Implementing Synchronized Multi-GPU Batch Normalization
=======================================================

In this tutorial, we discuss the implementation detail of Multi-GPU Batch Normalization (BN) (classic implementation: :class:`encoding.nn.BatchNorm2d`. We will provide the training example in a later version.

How BN works?
-------------

BN layer was introduced in the paper `Batch Normalization: Accelerating Deep Network Training by Reducing Internal Covariate Shift <https://arxiv.org/abs/1502.03167>`_, which dramatically speed up the training process of the network (enables larger learning rate) and makes the network less sensitive to the weight initialization. 

- Forward Pass: 
    For the input data :math:`X={x_1, ...x_N}`, the data are normalized to be zero-mean and unit-variance, then scale and shit:

    .. math::
        y_i = \gamma\cdot\frac{x_i-\mu}{\sigma} + \beta ,

    where :math:`\mu=\frac{\sum_i^N x_i}{N} , \sigma = \sqrt{\frac{\sum_i^N (x_i-\mu)^2}{N}+\epsilon}` and :math:`\gamma, \beta` are the learnable parameters.
        
- Backward Pass:
    For calculating the gradient :math:`\frac{d_\ell}{d_{x_i}}`, we need to consider the partial gradient from :math:`\frac{d_\ell}{d_y}` and the gradients from :math:`\frac{d_\ell}{d_\mu}` and :math:`\frac{d_\ell}{d_\sigma}`, since the :math:`\mu \text{ and } \sigma` are the function of the input :math:`x_i`. We use patial direvative in the notations:

    .. math::

        \frac{d_\ell}{d_{x_i}} = \frac{d_\ell}{d_{y_i}}\cdot\frac{\partial_{y_i}}{\partial_{x_i}} + \frac{d_\ell}{d_\mu}\cdot\frac{d_\mu}{d_{x_i}} + \frac{d_\ell}{d_\sigma}\cdot\frac{d_\sigma}{d_{x_i}}

    where :math:`\frac{\partial_{y_i}}{\partial_{x_i}}=\frac{\gamma}{\sigma}, \frac{d_\ell}{d_\mu}=-\frac{\gamma}{\sigma}\sum_i^N\frac{d_\ell}{d_{y_i}}
    \text{ and } \frac{d_\sigma}{d_{x_i}}=-\frac{1}{\sigma}(\frac{x_i-\mu}{N})`.

Why Synchronize BN?
-------------------

- Standard Implementations of BN in public frameworks (suck as Caffe, MXNet, Torch, TF, PyTorch) are unsynchronized, which means that the data are normalized within each GPU. Therefore the `working batch-size` of the BN layer is `BatchSize/nGPU` (batch-size in each GPU). 

- Since the `working batch-size` is typically large enough for standard vision tasks, such as classification and detection, there is no need to synchronize BN layer during the training. The synchronization will slow down the training.

- However, for the Semantic Segmentation task, the state-of-the-art approaches typically adopt dilated convoluton, which is very memory consuming. The `working bath-size` can be too small for BN layers (2 or 4 in each GPU) when using larger/deeper pre-trained networks, such as :class:`encoding.dilated.ResNet` or :class:`encoding.dilated.DenseNet`. 

How to Synchronize?
-------------------

Suppose we have :math:`K` number of GPUs, :math:`sum(x)_k` and :math:`sum(x^2)_k` denotes the sum of elements and sum of element squares in :math:`k^{th}` GPU.

- Forward Pass:
    We can calculate the sum of elements :math:`sum(x)=\sum x_i \text{ and sum of squares } sum(x^2)=\sum x_i^2` in each GPU, then apply :class:`encoding.parallel.allreduce` operation to sum accross GPUs. Then calculate the global mean :math:`\mu=\frac{sum(x)}{N} \text{ and global variance } \sigma=\sqrt{\frac{sum(x^2)}{N}-\mu^2+\epsilon}`. 

- Backward Pass:
    * :math:`\frac{d_\ell}{d_{x_i}}=\frac{d_\ell}{d_{y_i}}\frac{\gamma}{\sigma}` can be calculated locally in each GPU.
    * Calculate the gradient of :math:`sum(x)` and :math:`sum(x^2)` individually in each GPU :math:`\frac{d_\ell}{d_{sum(x)_k}}` and :math:`\frac{d_\ell}{d_{sum(x^2)_k}}`. 

    * Then Sync the gradient (automatically handled by :class:`encoding.parallel.allreduce`) and continue the backward.


Citation
--------

.. note::
    This code is provided together with the paper, please cite our work.

        * Hang Zhang, Kristin Dana, Jianping Shi, Zhongyue Zhang, Xiaogang Wang, Ambrish Tyagi, Amit Agrawal. "Context Encoding for Semantic Segmentation"  *The IEEE Conference on Computer Vision and Pattern Recognition (CVPR) 2018*::

            @InProceedings{Zhang_2018_CVPR,
            author = {Zhang, Hang and Dana, Kristin and Shi, Jianping and Zhang, Zhongyue and Wang, Xiaogang and Tyagi, Ambrish and Agrawal, Amit},
            title = {Context Encoding for Semantic Segmentation},
            booktitle = {The IEEE Conference on Computer Vision and Pattern Recognition (CVPR)},
            month = {June},
            year = {2018}
            }
