Dimension Independent Generalization Error with Regularized Online Optimization

One classical canon of statistics is that large models are prone to overfitting and model selection procedures are necessary for high-dimensional data. However, many overparameterized models such as neural networks, which are often trained with simple online methods and regularization, perform very well in practice. The empirical success of overparameterized models, which is often known as benign overfitting, motivates us to have a new look at the statistical generalization theory for online optimization. In particular, we present a general theory on the generalization error of stochastic gradient descent (SGD) for both convex and non-convex loss functions. We further provide the definition of "low effective dimension" so that the generalization error either does not depend on the ambient dimension $p$ or depends on $p$ via a poly-logarithmic factor. We also demonstrate on several widely used statistical models that the "low effect dimension" arises naturally in overparameterized settings. The studied statistical applications include both convex models such as linear regression and logistic regression, and non-convex models such as $M$-estimator and two-layer neural networks.