Batch Normalization Provably Avoids Rank Collapse for Randomly Initialised Deep Networks
This addresses a fundamental training instability in deep learning, providing theoretical and empirical insights for practitioners, though it builds on existing knowledge about batch normalization.
The paper tackles the problem of rank collapse in randomly initialized deep networks, showing that batch normalization effectively avoids this issue, with derived lower rank bounds for linear networks and empirical confirmation for ReLU networks.
Randomly initialized neural networks are known to become harder to train with increasing depth, unless architectural enhancements like residual connections and batch normalization are used. We here investigate this phenomenon by revisiting the connection between random initialization in deep networks and spectral instabilities in products of random matrices. Given the rich literature on random matrices, it is not surprising to find that the rank of the intermediate representations in unnormalized networks collapses quickly with depth. In this work we highlight the fact that batch normalization is an effective strategy to avoid rank collapse for both linear and ReLU networks. Leveraging tools from Markov chain theory, we derive a meaningful lower rank bound in deep linear networks. Empirically, we also demonstrate that this rank robustness generalizes to ReLU nets. Finally, we conduct an extensive set of experiments on real-world data sets, which confirm that rank stability is indeed a crucial condition for training modern-day deep neural architectures.