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.
Normalization and geometric characterization. Batch normalization is believed to perform well because it suppresses the internal covariate shift @cite_26 . Recent extensive studies, however, have reported alternative explanations on how batch normalization works @cite_5 @cite_17 . empirically found that batch normalization decreases a sharp change of the loss function and makes the loss landscape smoother. reported that batch normalization works to prevent an explosion of the loss and gradients. While some theoretical studies analyzed FIMs in un-normalized DNNs @cite_10 @cite_30 @cite_21 , analysis in normalized DNNs has been limited. analyzed gradients and Hessian under batch normalization in a single layer and theoretically evaluated their worst case bounds, but its inequality was too general to quantify the decrease of sharpness. In particular, it misses the special effect of the last layer, as we found in this study. The original paper @cite_0 of layer normalization analyzed the FIM in generalized linear models (GLMs) and argued that the normalization could decrease curvature of the parameter space. While a GLM corresponds to the single layer model, shallow and deep networks have hidden layers. As the hidden layers become wide, pathological sharpness appears and layer normalization suffers from it.
@cite_30 @cite_26 @cite_21 @cite_0 @cite_5 @cite_10 @cite_17
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As an indispensable component, Batch Normalization (BN) has successfully improved the training of deep neural networks (DNNs) with mini-batches, by normalizing the distribution of the internal representation for each hidden layer. However, the effectiveness of BN would diminish with the scenario of micro-batch (e.g. less than 4 samples in a mini-batch), since the estimated statistics in a mini-batch are not reliable with insufficient samples. This limits BN's room in training larger models on segmentation, detection, and video-related problems, which require small batches constrained by memory consumption. In this paper, we present a novel normalization method, called Kalman Normalization (KN), for improving and accelerating the training of DNNs, particularly under the context of micro-batches. Specifically, unlike the existing solutions treating each hidden layer as an isolated system, KN treats all the layers in a network as a whole system, and estimates the statistics of a certain layer by considering the distributions of all its preceding layers, mimicking the merits of Kalman Filtering. On ResNet50 trained in ImageNet, KN has 3.4 lower error than its BN counterpart when using a batch size of 4; Even when using typical batch sizes, KN still maintains an advantage over BN while other BN variants suffer a performance degradation. Moreover, KN can be naturally generalized to many existing normalization variants to obtain gains, e.g. equipping Group Normalization with Group Kalman Normalization (GKN). KN can outperform BN and its variants for large scale object detection and segmentation task in COCO 2017. While the authors of Batch Normalization (BN) identify and address an important problem involved in training deep networks- Internal Covariate Shift- the current solution has certain drawbacks. For instance, BN depends on batch statistics for layerwise input normalization during training which makes the estimates of mean and standard deviation of input (distribution) to hidden layers inaccurate due to shifting parameter values (especially during initial training epochs). Another fundamental problem with BN is that it cannot be used with batch-size 1 during training. We address these drawbacks of BN by proposing a non-adaptive normalization technique for removing covariate shift, that we call Normalization Propagation. Our approach does not depend on batch statistics, but rather uses a data-independent parametric estimate of mean and standard-deviation in every layer thus being computationally faster compared with BN. We exploit the observation that the pre-activation before Rectified Linear Units follow Gaussian distribution in deep networks, and that once the first and second order statistics of any given dataset are normalized, we can forward propagate this normalization without the need for recalculating the approximate statistics for hidden layers. While the authors of Batch Normalization (BN) identify and address an important problem involved in training deep networks-- Internal Covariate Shift-- the current solution has certain drawbacks. Specifically, BN depends on batch statistics for layerwise input normalization during training which makes the estimates of mean and standard deviation of input (distribution) to hidden layers inaccurate for validation due to shifting parameter values (especially during initial training epochs). Also, BN cannot be used with batch-size 1 during training. We address these drawbacks by proposing a non-adaptive normalization technique for removing internal covariate shift, that we call Normalization Propagation. Our approach does not depend on batch statistics, but rather uses a data-independent parametric estimate of mean and standard-deviation in every layer thus being computationally faster compared with BN. We exploit the observation that the pre-activation before Rectified Linear Units follow Gaussian distribution in deep networks, and that once the first and second order statistics of any given dataset are normalized, we can forward propagate this normalization without the need for recalculating the approximate statistics for hidden layers. Batch normalization is a ubiquitous deep learning technique that normalizes activations in intermediate layers. It is associated with improved accuracy and faster learning, but despite its enormous success there is little consensus regarding why it works. We aim to rectify this and take an empirical approach to understanding batch normalization. Our primary observation is that the higher learning rates that batch normalization enables have a regularizing effect that dramatically improves generalization of normalized networks, which is both demonstrated empirically and motivated theoretically. We show how both activations and gradient information becomes less input dependent across spatial dimensions and examples within a mini-batch for deep unnormalized networks, and show how this limit possible learning rates. Motivated by recent results in random matrix theory, we argue that this ill-conditioning is due to fluctuations in random initialization, shedding new light on classical initialization schemes and their consequences.