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

Karthik Duraisamy

TL;DR

The paper investigates finite-sample generalization bounds for a single gradient-descent step in in-context linear regression under a random-design setting, deriving non-asymptotic bounds on the generalization error $E_{gen}$ that avoid arbitrary constants. It contrasts these bounds with classical least squares results, decomposing error into systematic and noise components and identifying an optimal step size $\u03b7^*$, with scaling relations in terms of $n$ and $d$. The analysis yields explicit, robust expressions and highlights the practical implications for in-context learning and gradient-based updates in linear models. A byproduct of the work is a set of identities involving high-order products of Gaussian random matrices, enriching the toolbox for random-matrix calculations in statistical learning.

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.

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

TL;DR

The paper investigates finite-sample generalization bounds for a single gradient-descent step in in-context linear regression under a random-design setting, deriving non-asymptotic bounds on the generalization error that avoid arbitrary constants. It contrasts these bounds with classical least squares results, decomposing error into systematic and noise components and identifying an optimal step size , with scaling relations in terms of and . The analysis yields explicit, robust expressions and highlights the practical implications for in-context learning and gradient-based updates in linear models. A byproduct of the work is a set of identities involving high-order products of Gaussian random matrices, enriching the toolbox for random-matrix calculations in statistical learning.

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.
Paper Structure (30 sections, 2 theorems, 7 equations, 2 figures, 2 tables, 1 algorithm)

This paper contains 30 sections, 2 theorems, 7 equations, 2 figures, 2 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Class options
  3. Front matter
  4. Cross references and hyperlinks
  5. Cleveref
  6. Hyperef
  7. Math and equations
  8. Theorem-like environments
  9. Tables
  10. Figures
  11. Algorithms
  12. Sections
  13. Supplemental material
  14. Template
  15. Bibliography
  16. ...and 15 more sections

Key Result

Theorem 6.1

\newlabelthm:mvt0 Suppose $f$ is a function that is continuous on the closed interval $[a,b]$. and differentiable on the open interval $(a,b)$. Then there exists a number $c$ such that $a < c < b$ and In other words, $f(b)-f(a) = f'(c)(b-a)$.

Figures (2)

  • Figure 1: Example figure using external image files.
  • Figure 2: Example PGFPLOTS figure.

Theorems & Definitions (5)

  • Theorem 6.1: Mean Value Theorem
  • Corollary 6.2
  • Proof 1
  • Claim 6.3
  • Proof 2: Proof of main theorem