P-BN: Towards Effective Batch Normalization in the Path Space

Neural networks with ReLU activation functions have demonstrated their success in many applications. Recently, researchers noticed a potential issue with the optimization of ReLU networks: the ReLU activation functions are positively scale-invariant (PSI), while the weights are not. This mismatch may lead to undesirable behaviors in the optimization process. Hence, some new algorithms that conduct optimizations directly in the path space (the path space is proven to be PSI) were developed, such as Stochastic Gradient Descent (SGD) in the path space, and it was shown that SGD in the path space is superior to that in the weight space. However, it is still unknown whether other deep learning techniques beyond SGD, such as batch normalization (BN), could also have their counterparts in the path space. In this paper, we conduct a formal study on the design of BN in the path space. According to our study, the key challenge is how to ensure the forward propagation in the path space, because BN is utilized during the forward process. To tackle such challenge, we propose a novel re-parameterization of ReLU networks, with which we replace each weight in the original neural network, with a new value calculated from one or several paths, while keeping the outputs of the network unchanged for any input. Then we show that BN in the path space, namely P-BN, is just a slightly modified conventional BN on the re-parameterized ReLU networks. Our experiments on two benchmark datasets, CIFAR and ImageNet, show that the proposed P-BN can significantly outperform the conventional BN in the weight space.