The Directional Bias Helps Stochastic Gradient Descent to Generalize in Kernel Regression Models

TL;DR

This paper investigates how stochastic gradient descent (SGD) in kernel regression models exhibits a directional bias towards the dominant eigenvector, which may explain its superior generalization compared to gradient descent (GD).

Contribution

It generalizes the known directional bias property of SGD from linear regression to kernel regression, providing theoretical insights into its convergence behavior.

Findings

SGD converges along the eigenvector of the largest eigenvalue of the Gram matrix.

GD converges along the eigenvector of the smallest eigenvalue.

Simulation and case studies demonstrate the impact on generalization error.

Abstract

We study the Stochastic Gradient Descent (SGD) algorithm in nonparametric statistics: kernel regression in particular. The directional bias property of SGD, which is known in the linear regression setting, is generalized to the kernel regression. More specifically, we prove that SGD with moderate and annealing step-size converges along the direction of the eigenvector that corresponds to the largest eigenvalue of the Gram matrix. In addition, the Gradient Descent (GD) with a moderate or small step-size converges along the direction that corresponds to the smallest eigenvalue. These facts are referred to as the directional bias properties; they may interpret how an SGD-computed estimator has a potentially smaller generalization error than a GD-computed estimator. The application of our theory is demonstrated by simulation studies and a case study that is based on the FashionMNIST dataset.