Batch Normalization: Accelerating Deep Network Training by Reducing
Internal Covariate ShiftBatch Normalization: Accelerating Deep Network Training by Reducing
Internal Covariate ShiftSergey Ioffe and Christian Szegedy2015

Paper summarymartinthomaOne problem of training deep networks is that the features of lower-layer networks change while the upper-layer networks have already been adjusted to the previous lower-layer features. The phenomenon of changing inputs while optimizing is called *internal covariate shift*.
Batch normalization is done at training time for each mini batch.
## Ideas
* Training converges faster, if input is whitened (zero means, unit variances, decorrelated).
* Normalization parameters have to be computed within the gradient calculation step to prevent the model from blowing up
## What Batch Normalization is
For a layer with $d$-dimensional input $x = (x^{(1)}, \dots, x^{(d)})$, we will normalize
each dimension
$$\hat{x}^{(k)} = \frac{x^{(k)} - \mathbb{E}[x^{(k)}]}{\sqrt{Var[x^{(k)}]}}$$
where the expectation and the variance are computed over the training data set. This does *not* decorrelate the features, though.
Additionally, for each activation $x^{(k)}$ two paramters $\gamma^{(k)}, \beta^{(k)}$ are introduced which scale and shift the feature:
$$y^{(k)} = \gamma^{(k)} \cdot \hat{x}^{(k)} + \beta^{(k)}$$
Those two parameters (per feature) are learnable!
## Effect of Batch normalization
* Higher learning rates can be used
* Initialization is less important
* Acts as a regularizer, eliminating the need for dropout in some cases
* Faster training
## Datasets
* reaching 4.9% top-5 validation error (and 4.8% test error) on ImageNet classification
## Used by
* [Going Deeper with Convolutions](http://www.shortscience.org/paper?bibtexKey=journals/corr/SzegedyLJSRAEVR14)
* [Deep Residual Learning for Image Recognition](http://www.shortscience.org/paper?bibtexKey=journals/corr/HeZRS15#martinthoma)
## See also
* [other summaries](http://www.shortscience.org/paper?bibtexKey=conf/icml/IoffeS15)

First published: 2015/02/11 (2 years ago) Abstract: Training Deep Neural Networks is complicated by the fact that the
distribution of each layer's inputs changes during training, as the parameters
of the previous layers change. This slows down the training by requiring lower
learning rates and careful parameter initialization, and makes it notoriously
hard to train models with saturating nonlinearities. We refer to this
phenomenon as internal covariate shift, and address the problem by normalizing
layer inputs. Our method draws its strength from making normalization a part of
the model architecture and performing the normalization for each training
mini-batch. 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. Applied to a state-of-the-art image
classification model, Batch Normalization achieves the same accuracy with 14
times fewer training steps, and beats the original model by a significant
margin. Using an ensemble of batch-normalized networks, we improve upon the
best published result on ImageNet classification: reaching 4.9% top-5
validation error (and 4.8% test error), exceeding the accuracy of human raters.

One problem of training deep networks is that the features of lower-layer networks change while the upper-layer networks have already been adjusted to the previous lower-layer features. The phenomenon of changing inputs while optimizing is called *internal covariate shift*.
Batch normalization is done at training time for each mini batch.
## Ideas
* Training converges faster, if input is whitened (zero means, unit variances, decorrelated).
* Normalization parameters have to be computed within the gradient calculation step to prevent the model from blowing up
## What Batch Normalization is
For a layer with $d$-dimensional input $x = (x^{(1)}, \dots, x^{(d)})$, we will normalize
each dimension
$$\hat{x}^{(k)} = \frac{x^{(k)} - \mathbb{E}[x^{(k)}]}{\sqrt{Var[x^{(k)}]}}$$
where the expectation and the variance are computed over the training data set. This does *not* decorrelate the features, though.
Additionally, for each activation $x^{(k)}$ two paramters $\gamma^{(k)}, \beta^{(k)}$ are introduced which scale and shift the feature:
$$y^{(k)} = \gamma^{(k)} \cdot \hat{x}^{(k)} + \beta^{(k)}$$
Those two parameters (per feature) are learnable!
## Effect of Batch normalization
* Higher learning rates can be used
* Initialization is less important
* Acts as a regularizer, eliminating the need for dropout in some cases
* Faster training
## Datasets
* reaching 4.9% top-5 validation error (and 4.8% test error) on ImageNet classification
## Used by
* [Going Deeper with Convolutions](http://www.shortscience.org/paper?bibtexKey=journals/corr/SzegedyLJSRAEVR14)
* [Deep Residual Learning for Image Recognition](http://www.shortscience.org/paper?bibtexKey=journals/corr/HeZRS15#martinthoma)
## See also
* [other summaries](http://www.shortscience.org/paper?bibtexKey=conf/icml/IoffeS15)