The Directional Bias Helps Stochastic Gradient Descent to Generalize in Kernel Regression Models
This work provides theoretical insights into SGD's generalization properties for researchers in machine learning and statistics, though it is incremental as it extends known directional bias from linear to kernel regression.
The authors tackled the problem of understanding why Stochastic Gradient Descent (SGD) generalizes better than Gradient Descent (GD) in kernel regression models, showing that SGD converges along the direction of the largest eigenvalue of the Gram matrix, while GD converges along the smallest eigenvalue, which may explain SGD's smaller generalization error, as demonstrated in simulations and on the FashionMNIST dataset.
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.