  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.
