Fixing the gradient check writeup (#8057)

doc/design/auto_gradient_check.md

-## Auto Gradient Checker Design
+## Auto Gradient Check Design
 
-## Backgraound：
-- Generally, it is easy to check whether the forward computation of an Operator is correct or not. However, backpropagation is a notoriously difficult algorithm to debug and get right:
-  1. you should get the right backpropagation formula according to the forward computation.
-  2. you should implement it right in CPP.
-  3. it's difficult to prepare test data.
+## Background：
+- Generally, it is easy to check whether the forward computation of an Operator is correct or not. However, backpropagation is a notoriously difficult algorithm to debug and get right because of the following challenges:
+  1. The formula for backpropagation formula should be correct according to the forward computation.
+  2. The Implementation of the above shoule be correct in CPP.
+  3. It is difficult to prepare an unbiased test data.
 
-- Auto gradient checking gets a numerical gradient by forward Operator and use it as a reference of the backward Operator's result. It has several advantages:
-  1. numerical gradient checker only need forward operator.
-  2. user only need to prepare the input data for forward Operator.
+- Auto gradient checking gets a numerical gradient using forward Operator and uses it as a reference for the backward Operator's result. It has several advantages:
+  1. Numerical gradient checker only needs the forward operator.
+  2. The user only needs to prepare the input data for forward Operator and not worry about the backward Operator.
 
 ## Mathematical Theory
-The following two document from Stanford has a detailed explanation of how to get numerical gradient and why it's useful.
+The following documents from Stanford have a detailed explanation of how to compute the numerical gradient and why it is 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
+## Numerical Gradient Implementation
 ### Python Interface
 ```python
 def get_numerical_gradient(op,
@@ -27,73 +27,76 @@ def get_numerical_gradient(op,
                          delta=0.005,
                          local_scope=None):
     """
-    Get Numeric Gradient for an operator's input.
+    Get Numerical Gradient for the input of an operator.
 
-    :param op: C++ operator instance, could be an network
+    :param op: C++ operator instance, could be an network.
     :param input_values: The input variables. Should be an dictionary, whose key is
-    variable name, and value is numpy array.
+    variable name, and value is a numpy array.
     :param output_name: The final output variable name.
-    :param input_to_check: The input variable with respect to which to compute the 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 will suffer from numerical stability problem.
+    :param input_to_check: The input variable with respect to which the gradient has to be computed.
+    :param delta: The perturbation value for numerical gradient method. The
+    smaller the delta, the more accurate the result. But if the delta is too
+    small, it will suffer from the numerical stability problem.
     :param local_scope: The local scope used for get_numeric_gradient.
     :return: The gradient array in numpy format.
     """
 ```
 
-### Explaination:
+### Explanation:
 
-- Why need `output_name`
-  - An Operator may have multiple Output, one can get independent gradient from each Output. So caller should specify the name of the output variable.
+- Why do we need an `output_name`
+  - An Operator may have multiple Outputs, one can compute an independent gradient from each Output. So the caller should specify the name of the output variable.
 
-- 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.
+- Why do we need `input_to_check`
+  - One operator can have multiple inputs. Gradient Op can calculate the gradient of these inputs at the same time. But Numerical 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 each with a different input.
 
 
 ### Core Algorithm Implementation
 
 
 ```python
-    # we only compute gradient of one element a time.
+    # we only compute the gradient of one element a time.
     # we use a for loop to compute the gradient of each element.
     for i in xrange(tensor_size):
-        # get one input element by its index i.
-        origin = tensor_to_check.get_float_element(i)
+        # get one input element using the index i.
+        original = tensor_to_check.get_float_element(i)
 
-        # add delta to it, run op and then get the new value of the result tensor.
-        x_pos = origin + delta
+        # add delta to it, run the forward op and then
+        # get the new value of the result tensor.
+        x_pos = original + delta
         tensor_to_check.set_float_element(i, x_pos)
         y_pos = get_output()
 
-        # plus delta to this element, run op and get the new value of the result tensor.
-        x_neg = origin - delta
+        # Subtract delta from this element, run the op again
+        # and get the new value of the result tensor.
+        x_neg = original - 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)
+        tensor_to_check.set_float_element(i, original)
 
-        # compute the gradient of this element and store it into a numpy array.
+        # 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 Graident Checker Framework
+## Auto Gradient Check Framework
 
 Each Operator Kernel has three kinds of Gradient:
 
 1. Numerical gradient
 2. CPU kernel gradient
-3. GPU kernel gradient (if supported)
+3. GPU kernel gradient (if supported by the device)
 
-The numerical gradient only relies on forward Operator. So we use the numerical gradient as the reference value. And the gradient checking is performed in the following three steps:
+The numerical gradient only relies on the forward Operator, so we use the numerical gradient as the reference value. The gradient checking is performed in the following three steps:
 
-1. calculate the numerical gradient
-2. calculate CPU kernel gradient with the backward Operator and compare it with the numerical gradient
-3. calculate GPU kernel gradient with the backward Operator and compare it with the numeric gradient (if supported)
+1. Calculate the numerical gradient
+2. Calculate CPU kernel gradient with the backward Operator and compare it with the numerical gradient.
+3. Calculate GPU kernel gradient with the backward Operator and compare it with the numeric gradient. (if supported)
 
 #### Python Interface
 
@@ -110,25 +113,26 @@ The numerical gradient only relies on forward Operator. So we use the numerical
         :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: the input variable with respect to which to compute the gradient.
+        :param inputs_to_check: the input variable with respect to which the
+          gradient will be computed.
         :param output_name: The final output variable name.
         :param max_relative_error: The relative tolerance parameter.
-        :param no_grad_set: used when create backward ops
+        :param no_grad_set: used to create backward ops
         :param only_cpu: only compute and check gradient on cpu kernel.
         :return:
         """
 ```
 
-### How to check if two numpy array is close enough?
-if `abs_numerical_grad` is nearly zero, then use abs error for numerical_grad
+### How to check if two numpy arrays are close enough?
+if `abs_numerical_grad` is nearly zero, then use absolute error for numerical_grad.
 
 ```python
 numerical_grad = ...
 operator_grad = numpy.array(scope.find_var(grad_var_name(name)).get_tensor())
 
 abs_numerical_grad = numpy.abs(numerical_grad)
-# if abs_numerical_grad is nearly zero, then use abs error for numeric_grad, not relative
-# error.
+# if abs_numerical_grad is nearly zero, then use abs error for
+# numeric_grad, instead of relative error.
 abs_numerical_grad[abs_numerical_grad < 1e-3] = 1
 
 diff_mat = numpy.abs(abs_numerical_grad - operator_grad) / abs_numerical_grad
@@ -140,7 +144,7 @@ max_diff = numpy.max(diff_mat)
 The Input data for auto gradient checker should be reasonable to avoid numerical stability problem.
 
 
-#### Refs:
+#### References:
 
 - [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)