## auto gradient check Design

## Backgraound：
- Operator forward computing is easy to check if the result is right because it has a clear definition. **But** backpropagation is a notoriously difficult algorithm to debug and get right:
  - **Firstly** you should get the right backpropagation formula according to the forward computation.
  - **Secondly** you should implement it right in CPP.
  - **Thirdly** it's difficult to prepare test data.

- Auto gradient check gets a numeric gradient by forward Operator and use it as a reference of the backward Operator's result. It has several advantages:
  - **Firstly** numeric gradient checker only need forward operator.
  - **Secondly** user only need to prepare the input data for forward Operator.

## mathematical theory
The following two document from stanford has a detailed explanation of how to get numeric gradient and why it's useful.

- [Gradient checking and advanced optimization(en)](http://deeplearning.stanford.edu/wiki/index.php/Gradient_checking_and_advanced_optimization)
- [Gradient checking and advanced optimization(cn)](http://ufldl.stanford.edu/wiki/index.php/%E6%A2%AF%E5%BA%A6%E6%A3%80%E9%AA%8C%E4%B8%8E%E9%AB%98%E7%BA%A7%E4%BC%98%E5%8C%96)


## Numeric Gradient Implementation
### Interface
```python
def get_numeric_gradient(op,
                         input_values,
                         output_name,
                         input_to_check,
                         delta=0.005,
                         local_scope=None):
    """
    Get Numeric Gradient for an operator's input.

    :param op: C++ operator instance, could be an network
    :param input_values: The input variables. Should be an dictionary, key is
    variable name. Value is numpy array.
    :param output_name: The final output variable name.
    :param input_to_check: The input variable need to get gradient.
    :param delta: The perturbation value for numeric gradient method. The
    smaller delta is, the more accurate result will get. But if that delta is
     too small, it could occur numerical stability problem.
    :param local_scope: The local scope used for get_numeric_gradient.
    :return: The gradient array in numpy format.
    """
```

### Explaination:

1. Why need `output_name`
  - One Operator may have multiple Output, you can get independent gradient from each Output. So user should set one output to calculate.

1. Why need `input_to_check`
  - One operator may have multiple inputs. Gradient Op can calculate the gradient of these Inputs at the same time. But Numeric Gradient needs to calculate them one by one. So `get_numeric_gradient` is designed to calculate the gradient for one input. If you need to compute multiple inputs, you can call `get_numeric_gradient` multiple times.


### Core algorithm implement


```python
    # we only compute gradient of one element each time.
    # we use a for loop to compute the gradient of every element.
    for i in xrange(tensor_size):
        # get one input element throw it's index i.
        origin = tensor_to_check.get_float_element(i)

        # add delta to it, run op and then get the sum of the result tensor.
        x_pos = origin + delta
        tensor_to_check.set_float_element(i, x_pos)
        y_pos = get_output()

        # plus delta to this element, run op and get the sum of the result tensor.
        x_neg = origin - delta
        tensor_to_check.set_float_element(i, x_neg)
        y_neg = get_output()

        # restore old value
        tensor_to_check.set_float_element(i, origin)

        # compute the gradient of this element and store it into a numpy array.
        gradient_flat[i] = (y_pos - y_neg) / delta / 2

    # reshape the gradient result to the shape of the source tensor.
    return gradient_flat.reshape(tensor_to_check.get_dims())
```

## auto check framework design

Each Operator Kernel has three kinds of Gradient:

- 1. Numeric Gradient
- 2. CPU Operator Gradient
- 3. GPU Operator Gradient(if supported)

Numeric Gradient Only relies on forward Operator. So we use Numeric Gradient as the reference value.

- **Firstly** calculate the numeric gradient.
- **Secondly** calculate CPU kernel Gradient with the backward Operator and compare it with the numeric gradient.
- **Thirdly** calculate GPU kernel Gradient with the backward Operator and compare it with the numeric gradient.(if support GPU)

#### auto check python Interface

```python
    def check_grad(self,
                   forward_op,
                   input_vars,
                   inputs_to_check,
                   output_name,
                   no_grad_set=None,
                   only_cpu=False,
                   max_relative_error=0.005):
        """
        :param forward_op: used to create backward_op
        :param input_vars: numpy value of input variable. The following
            computation will use these variables.
        :param inputs_to_check: inputs var names that should check gradient.
        :param output_name: output name that used to
        :param max_relative_error: The relative tolerance parameter.
        :param no_grad_set: used when create backward ops
        :param only_cpu: only compute and check gradient on cpu kernel.
        :return:
        """
```

### How two check two numpy array is close enough?
if `abs_numeric_grad` is nearly zero, then use abs error for numeric_grad, not relative

```python
numeric_grad = ...
operator_grad = numpy.array(scope.find_var(grad_var_name(name)).get_tensor())

abs_numeric_grad = numpy.abs(numeric_grad)
# if abs_numeric_grad is nearly zero, then use abs error for numeric_grad, not relative
# error.
abs_numeric_grad[abs_numeric_grad < 1e-3] = 1

diff_mat = numpy.abs(abs_numeric_grad - operator_grad) / abs_numeric_grad
max_diff = numpy.max(diff_mat)
```


#### Notes：
1，The Input data for auto gradient checker should be reasonable to avoid numeric problem.


#### refs:

- [Gradient checking and advanced optimization(en)](http://deeplearning.stanford.edu/wiki/index.php/Gradient_checking_and_advanced_optimization)
- [Gradient checking and advanced optimization(cn)](http://ufldl.stanford.edu/wiki/index.php/%E6%A2%AF%E5%BA%A6%E6%A3%80%E9%AA%8C%E4%B8%8E%E9%AB%98%E7%BA%A7%E4%BC%98%E5%8C%96)