Uncertainty Quantification in Large Language Models Through Convex Hull Analysis

TL;DR

This paper introduces a geometric method using convex hull analysis to quantify uncertainty in large language models by analyzing response embeddings and their dispersion across different prompt complexities and settings.

Contribution

It presents a novel approach leveraging convex hulls and clustering of embeddings to measure uncertainty in LLM outputs, addressing limitations of traditional probabilistic methods.

Findings

Uncertainty varies with prompt complexity, model, and temperature.

Convex hull analysis effectively captures output variability.

Embedding dispersion correlates with response confidence.

Abstract

Uncertainty quantification approaches have been more critical in large language models (LLMs), particularly high-risk applications requiring reliable outputs. However, traditional methods for uncertainty quantification, such as probabilistic models and ensemble techniques, face challenges when applied to the complex and high-dimensional nature of LLM-generated outputs. This study proposes a novel geometric approach to uncertainty quantification using convex hull analysis. The proposed method leverages the spatial properties of response embeddings to measure the dispersion and variability of model outputs. The prompts are categorized into three types, i.e., `easy', `moderate', and `confusing', to generate multiple responses using different LLMs at varying temperature settings. The responses are transformed into high-dimensional embeddings via a BERT model and subsequently projected into a two-dimensional space using Principal Component Analysis (PCA). The Density-Based Spatial Clustering of Applications with Noise (DBSCAN) algorithm is utilized to cluster the embeddings and compute the convex hull for each selected cluster. The experimental results indicate that the uncertainty of the model for LLMs depends on the prompt complexity, the model, and the temperature setting.