Semi-Implicit Back Propagation
Authors: Ren Liu, Xiaoqun Zhang
Keywords: Optimization, Neural Network, Proximal mapping, Back propagation, Implicit
TL;DR: A new scheme in the spirit of error back propagation, with an implicit updates on the parameters sets and semi-implicit updates on the hidden neurons.
Abstract: Neural network has attracted great attention for a long time and many researchers are devoted  to improve the effectiveness of  neural network training algorithms. Though stochastic gradient descent (SGD) and other explicit gradient-based methods are widely adopted, there are still many challenges such as gradient vanishing and small step sizes, which leads to slow convergence and instability of SGD algorithms. Motivated by error back propagation (BP)  and proximal methods, we propose a semi-implicit back propagation method for neural network training. Similar to BP, the difference on the neurons are propagated in a backward fashion and the parameters are updated with proximal mapping. The implicit update for both hidden neurons and parameters allows to choose large step size in the training algorithm. Finally, we also show that any fixed point of  convergent sequences produced by this algorithm is a stationary point of the objective loss function. The experiments on both MNIST and CIFAR-10 demonstrate that the proposed semi-implicit BP algorithm leads to  better performance in terms of both loss decreasing and training/validation accuracy, compared to SGD and a similar algorithm ProxBP.