The Normalization Method for Alleviating Pathological Sharpness in Wide Neural Networks
This work addresses a theoretical bottleneck in understanding normalization methods for deep learning practitioners, but it is incremental as it builds on existing knowledge of network geometry.
The paper tackled the problem of pathological sharpness in wide neural networks by analyzing the Fisher information matrix, finding that batch normalization in the last layer drastically reduces this sharpness under specific width and sample conditions, while other normalization methods do not.
Normalization methods play an important role in enhancing the performance of deep learning while their theoretical understandings have been limited. To theoretically elucidate the effectiveness of normalization, we quantify the geometry of the parameter space determined by the Fisher information matrix (FIM), which also corresponds to the local shape of the loss landscape under certain conditions. We analyze deep neural networks with random initialization, which is known to suffer from a pathologically sharp shape of the landscape when the network becomes sufficiently wide. We reveal that batch normalization in the last layer contributes to drastically decreasing such pathological sharpness if the width and sample number satisfy a specific condition. In contrast, it is hard for batch normalization in the middle hidden layers to alleviate pathological sharpness in many settings. We also found that layer normalization cannot alleviate pathological sharpness either. Thus, we can conclude that batch normalization in the last layer significantly contributes to decreasing the sharpness induced by the FIM.