We study approximation and learning capacities of convolutional neural networks (CNNs) with one-side zero-padding and multiple channels. Our first result proves a new approximation bound for CNNs with certain constraint on the weights. Our second result gives new analysis on the covering number of feed-forward neural networks with CNNs as special cases. The analysis carefully takes into account the size of the weights and hence gives better bounds than the 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 for classification are minimax optimal in some common settings.

翻译：我们研究了具有单侧零填充和多通道的卷积神经网络（CNN）的逼近与学习能力。首个结果证明了在特定权重约束下CNN的新逼近界。第二个结果给出了以前馈神经网络（含CNN特例）为对象的覆盖数新分析方法。该分析细致考虑了权重规模，因此在某些情况下比现有文献给出了更优的界。基于这两个结果，我们能够推导出基于CNN的估计量在许多学习问题中的收敛速率。特别地，我们在非参数回归场景中建立了基于CNN的最小二乘估计学习光滑函数的极小极大最优收敛速率。针对二分类问题，我们推导了采用铰链损失和逻辑损失的CNN分类器的收敛速率。同时证明，在若干常见设定下，所获得的分类速率具有极小极大最优性。