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Title: Towards Stabilizing Batch Statistics in Backward Propagation of Batch Normalization

Paper content:
Batch Normalization (BN) is one of the most widely used techniques in Deep Learning field.
But its performance can awfully degrade with insufficient batch size.
This weakness limits the usage of BN on many computer vision tasks like detection or segmentation, where batch size is usually small due to the constraint of memory consumption.
Therefore many modified normalization techniques have been proposed, which either fail to restore the performance of BN completely, or have to introduce additional nonlinear operations in inference procedure and increase huge consumption.
In this paper, we reveal that there are two extra batch statistics involved in backward propagation of BN, on which has never been well discussed before.
The extra batch statistics associated with gradients also can severely affect the training of deep neural network.
Based on our analysis, we propose a novel normalization method, named Moving Average Batch Normalization (MABN).
MABN can completely restore the performance of vanilla BN in small batch cases, without introducing any additional nonlinear operations in inference procedure.
We prove the benefits of MABN by both theoretical analysis and experiments.
Our experiments demonstrate the effectiveness of MABN in multiple computer vision tasks including ImageNet and COCO.
The code has been released in https://github.com/megvii-model/MABN.
Batch Normalization (BN) (Ioffe & Szegedy, 2015) is one of the most popular techniques for training neural networks.
It has been widely proven effective in many applications, and become the indispensable part of many state of the art deep models.
Despite the success of BN, it's still challenging to utilize BN when batch size is extremely small 1 .
The batch statistics with small batch size are highly unstable, leading to slow convergence during training and bad performance during inference.
For example, in detection or segmentation tasks, the batch size is often limited to 1 or 2 per GPU due to the requirement of high resolution inputs or complex structure of the model.
Directly computing batch statistics without any modification on each GPU will make performance of the model severely degrade.
To address such issues, many modified normalization methods have been proposed.
They can be roughly divided into two categories: some of them try to improve vanilla BN by correcting batch statistics (Ioffe, 2017; Singh & Shrivastava, 2019) , but they all fail to completely restore the performance of vanilla BN; Other methods get over the instability of BN by using instance-level normalization (Ulyanov et al., 2016; Ba et al., 2016; Wu & He, 2018) , therefore models can avoid the affect of batch statistics.
This type of methods can restore the performance in small batch cases to some extent.
However, instance-level normalization hardly meet industrial or commercial needs so far, for this type of methods have to compute instance-level statistics both in training and inference, which will introduce additional nonlinear operations in inference procedure and dramatically increase consumption Shao et al. (2019) .
While vanilla BN uses the statistics computed over the whole training data instead of batch of samples when training finished.
Thus BN is a linear operator and can be merged with convolution layer during inference procedure.
Figure 1 (a) shows with ResNet-50 (He et al., 2016) , instance-level normalization almost double the inference time compared with vanilla BN.
Therefore, it's a tough but necessary task to restore the performance of BN in small batch training without introducing any nonlinear operations in inference procedure.
In this paper, we first analysis the formulation of vanilla BN, revealing there are actually not only 2 but 4 batch statistics involved in normalization during forward propagation (FP) as well as backward propagation (BP).
The additional 2 batch statistics involved in BP are associated with gradients of the model, and have never been well discussed before.
They play an important role in regularizing gradients of the model during BP.
In our experiments (see Figure 2) , variance of the batch statistics associated with gradients in BP, due to small batch size, is even larger than that of the widelyknown batch statistics (mean, variance of feature maps).
We believe the instability of batch statistics associated with gradients is one of the key reason why BN performs poorly in small batch cases.
Based on our analysis, we propose a novel normalization method named Moving Average Batch Normalization (MABN).
MABN can completely get over small batch issues without introducing any nonlinear manipulation in inference procedure.
The core idea of MABN is to replace batch statistics with moving average statistics.
We substitute batch statistics involved in BP and FP with different type of moving average statistics respectively, and theoretical analysis is given to prove the benefits.
However, we observed directly using moving average statistics as substitutes for batch statistics can't make training converge in practice.
We think the failure takes place due to the occasional large gradients during training, which has been mentioned in Ioffe (2017) .
To avoid training collapse, we modified the vanilla normalization form by reducing the number of batch statistics, centralizing the weights of convolution kernels, and utilizing renormalizing strategy.
We also theoretically prove the modified normalization form is more stable than vanilla form.
MABN shows its effectiveness in multiple vision public datasets and tasks, including ImageNet (Russakovsky et al., 2015) , COCO (Lin et al., 2014) .
All results of experiments show MABN with small batch size (1 or 2) can achieve comparable performance as BN with regular batch size (see Figure 1(b
) ). Besides
, it has same inference consumption as vanilla BN (see Figure 1(a) ).
We also
conducted sufficient ablation experiments to verify the effectiveness of MABN further.

assistant: text: We propose a novel normalization method to handle small batch size cases.