Finite Sample Analysis and Bounds of Generalization Error of Gradient Descent in In-Context Linear Regression

TL;DR

This paper provides finite sample bounds and analytical insights into the generalization error of a single gradient descent step in linear regression, with implications for understanding in-context learning in transformers.

Contribution

It derives non-asymptotic bounds for gradient descent in linear regression, avoiding arbitrary constants and comparing with classical least squares, enhancing understanding of generalization.

Findings

Explicit finite sample bounds for gradient descent generalization error.

Comparison between gradient descent and least squares regression.

Insights into optimal step sizes and noise components.

Abstract

Recent studies show that transformer-based architectures emulate gradient descent during a forward pass, contributing to in-context learning capabilities - an ability where the model adapts to new tasks based on a sequence of prompt examples without being explicitly trained or fine tuned to do so. This work investigates the generalization properties of a single step of gradient descent in the context of linear regression with well-specified models. A random design setting is considered and analytical expressions are derived for the statistical properties and bounds of generalization error in a non-asymptotic (finite sample) setting. These expressions are notable for avoiding arbitrary constants, and thus offer robust quantitative information and scaling relationships. These results are contrasted with those from classical least squares regression (for which analogous finite sample bounds are also derived), shedding light on systematic and noise components, as well as optimal step sizes. Additionally, identities involving high-order products of Gaussian random matrices are presented as a byproduct of the analysis.