  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, perform
very well in practice, although they are often trained with simple online
methods and regularization. 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) solutions for both convex and locally convex loss functions. We
further discuss data and model conditions that lead to a ``low effective
dimension". Under these conditions, we show 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 that in several widely used
statistical models, the ``low effective 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.
