Unveiling the Statistical Foundations of Chain-of-Thought Prompting Methods

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.

Graph Convolution Based Cross-Network Multi-Scale Feature Fusion for Deep Vessel Segmentation

Vessel segmentation is widely used to help with vascular disease diagnosis. Vessels reconstructed using existing methods are often not sufficiently accurate to meet clinical use standards. This is because 3D vessel structures are highly complicated and exhibit unique characteristics, including sparsity and anisotropy. In this paper, we propose a novel hybrid deep neural network for vessel segmentation. Our network consists of two cascaded subnetworks performing initial and refined segmentation respectively. The second subnetwork further has two tightly coupled components, a traditional CNN-based U-Net and a graph U-Net. Cross-network multi-scale feature fusion is performed between these two U-shaped networks to effectively support high-quality vessel segmentation. The entire cascaded network can be trained from end to end. The graph in the second subnetwork is constructed according to a vessel probability map as well as appearance and semantic similarities in the original CT volume. To tackle the challenges caused by the sparsity and anisotropy of vessels, a higher percentage of graph nodes are distributed in areas that potentially contain vessels while a higher percentage of edges follow the orientation of potential nearbyvessels. Extensive experiments demonstrate our deep network achieves state-of-the-art 3D vessel segmentation performance on multiple public and in-house datasets.