Optimizing Neural Networks in the Equivalent Class Space

It has been widely observed that many activation functions and pooling methods of neural network models have (positive-) rescaling-invariant property, including ReLU, PReLU, max-pooling, and average pooling, which makes fully-connected neural networks (FNNs) and convolutional neural networks (CNNs) invariant to (positive) rescaling operation across layers. This may cause unneglectable problems with their optimization: (1) different NN models could be equivalent, but their gradients can be very different from each other; (2) it can be proven that the loss functions may have many spurious critical points in the redundant weight space. To tackle these problems, in this paper, we first characterize the rescaling-invariant properties of NN models using equivalent classes and prove that the dimension of the equivalent class space is significantly smaller than the dimension of the original weight space. Then we represent the loss function in the compact equivalent class space and develop novel algorithms that conduct optimization of the NN models directly in the equivalent class space. We call these algorithms Equivalent Class Optimization (abbreviated as EC-Opt) algorithms. Moreover, we design efficient tricks to compute the gradients in the equivalent class, which almost have no extra computational complexity as compared to standard back-propagation (BP). We conducted experimental study to demonstrate the effectiveness of our proposed new optimization algorithms. In particular, we show that by using the idea of EC-Opt, we can significantly improve the accuracy of the learned model (for both FNN and CNN), as compared to using conventional stochastic gradient descent algorithms.