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.

翻译：我们研究了卷积神经网络（CNN）的逼近能力与学习能力。首先，我们证明了在特定权重约束条件下CNN的全新逼近误差界。其次，针对包含CNN作为特殊案例的前馈神经网络，我们提出了覆盖数的新分析方法。该方法通过精细考量权重规模，在特定情形下获得了优于现有文献的误差界。基于这两项成果，我们推导出基于CNN的估计量在多种学习问题中的收敛速率。具体而言，在非参数回归框架下，我们建立了基于CNN的最小二乘估计在光滑函数学习中的极小极大最优收敛速率；对于二分类问题，我们推导了采用铰链损失与逻辑损失的CNN分类器的收敛速率，并证明在若干场景下这些速率达到极小极大最优。