We study the approximation and learning capacities of convolutional neural
networks (CNNs). Our first result proves a new approximation bound for CNNs
with certain constraint on the weights. Our second result gives a new analysis
on the covering number of feed-forward neural networks, which include CNNs as
special cases. The analysis carefully takes into account the size of the
weights and hence gives better bounds than existing literature in some
situations. Using these two results, we are able to derive rates of convergence
for estimators based on CNNs in many learning problems. In particular, we
establish minimax optimal convergence rates of the least squares based on CNNs
for learning smooth functions in the nonparametric regression setting. For
binary classification, we derive convergence rates for CNN classifiers with
hinge loss and logistic loss. It is also shown that the obtained rates are
minimax optimal in several settings.

卷积神经网络的近似和学习能力的研究，证明了满足权重约束的卷积神经网络的新逼近界限，并给出了覆盖神经网络的新分析，从而得到了更好的收敛界限，并利用这两个结果在许多学习问题中推导了基于卷积神经网络的估计器的收敛速度，对于学习平滑函数的非参数回归设置和二元分类中的卷积神经网络分类器的 Hinge 损失和 Logistic 损失，得到了最优的收敛速度。