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+# Batch Normalization
+
+## What is batch normalization
+
+Batch normalization is a frequently-used method in deep network training. It adjusts the mean and variance of a layer's output, and make the data distribution easier for next layer's training.
+
+The principle of batch normalization can be summarized into a simple function:
+
+```
+y = (x - E[x]) / STD[x]) * scale + bias
+```
+
+`x` is a batch of output data of a certain layer. `E[x]` and `STD[x]` is the mean and standard deviation of `x`, respectively。 `scale` and `bias` are two trainable parameters. The training of batch normalization layer equals to the learning of best values of `scale` and `bias`.
+
+In our design, we use a single operator(`batch_norm_op`) to implement the whole batch normalization in C++, and wrap it as a layer in Python.
+
+## Differences with normal operators
+
+`batch_norm_op` is a single operator. However, there are a few differences between `BatchNormOp` and normal operators, which we shall take into consideration in our design.
+
+1. `batch_norm_op` shall behave differently in training and inferencing. For example, during inferencing, there is no batch data and it's impossible to compute `E[x]` and `STD[x]`, so we have to use an `estimated_mean` and an `estimated_variance` instead of them. These require our framework to be able to inform operators current running type (training/inferencing), then operators can switch their behaviors.
+
+2. `batch_norm_op` shall have the ability to maintain `estimated_mean` and `estimated_variance` across mini-batch. In each mini-batch, `estimated_mean` is iterated by the following equations:
+
+```
+if batch_id == 0
+ estimated_mean = E[x]
+else
+ estimated_mean = estimated_mean * momentum + (1.0 - momentum_) * E[x]
+```
+
+The iterating of `estimated_variance` is similar. `momentum` is an attribute, which controls estimated_mean updating speed.
+
+## Implementation
+
+Batch normalization is designed as a single operator is C++, and then wrapped as a layer in Python.
+
+### C++
+
+As most C++ operators do, `batch_norm_op` is defined by inputs, outputs, attributes and compute kernels.
+
+#### Inputs
+
+- `x`: The inputs data, which is generated by the previous layer.
+- `estimated_mean`: The estimated mean of all previous data batches. It is updated in each forward propagation and will be used in inferencing to take the role of `E[x]`.
+- `estimated_var`: The estimated standard deviation of all previous data batches. It is updated in each forward propagation and will be used in inferencing to take the role of `STD[x]`.
+- `scale`: trainable parameter 'scale'
+- `bias`: trainable parameter 'bias'
+
+#### Outputs
+
+- `y`: The output data.
+- `batch_mean`: The mean value of batch data.
+- `batch_var`: The standard deviation value of batch data.
+- `saved_mean`: Updated `estimated_mean` with current batch data. It's supposed to share the memory with input `estimated_mean`.
+- `saved_var`: Updated `estimated_var` with current batch data. It's supposed to share the memory with input `estimated_var`.
+
+#### Attributes
+
+- `is_infer`: *bool*. If true, run `batch_norm_op` in inferencing mode.
+- `use_global_est`: *bool*. If true, use `saved_mean` and `saved_var` instead of `E[x]` and `STD[x]` in trainning.
+- `epsilon`: *float*. The epsilon value to avoid division by zero.
+- `momentum`: *float*. Factor used in `estimated_mean` and `estimated_var` updating. The usage is shown above.
+
+#### Kernels
+
+The following graph showes the training computational process of `batch_norm_op`:
+
+
+
+cudnn provides APIs to finish the whole series of computation, we can use them in our GPU kernel.
+
+### Python
+
+`batch_norm_op` is warpped as a layer in Python:
+
+```python
+def batch_norm_layer(net,
+ input,
+ output,
+ scale,
+ bias,
+ use_global_est = False,
+ epsilon = 1e-6,
+ momentum = 0.99):
+ mean_cache = scope.new_var(name = 'estimated_mean', trainable = False)
+ var_cache = scop.new_var(name = 'estimated_var', trainable = False)
+ batch_mean = scope.new_var(name = 'batch_mean')
+ batch_var = scope.new_var(name = 'batch_var')
+ batch_norm_op = Operator('batch_norm_op',
+ x = input,
+ estimated_mean = mean_cache,
+ estimated_mean = var_cache,
+ scale = scale,
+ bias = bias,
+ y = output,
+ batch_mean = batch_mean,
+ batch_var = batch_var,
+ saved_mean = mean_cache,
+ saved_var = var_cache,
+ is_infer = False,
+ use_global_est = use_global_est,
+ epsilon = epsilon,
+ momentum = momentum)
+ net.append_op(batch_norm_op)
+ return output
+```
+
+Because Python API has not been finally decided, the code above can be regarded as pseudo code. There are a few key points we shall note:
+
+1. `estimated_mean` and `estimated_var` are assigned the same variables with `saved_mean` and `saved_var` respectively. So they share same the memories. The output mean and variance values(`saved_mean` and `saved_var`) of a certain batch will be the inputs(`estimated_mean` and `estimated_var`) of the next batch.
+
+2. `is_infer` decided whether `batch_norm_op` will run in training mode or inferencing mode. However, a network may contains both training and inferencing parts. And user may switch `batch_norm_op`'s running mode in Python `for` loop like this:
+
+```python
+for pass_id in range(PASS_NUM):
+ # ...
+ net.train() # run training model
+ if pass_id % 100 == 0:
+ net.infer(test_image) # run inferencing model
+ # ...
+```
+
+`is_infer` is an attribute. Once an operator is created, its attributes can not be changed. It suggests us that we shall maintain two `batch_norm_op` in the model, one's `is_infer` is `True`(we call it `infer_batch_norm_op`) and the other one's is `False`(we call it `train_batch_norm_op`). They share all parameters and variables, but be placed in two different branches. That is to say, if a network contains a `batch_norm_op`, it will fork into two branches, one go through `train_batch_norm_op` and the other one go through `infer_batch_norm_op`:
+
+
+
+