Test your knowledge 

Batch Normalization (BatchNorm) is a very frequently used technique in Deep Learning due to its power to not only enhance model performance but also reduce training time. However, the reason why it works remains a mystery to most of us. Furthermore, many tutorials and explanations on the Internet interpret it ambiguously, leaving readers with a false perception of the problem.
In this article, we attempt to give an introduction to BatchNorm, including what it is and why it works in the objective and uptodate point of view.
Batch Normalization
BatchNorm was first proposed by Sergey and Christian in 2015. In their paper, the authors stated:
Batch Normalization allows us to use much higher learning rates and be less careful about initialization. It also acts as a regularizer, in some cases eliminating the need for Dropout.
which was then verified by many other researchers, building the popularity of BatchNorm.
Batch?
Deep Neural Networks are trained with Gradient Descent optimization, most commonly, the Stochastic Gradient Descent variant.
With Gradient Descent, the process operates in steps, with each step involves feeding data to the network. There are 3 standard strategies of inputting data:
To date, the MiniBatch SGD is preferred in the Deep Learning community as it has some advantages over the other two:
Normalization?
Normalization is the process of bringing values to a standard form.
In the scope of Machine Learning, normalization often means returning the values to a distribution of 0mean and unit variance.
Batch Normalization?
Batch Normalization is the act of applying normalizations to each batch of the MiniBatch SGD. These normalizations are NOT just applied before giving the data to the network but may be applied at many layers of the network.
For a layer with ddimensional input , we apply normalization to each of the dimension separately. In other words, is transformed independently from other dimensions. For readability, from now on, we will omit the in the nomination of .
Given that the input is a vector of size m (), with being the batch size. The BatchNorm is described as below:
where:
 and are 2 additional parameters for each dimension of the input. They are also learned along with the other parameters in the Back Propagation process. and , as outlined by the authors, attempt to restore the representation power of the networks, since simply transforming input data to a 0mean unitvariance distribution may change what the layer represents (e.g. in Sigmoid nonlinearity, this transformation may limit the output to the linear regime).
 is a small constant added for the numerical stability of the computation (i.e. cases when equals 0).
The authors suggested that BatchNorm may be used either immediatebefore or immediateafter the activation functions. In their experiments, BatchNorm is added rightbefore the nonlinearities. However, some other researchers found using BatchNorm after nonlinearities can produce better outputs. The same test even indicates that removing the scaleandshift phase (using and ) may increase the model’s performance.
The benefits of using BatchNorm
As described in the paper, BatchNorm helps in both speed and accuracy. To be more specific, it:
Even though the effectiveness of BatchNorm is inarguable, some of the reasoning from the authors is not agreed by many other researchers. We will elaborate on this in the subsequent sections.
Good practices with BatchNorm
Test your understanding 

Why does BatchNorm work?
The authors’ reasoning
Back Propagation (BackProp) is a powerful technique to train Neural Networks, however, it has the disadvantage of assuming all the previous layers are fixed when updating each specific layer. Put it differently, at each backward step, BackProp updates all layers simultaneously, moreover, the updating value of layer ith assumes all previous layers (all layer jth, with j < i), are fixed, which is not true as all the network is changing. The continuous change in the distribution of each layer’s input is called the Internal covariate shift (ICS).
While this assumption simplifies the learning problem and makes the BackProp works, it is always violated, making the optimization process suboptimal.
A solution to this problem is to decrease the learning rate, which lessens the changes in weights, which, in turn, makes the assumption more valid. Nonetheless, small learning rates diminish the speed of training, thus do not contribute any good to the process.
Another means is to be careful with parameter initialization, which is the direction taken by many previous studies.
Batch Normalization is introduced to partly solve this problem. It “standardize” the input of each layer in term of mean and variance in order to keep the ICS low.
In short, it is mainly:
BatchNorm Low ICS efficient learning
A sideeffect that also turned out to be good is that the noise generated through the process helps with regularization, enhancing the generalization ability of the networks.
The explanation of reducing ICS, however, is not concrete as there are some questions still remain unanswered:
Researches are conducted to answer the above questions. For example, this paper from Johan et al. argues the success of BatchNorm can be explained without ICS.
Ian Goodfellow’s lecture
Ian Goodfellow is the creator of GAN (Generative Adversarial Network), author of Deep Learning Book, and is also one of the pioneers of Deep Learning.
In his lecture on Batch Normalization, Ian stated his view on the reason why it works: because and help control the higherorder statistics.
As the Gradient Descent algorithm uses only firstorder statistics to compute the optimization updates, we are unable to track the higherorder terms, which becomes increasingly important as the networks getting deeper. A change in the weights of one layer does not only influence the immediate successive layer but also the further layers on the net. (This matter is already wellknown in the community of Deep Learning. )
Alternatively, the effects of higherorder derivation can be thought of from the secondaryschoolPhysics point of view. In Physics, the velocity is the derivative of position, while the acceleration is the derivative of velocity (or says, acceleration is the secondorder derivative of position). By knowing velocity, we can somehow estimate the value of the position after an amount of time. Nonetheless, if the amount of time is large, the acceleration may change the velocity in the meantime, making our estimation further inaccurate. Hence, we are forced to set a small time frame. In this example, the position is analogous for the weights, velocity for the first derivative, the time frame for the learning rate, and acceleration for the higherorder derivatives. As we only know the first derivative (velocity), we have to set a small learning rate (time frame) in order to have an accurate estimation without being affected too much by the higherorder derivatives (acceleration).
According to Ian, by using BatchNorm, the statistics of a layer are controlled by only two parameters and , which is much easier for the subsequent layers to adapt to, instead of all the complex sets of weights. Furthermore, by letting these 2 parameters be learned, the original distribution of the input can be recovered if necessary.
Ian’s point of view, although not related to ICS, still mostly agrees with the authors of BatchNorm.
However, this interpretation remains a hypothesis without solid evidence. The abovementioned experiment from DuchaAiki is, in one way or another, opposite to Ian’s thought when showing that BatchNorm without and gives a stronger performance over the conventional one in his case.
Shibani et al.’s view
Shibani et al. published a paper with the name being literally “How Does Batch Normalization Help Optimization?“.
In their research, they suggested 4 crucial points:
Conclusion
In this article, we have a discussion of Batch Normalization, what it is, how it works and the reasons that make up its reputation.
Sergey Ioffe and Christian Szegedy, the authors of BatchNorm, proposed a wonderful technique to boost both the performance and speed of Deep Neural Networks. However, as concurred by the majority, their reasoning about the work is obscure, leaving a lot of confusion and disagreements.
Recently, there are a number of researches attempting to demystify the mystery behind BatchNorm. One of them is from Shibani et al., who conducted an indepth experiment, proving that ICS is uncorrelated with BatchNorm’s success. Furthermore, they figured out the root cause that makes BatchNorm work, which is mainly from its ability to smoothen the optimization landscape by making the gradients of the loss more Lipschitz. To date, this seems to be the most plausible explanation provided being backed up by experiments and mathematical proofs.
References:
 Batch Normalization original paper: link
 A test of BatchNorm by DuchaAiki on Github: link
 Xiang Li et al.’s paper about Dropout and BatchNorm: link
 Ian Goodfellow’s lecture on BatchNorm: link
 Shibani et al.’s paper about how BatchNorm helps: link, discussion: link
 Johan et al.’s paper about Understanding BatchNorm: link
 Mlexplained’s post about BatchNorm: link
 Paperspace’s post about BatchNorm: link
 Rutgers university’s lecture on Lipschitz continuity: link and link
 Elad et al.’s paper about Norm in deep networks: link