  In this paper, we consider the problem of binary classification with a class
of general deep convolutional neural networks, which includes fully-connected
neural networks and fully convolutional neural networks as special cases. We
establish non-asymptotic excess risk bounds for a class of convex surrogate
losses and target functions with different modulus of continuity. An important
feature of our results is that we clearly define the prefactors of the risk
bounds in terms of the input data dimension and other model parameters and show
that they depend polynomially on the dimensionality in some important models.
We also show that the classification methods with CNNs can circumvent the curse
of dimensionality if the input data is supported on an approximate
low-dimensional manifold. To establish these results, we derive an upper bound
for the covering number for the class of general convolutional neural networks
with a bias term in each convolutional layer, and derive new results on the
approximation power of CNNs for any uniformly-continuous target functions.
These results provide further insights into the complexity and the
approximation power of general convolutional neural networks, which are of
independent interest and may have other applications. Finally, we apply our
general results to analyze the non-asymptotic excess risk bounds for four
widely used methods with different loss functions using CNNs, including the
least squares, the logistic, the exponential and the SVM hinge losses.
