backward.md

# Backward Building

## Motivation

In Neural Network, most models are solved by the backpropagation algorithm(known as **BP**) at present. Technically, BP calculates the gradient of the loss function, then propagates it back through the networks following the chain rule. However, when configuring the model structure, users do not need to definate the backward part. So a mechanism is required by the framework which is able to complete the model's backward part automatically acoording to the given forward part.

When implementing a certain `op`, the developer is also asked to implement its backward version, called `grad_op`. A `grad_op` takes gradients of its corresponding `op`'s outputs, and calculate gradients of the `op`'s inputs. During the building of a model's backward part, the framework creates each forward `op`'s `grad_op`, and then string them together in reverse order of forward part. In this way, gradients spread from the end to the beginning of the model, in other word, from the loss to parameters.

## Challenges

The motivation of backward building is obvious. However, to implement it correctly is not so easy. In the **Fluid** design, a deep learning model is described by `Program`, `Block`, `Op` and `Variable`. The `Block` itself can be nested. It means that the `op`s and `variable`s are scattered across different blocks rather than all be gathered in a single graph. Our backward building algorithm shall visit blocks in recursive order and be able to insert `grad_op`s and new created `variable`s into right place. 

## Usage

Although the whole algorithm is comprised of many functions, only one is exposed as API:

```python
def append_backward(loss, parameter_list=None, no_grad_set=None):
    """
    Append backward part to main_program

    Args:
        loss(Variable): The variable generated by cost function.
        parameter_list(list): Parameters that need to be updated by optimizer.
            If None, it means all parameters need to be updated.

        no_grad_set(set): Variables that have no gradients in Block 0. 
            If None, the set will be generated inside the function and 
            contains all variables with `step_gradient=True` from all blocks.
        
    Return:
        (list[Variable]): list of (parameters, gradients) pair.
    """
```

By invoking this API, the framework appends backward part for the program where the `loss` is. It takes three arguments. `loss` means the final loss value. It must be a scalar and is usually the output of the loss layer. It is also where the gradient generated and backpropagation starts. `parameter_list` marks all parameters needs updating. If it's `None`, all parameter will be updated by optimizers. `no_grad_set` marks variables without gradient. if all outputs of some `grad_op` are in `no_grad_set`, the `grad_op` will not be run.

This API will be invoked automatically before optimizer building. 
As a result, in most cases users do not need to invoke the API by themselves to append backward part.

## Implementation

The implementation of backward building algorithm is in `backward.py` file. The whole algorithm can be divided to two independent parts: creating of `grad_op`s and creating of new variables. 

### Creating `grad_op`s

The creating of `grad_op`s is implemented by:

```python
def _append_backward_ops_(target,
                          block,
                          target_block,
                          no_grad_dict,
                          grad_to_var):
    """
    Create all grad ops, and insert them into given block

    Args:
        target(Variable): the target variable of forward pass
        block(Block): the block where forward ops are
        target_block(Block): the block which is going to hold new generated grad ops
        no_grad_dict(dict): 
            key(int)  block index
            val(set) a set of varibale names. These varibales have no gradient
        grad_to_var(dict)(output argument):
            key(str): grad variable name
            val(str): corresponding forward variable name
    """
```

Given a `block`, the function will traverses all `op`s in this block in reverse order, gets corresponding `grad_op` from the C++ core via `core.get_grad_op_desc()`, then append it to `target_block`. 

However, some specific `op`(e.g. `while_op`, `if_else_op`) can hold its own sub-block. For these sub-blocks contains `op`s as well, the `grad_op` creating should be recursive.

During the reverse traversal, we check each `op` whether it has an attribute named `sub_block`. If so, it means there is a sub-block and we need to deal with it first. After creating a new block whose father is the one in `op`'s attribute, we invoke `_append_backward_ops_()` recursively, assigning the new block to parameter `target_block` and the one in `op`'s attribute to `block`. The *pseudo-code* shows this process:

```
******* pseudo-code ********
for op in reversed(block.ops):
    if op has an attribute named 'sub_block':
        Get the sub-block(`s_block`) from op's attribute.
        Create a new block(`grad_s_block`), whose father is `s_block`.
        Invoke _append_backward_ops_(), with `block=s_block` and `target_block=grad_s_block`
    
    Invoke `core.get_grad_op_desc()` to get op's grad_op.
    Insert name correspondings between variables and their gradients of the grad_op to grad_to_var
    Assign grad_s_block to grad_op as it's 'sub_block' attribute.
    Append grad_op to current target_block.
```

The first invoking of `_append_backward_ops_()` is initiated by `append_backward()`, in which parameters `block` and `target_block` are all assigned with root block(the block with index 0).

### Corner Cases of `grad_op` Creating

In the previous section, we show the regular process of `grad_op` creating. However, in some corner cases, regular algorithm is not enough to get the correct result and appending handling is required. These addtional processes run after the above-mentioned algorithm and do some special adjusts on its output `grad_op`s.

#### Shared Variables

If a variable is readed by more than one `op` in the forward pass, its gradient is likey to be written by more than one `grad_op`s in the following backward pass. To make the gradient result being the sum of all `grad_op`s' outputs instead of the last running one, we assign each output with a temporary variables, and then add a `sum_op` to add them up. 

For the debug convinience, if the final gradient name is `w@GRAD`, it's corresponding temporary variables will be named as `w@GRAD@RENAME@0`, `w@GRAD@RENAME@1`...

<figure class="center">
<img src="./images/duplicate_op.png" width="45%" >
<img src="images/duplicate_op2.png" width="45%" >
</figure>

See function `_addup_repetitive_outputs_` in `backward.py` for implementation details.

#### No Gradient Variables

In our framework, variables can be marked as *no_gradient*, it means that the gradient of this variable is unnecessary and can be considered as zero in model training. Obviously, when all the outputs of some `grad_op` is marked as *no_gradient*, the `grad_op` itself can be skipped in backward pass. 

But these unnecessary gradients still need to be creating and initialized by something, otherwise following `grad_op`s who take these gradients as inputs take the risk of using uninitialized memory. In our code, we employ `fill_zeros_like_op` to initialize them as all zeros. 

This features are implemented in function `_remove_no_grad_branch_`. It checks new created `grad_op`'s one-by-one, removes whose outputs are all in `no_grad_set` or inserts `fill_zeros_like_op` when its necessary. We can get the `no_grad_set` from the `_append_backward_ops_` argument `no_grad_dict` or generate it on fly by scanning all variables' `no_gradient` attribute(True or False). 

### Creating Backward Variables

Up to now, we have completed all creating and adjusting jobs of `grad_op`s. However, backward variables have not been created. Now they are only represented by `grad_op`'s input and output arguments. The backward variable creating job will be done by:

```python
def _append_backward_vars_(block, 
                           start_op_idx, 
                           grad_to_var, 
                           grad_info_map):
    """
    Create new variables required by backward pass.

    Args:
        block(Block): the block where new variables will be created
        start_op_idx(int): Only variables required by ops in block.ops[start_op_idx : ] will be created
        grad_to_var(dict):
            key(str): grad variable name
            val(str): corresponding forward variable name
            In most cases, this dict is generated by _append_backward_ops_()
        grad_info_map(dict)(output argument):
            key(str): forward variable name
            val(tuple): a tuple of (str, int), str is the corresponding grad name, int is the block index
    """
```

Given a `block`, this function traverses all the `grad_op`s in it(The argument `start_op_idx` indicates where the grad_op sequence starts.) and creates all the uncreated outputs. The *pseudo-code* shows this process:

```
for op in block.ops[start_op_idx : ]:

    if op has an attribute named 'sub_block':
        Get the sub-block(`s_block`) from op's attribute.
        Invoke _append_backward_vars_(), with `block=s_block`
        
    for var_name in op.all_output_names():
        if block.has_var_recursive(var_name) or var_name is the name of empty variable:
            continue
        create a new variable named 'var_name' in block
        if grad_to_var.has_key(var_name):
            set grad_info_map[grad_to_var[var_name]] as a tuple of (var_name. block)
            
    do op's var type inference
    do op's shape inference
```