  Modern deep learning models are usually highly over-parameterized so that
they can overfit the training data. Surprisingly, such overfitting neural
networks can usually still achieve high prediction accuracy. To study this
"benign overfitting" phenomenon, a line of recent works has theoretically
studied the learning of linear models and two-layer neural networks. However,
most of these analyses are still limited to the very simple learning problems
where the Bayes-optimal classifier is linear. In this work, we investigate a
class of XOR-type classification tasks with label-flipping noises. We show
that, under a certain condition on the sample complexity and signal-to-noise
ratio, an over-parameterized ReLU CNN trained by gradient descent can achieve
near Bayes-optimal accuracy. Moreover, we also establish a matching lower bound
result showing that when the previous condition is not satisfied, the
prediction accuracy of the obtained CNN is an absolute constant away from the
Bayes-optimal rate. Our result demonstrates that CNNs have a remarkable
capacity to efficiently learn XOR problems, even in the presence of highly
correlated features.
