Optimal Approximation and Learning Rates for Deep Convolutional Neural
  Networks
Shao-Bo Lin
This paper focuses on approximation and learning performance analysis for
deep convolutional neural networks with zero-padding and max-pooling. We prove
that, to approximate $r$-smooth function, the approximation rates of deep
convolutional neural networks with depth $L$ are of order $ (L^2/\log
L)^{-2r/d} $, which is optimal up to a logarithmic factor. Furthermore, we
deduce almost optimal learning rates for implementing empirical risk
minimization over deep convolutional neural networks.