[2408.14511] Unveiling the Statistical Foundations of Chain-of-Thought Prompting Strategies

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Summary:Chain-of-Thought (CoT) prompting and its variants have gained recognition as efficient strategies for fixing multi-step reasoning issues utilizing pretrained massive language fashions (LLMs). On this work, we analyze CoT prompting from a statistical estimation perspective, offering a complete characterization of its pattern complexity. To this finish, we introduce a multi-step latent variable mannequin that encapsulates the reasoning course of, the place the latent variable encodes the duty data. Beneath this framework, we display that when the pretraining dataset is sufficiently massive, the estimator shaped by CoT prompting is equal to a Bayesian estimator. This estimator successfully solves the multi-step reasoning drawback by aggregating a posterior distribution inferred from the demonstration examples within the immediate. Furthermore, we show that the statistical error of the CoT estimator may be decomposed into two fundamental elements: (i) a prompting error, which arises from inferring the true process utilizing CoT prompts, and (ii) the statistical error of the pretrained LLM. We set up that, below applicable assumptions, the prompting error decays exponentially to zero because the variety of demonstrations will increase. Moreover, we explicitly characterize the approximation and generalization errors of the pretrained LLM. Notably, we assemble a transformer mannequin that approximates the goal distribution of the multi-step reasoning drawback with an error that decreases exponentially within the variety of transformer blocks. Our evaluation extends to different variants of CoT, together with Self-Constant CoT, Tree-of-Thought, and Choice-Inference, providing a broad perspective on the efficacy of those strategies. We additionally present numerical experiments to validate the theoretical findings.