# Operator/expression 's Backward

## Motivation

In Neural Network, the backpropagation algorithm follows the chain rule, so we need to compound the fundmental gradient operators/expressions together with chain rule . Every forward network need a backward network to construct the full computation graph, the operator/expression's backward pass will be generated respect to forward pass.
  
## Backward Operator Registry

A backward network is built up with several backward operators. Backward operators take forward operators' inputs, outputs and output gradients and then calculate its input gradients.

|                        | forward operator | backward operator 
| ---------------------- | ---------------- |------------------------- |		
| **Operator::inputs_**  | Inputs       | Inputs, Outputs, OutputGradients |	
| **Operator::outputs_** | Outputs          | InputGradients            |

 In most cases, there is a one-to-one correspondence between forward and backward operators. These correspondences are recorded by a global hash map(`OpInfoMap`). To follow the philosophy of minimum core and make operators pluggable, the registry mechanism is introduced.

For example, we have got a `mul_op`, and we can register it's information and corresponding backward operator by the following macro:

```cpp
REGISTER_OP(mul, MulOp, MulOpMaker, mul_grad, MulOpGrad);
```

`mul` is the operator's type. `MulOp` and `MulOpMaker` are the operator class and the operator maker class respectively.

`mul_grad` is the type of backward operator, and `MulOpGrad` is its class name.

## Backward Opeartor Creating

Given a certain forward operator, we can get its corresponding backward opeartor by calling:

```cpp
OperatorBase* bwd_op = BuildGradOp(const OperatorBase* fwd_op);
``` 

The function `BuildGradOp` will sequentially execute following processes:

1. Get the `type_` of given forward operator, and then get the corresponding backward operator's type by looking up the `OpInfoMap`.

2. Build two maps named `inputs` and `outputs` to temporary storage backward operator's inputs and outputs. Copy forward operator's `inputs_` and `outputs_` to map `inputs`, except these are not necessary for gradient computing.

3. Add forward inputs' gradient variables into map `output`, adding forward outputs' gradient variables into map `input`.

4. Building backward operator with `inputs`, `outputs` and forward operator's attributes.

## Backward Network Building

A backward network is a series of backward operators. The main idea of building a backward network is creating backward operators in the inverted sequence and put them together.

In our design, the network itself is also a kind of operator. So the operators contained by a big network may be some small network. 

given a forward network, it generates the backward network. We only care about the Gradients—`OutputGradients`,`InputGradients`.

1. Op 

   when the input forward network is a Op, return its gradient Operator Immediately.

2. NetOp 

   when the input forward network is a NetOp, it need to call the sub NetOp/Operators backward function recursively. During the process, we need to collect the `OutputGradients` name according to forward NetOp.

   **shared variable**. As illustrated in the pictures, two operator's `Output` `Gradient` will overwirte their shared input variable.  

   <p align="center">
   <img src="./images/duplicate_op.png" width="70%" ><br/>

   1. shared variable in two operators. 

   </p>

   Share variable between operators or same input variable used in multiple operators lead to a duplicate gradient variable. As demo show above, we need to rename gradient name recursively, and add a generic add operator replace the overwirte links. 

   <p align="center">
   <img src="images/duplicate_op2.png" width="90%" ><br/>

   2. replace shared variable gradient with `Add` Operator

   </p>



​	Then collect the sub graph `OutputGradients`/`InputGradients` as the NetOp's and return it.