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.

通过梯度下降训练的过参数化的修正线性单元卷积神经网络在一定的样本复杂度和信噪比条件下能够实现接近贝叶斯最优准确率，并且能够高效地学习存在高度相关特征的异或类型分类任务。