- 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)
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.
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.
<spanid="auto-gradient-checker-design"></span><h1>Auto Gradient Checker Design<aclass="headerlink"href="#auto-gradient-checker-design"title="Permalink to this headline">¶</a></h1>
<spanid="auto-gradient-check-design"></span><h1>Auto Gradient Check Design<aclass="headerlink"href="#auto-gradient-check-design"title="Permalink to this headline">¶</a></h1>
</div>
<divclass="section"id="backgraound">
<spanid="backgraound"></span><h1>Backgraound：<aclass="headerlink"href="#backgraound"title="Permalink to this headline">¶</a></h1>
<divclass="section"id="background">
<spanid="background"></span><h1>Background：<aclass="headerlink"href="#background"title="Permalink to this headline">¶</a></h1>
<ulclass="simple">
<li>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:<ol>
<li>you should get the right backpropagation formula according to the forward computation.</li>
<li>you should implement it right in CPP.</li>
<li>it’s difficult to prepare test data.</li>
<li>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:<ol>
<li>The formula for backpropagation formula should be correct according to the forward computation.</li>
<li>The Implementation of the above shoule be correct in CPP.</li>
<li>It is difficult to prepare an unbiased test data.</li>
</ol>
</li>
<li>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:<ol>
<li>numerical gradient checker only need forward operator.</li>
<li>user only need to prepare the input data for forward Operator.</li>
<li>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:<ol>
<li>Numerical gradient checker only needs the forward operator.</li>
<li>The user only needs to prepare the input data for forward Operator and not worry about the backward Operator.</li>
</ol>
</li>
</ul>
</div>
<divclass="section"id="mathematical-theory">
<spanid="mathematical-theory"></span><h1>Mathematical Theory<aclass="headerlink"href="#mathematical-theory"title="Permalink to this headline">¶</a></h1>
<p>The following two document from Stanford has a detailed explanation of how to get numerical gradient and why it’s useful.</p>
<p>The following documents from Stanford have a detailed explanation of how to compute the numerical gradient and why it is useful.</p>
<divclass="toctree-wrapper compound">
<ul>
<liclass="toctree-l1"><aclass="reference external"href="http://deeplearning.stanford.edu/wiki/index.php/Gradient_checking_and_advanced_optimization">Gradient checking and advanced optimization(en)</a></li>
<spanid="numeric-gradient-implementation"></span><h1>Numeric Gradient Implementation<aclass="headerlink"href="#numeric-gradient-implementation"title="Permalink to this headline">¶</a></h1>
<spanid="numerical-gradient-implementation"></span><h1>Numerical Gradient Implementation<aclass="headerlink"href="#numerical-gradient-implementation"title="Permalink to this headline">¶</a></h1>
<divclass="section"id="python-interface">
<spanid="python-interface"></span><h2>Python Interface<aclass="headerlink"href="#python-interface"title="Permalink to this headline">¶</a></h2>
<li>An Operator may have multiple Output, one can get independent gradient from each Output. So caller should specify the name of the output variable.</li>
<li>Why do we need an<codeclass="docutils literal"><spanclass="pre">output_name</span></code><ul>
<li>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.</li>
</ul>
</li>
<li>Why need <codeclass="docutils literal"><spanclass="pre">input_to_check</span></code><ul>
<li>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 <codeclass="docutils literal"><spanclass="pre">get_numeric_gradient</span></code> is designed to calculate the gradient for one input. If you need to compute multiple inputs, you can call <codeclass="docutils literal"><spanclass="pre">get_numeric_gradient</span></code> multiple times.</li>
<li>Why do we need <codeclass="docutils literal"><spanclass="pre">input_to_check</span></code><ul>
<li>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 <codeclass="docutils literal"><spanclass="pre">get_numeric_gradient</span></code> is designed to calculate the gradient for one input. If you need to compute multiple inputs, you can call <codeclass="docutils literal"><spanclass="pre">get_numeric_gradient</span></code> multiple times each with a different input.</li>
<spanid="core-algorithm-implementation"></span><h2>Core Algorithm Implementation<aclass="headerlink"href="#core-algorithm-implementation"title="Permalink to this headline">¶</a></h2>
<divclass="highlight-python"><divclass="highlight"><pre><span></span><spanclass="c1"># we only compute gradient of one element a time.</span>
<divclass="highlight-python"><divclass="highlight"><pre><span></span><spanclass="c1"># we only compute the gradient of one element a time.</span>
<spanclass="c1"># we use a for loop to compute the gradient of each element.</span>
<spanid="auto-graident-checker-framework"></span><h1>Auto Graident Checker Framework<aclass="headerlink"href="#auto-graident-checker-framework"title="Permalink to this headline">¶</a></h1>
<spanid="auto-gradient-check-framework"></span><h1>Auto Gradient Check Framework<aclass="headerlink"href="#auto-gradient-check-framework"title="Permalink to this headline">¶</a></h1>
<p>Each Operator Kernel has three kinds of Gradient:</p>
<olclass="simple">
<li>Numerical gradient</li>
<li>CPU kernel gradient</li>
<li>GPU kernel gradient (if supported)</li>
<li>GPU kernel gradient (if supported by the device)</li>
</ol>
<p>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:</p>
<p>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:</p>
<olclass="simple">
<li>calculate the numerical gradient</li>
<li>calculate CPU kernel gradient with the backward Operator and compare it with the numerical gradient</li>
<li>calculate GPU kernel gradient with the backward Operator and compare it with the numeric gradient (if supported)</li>
<li>Calculate the numerical gradient</li>
<li>Calculate CPU kernel gradient with the backward Operator and compare it with the numerical gradient.</li>
<li>Calculate GPU kernel gradient with the backward Operator and compare it with the numeric gradient. (if supported)</li>
</ol>
<divclass="section"id="python-interface">
<spanid="id1"></span><h2>Python Interface<aclass="headerlink"href="#python-interface"title="Permalink to this headline">¶</a></h2>
...
...
@@ -329,25 +332,26 @@
<spanclass="sd"> :param forward_op: used to create backward_op</span>
<spanclass="sd"> :param input_vars: numpy value of input variable. The following</span>
<spanclass="sd"> computation will use these variables.</span>
<spanclass="sd"> :param inputs_to_check: the input variable with respect to which to compute the gradient.</span>
<spanclass="sd"> :param inputs_to_check: the input variable with respect to which the</span>
<spanclass="sd"> gradient will be computed.</span>
<spanclass="sd"> :param output_name: The final output variable name.</span>
<spanclass="sd"> :param max_relative_error: The relative tolerance parameter.</span>
<spanclass="sd"> :param no_grad_set: used when create backward ops</span>
<spanclass="sd"> :param no_grad_set: used to create backward ops</span>
<spanclass="sd"> :param only_cpu: only compute and check gradient on cpu kernel.</span>
<spanid="how-to-check-if-two-numpy-array-is-close-enough"></span><h2>How to check if two numpy array is close enough?<aclass="headerlink"href="#how-to-check-if-two-numpy-array-is-close-enough"title="Permalink to this headline">¶</a></h2>
<p>if <codeclass="docutils literal"><spanclass="pre">abs_numerical_grad</span></code> is nearly zero, then use abs error for numerical_grad</p>
<spanid="how-to-check-if-two-numpy-arrays-are-close-enough"></span><h2>How to check if two numpy arrays are close enough?<aclass="headerlink"href="#how-to-check-if-two-numpy-arrays-are-close-enough"title="Permalink to this headline">¶</a></h2>
<p>if <codeclass="docutils literal"><spanclass="pre">abs_numerical_grad</span></code> is nearly zero, then use absolute error for numerical_grad.</p>
- 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)
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.
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.
<li>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:<ol>
<li>you should get the right backpropagation formula according to the forward computation.</li>
<li>you should implement it right in CPP.</li>
<li>it’s difficult to prepare test data.</li>
<li>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:<ol>
<li>The formula for backpropagation formula should be correct according to the forward computation.</li>
<li>The Implementation of the above shoule be correct in CPP.</li>
<li>It is difficult to prepare an unbiased test data.</li>
</ol>
</li>
<li>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:<ol>
<li>numerical gradient checker only need forward operator.</li>
<li>user only need to prepare the input data for forward Operator.</li>
<li>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:<ol>
<li>Numerical gradient checker only needs the forward operator.</li>
<li>The user only needs to prepare the input data for forward Operator and not worry about the backward Operator.</li>
<li>An Operator may have multiple Output, one can get independent gradient from each Output. So caller should specify the name of the output variable.</li>
<li>Why do we need an<codeclass="docutils literal"><spanclass="pre">output_name</span></code><ul>
<li>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.</li>
</ul>
</li>
<li>Why need <codeclass="docutils literal"><spanclass="pre">input_to_check</span></code><ul>
<li>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 <codeclass="docutils literal"><spanclass="pre">get_numeric_gradient</span></code> is designed to calculate the gradient for one input. If you need to compute multiple inputs, you can call <codeclass="docutils literal"><spanclass="pre">get_numeric_gradient</span></code> multiple times.</li>
<li>Why do we need <codeclass="docutils literal"><spanclass="pre">input_to_check</span></code><ul>
<li>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 <codeclass="docutils literal"><spanclass="pre">get_numeric_gradient</span></code> is designed to calculate the gradient for one input. If you need to compute multiple inputs, you can call <codeclass="docutils literal"><spanclass="pre">get_numeric_gradient</span></code> multiple times each with a different input.</li>
<p>Each Operator Kernel has three kinds of Gradient:</p>
<olclass="simple">
<li>Numerical gradient</li>
<li>CPU kernel gradient</li>
<li>GPU kernel gradient (if supported)</li>
<li>GPU kernel gradient (if supported by the device)</li>
</ol>
<p>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:</p>
<p>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:</p>
<olclass="simple">
<li>calculate the numerical gradient</li>
<li>calculate CPU kernel gradient with the backward Operator and compare it with the numerical gradient</li>
<li>calculate GPU kernel gradient with the backward Operator and compare it with the numeric gradient (if supported)</li>
<li>Calculate the numerical gradient</li>
<li>Calculate CPU kernel gradient with the backward Operator and compare it with the numerical gradient.</li>
<li>Calculate GPU kernel gradient with the backward Operator and compare it with the numeric gradient. (if supported)</li>
<spanid="how-to-check-if-two-numpy-array-is-close-enough"></span><h2>How to check if two numpy array is close enough?<aclass="headerlink"href="#how-to-check-if-two-numpy-array-is-close-enough"title="永久链接至标题">¶</a></h2>
<p>if <codeclass="docutils literal"><spanclass="pre">abs_numerical_grad</span></code> is nearly zero, then use abs error for numerical_grad</p>
<spanid="how-to-check-if-two-numpy-arrays-are-close-enough"></span><h2>How to check if two numpy arrays are close enough?<aclass="headerlink"href="#how-to-check-if-two-numpy-arrays-are-close-enough"title="永久链接至标题">¶</a></h2>
<p>if <codeclass="docutils literal"><spanclass="pre">abs_numerical_grad</span></code> is nearly zero, then use absolute error for numerical_grad.</p>
