  This paper proposes a fractional order gradient method for the backward
propagation of convolutional neural networks. To overcome the problem that
fractional order gradient method cannot converge to real extreme point, a
simplified fractional order gradient method is designed based on Caputo's
definition. The parameters within layers are updated by the designed gradient
method, but the propagations between layers still use integer order gradients,
and thus the complicated derivatives of composite functions are avoided and the
chain rule will be kept. By connecting every layers in series and adding loss
functions, the proposed convolutional neural networks can be trained smoothly
according to various tasks. Some practical experiments are carried out in order
to demonstrate fast convergence, high accuracy and ability to escape local
optimal point at last.
