Density estimation with LLMs: a geometric investigation of in-context learning trajectories

TL;DR

This paper explores how large language models perform in-context density estimation, revealing their learning trajectories in a low-dimensional space and interpreting their behavior as an adaptive kernel density estimator.

Contribution

It introduces a geometric analysis of LLMs' in-context learning trajectories for density estimation, linking their behavior to an adaptive kernel density estimation model.

Findings

LLMs follow similar low-dimensional learning trajectories

LLaMA's in-context density estimation resembles an adaptive KDE

The proposed kernel model captures key behaviors with only two parameters

Abstract

Large language models (LLMs) demonstrate remarkable emergent abilities to perform in-context learning across various tasks, including time series forecasting. This work investigates LLMs' ability to estimate probability density functions (PDFs) from data observed in-context; such density estimation (DE) is a fundamental task underlying many probabilistic modeling problems. We leverage the Intensive Principal Component Analysis (InPCA) to visualize and analyze the in-context learning dynamics of LLaMA-2 models. Our main finding is that these LLMs all follow similar learning trajectories in a low-dimensional InPCA space, which are distinct from those of traditional density estimation methods like histograms and Gaussian kernel density estimation (KDE). We interpret the LLaMA in-context DE process as a KDE with an adaptive kernel width and shape. This custom kernel model captures a significant portion of LLaMA's behavior despite having only two parameters. We further speculate on why LLaMA's kernel width and shape differs from classical algorithms, providing insights into the mechanism of in-context probabilistic reasoning in LLMs. Our codebase, along with a 3D visualization of an LLM's in-context learning trajectory, is publicly available at https://github.com/AntonioLiu97/LLMICL_inPCA