the multi-step reasoning problem by aggregating a posterior distribution inferred from the demonstration examples in the prompt. Moreover, we prove that the statistical error of the CoT estimator can be decomposed into two main components: (i) a prompting error, which arises from inferring the true task using CoT prompts, and (ii) the statistical error of the pretrained LLM. We establish that, under appropriate assumptions, the prompting error decays exponentially to zero as the number of demonstrations increases. Additionally, we explicitly characterize the approximation and generalization errors of the pretrained LLM. Notably, we construct a transformer model that approximates the target distribution of the multi-step reasoning problem with an error that decreases exponentially in the number of transformer blocks. Our analysis extends to other variants of CoT, including Self-Consistent CoT, Tree-of-Thought, and Selection-Inference, offering a broad perspective on the efficacy of these methods. We also provide numerical experiments to validate the theoretical findings.
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