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develop/doc/_sources/design/auto_gradient_check.md.txt
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## Auto Gradient Checker Design ## Auto Gradient Check Design
## Backgraound： ## 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: - 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. you should get the right backpropagation formula according to the forward computation. 1. The formula for backpropagation formula should be correct according to the forward computation.
2. you should implement it right in CPP. 2. The Implementation of the above shoule be correct in CPP.
3. it's difficult to prepare test data. 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: - 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 need forward operator. 1. Numerical gradient checker only needs the forward operator.
2. user only need to prepare the input data for forward Operator. 2. The user only needs to prepare the input data for forward Operator and not worry about the backward Operator.
## Mathematical Theory ## 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(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) - [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 Interface
```python ```python
def get_numerical_gradient(op, def get_numerical_gradient(op,
...@@ -27,73 +27,76 @@ def get_numerical_gradient(op, ...@@ -27,73 +27,76 @@ def get_numerical_gradient(op,
delta=0.005, delta=0.005,
local_scope=None): 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 :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 output_name: The final output variable name.
:param input_to_check: The input variable with respect to which to compute the gradient. :param input_to_check: The input variable with respect to which the gradient has to be computed.
:param delta: The perturbation value for numeric gradient method. The :param delta: The perturbation value for numerical gradient method. The
smaller delta is, the more accurate result will get. But if that delta is smaller the delta, the more accurate the result. But if the delta is too
too small, it will suffer from numerical stability problem. small, it will suffer from the numerical stability problem.
:param local_scope: The local scope used for get_numeric_gradient. :param local_scope: The local scope used for get_numeric_gradient.
:return: The gradient array in numpy format. :return: The gradient array in numpy format.
""" """
``` ```
### Explaination: ### Explanation:
- Why need `output_name` - Why do we need an `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. - 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` - Why do we 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. - 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 ### Core Algorithm Implementation
```python ```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. # we use a for loop to compute the gradient of each element.
for i in xrange(tensor_size): for i in xrange(tensor_size):
# get one input element by its index i. # get one input element using the index i.
origin = tensor_to_check.get_float_element(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. # add delta to it, run the forward op and then
x_pos = origin + delta # get the new value of the result tensor.
x_pos = original + delta
tensor_to_check.set_float_element(i, x_pos) tensor_to_check.set_float_element(i, x_pos)
y_pos = get_output() y_pos = get_output()
# plus delta to this element, run op and get the new value of the result tensor. # Subtract delta from this element, run the op again
x_neg = origin - delta # and get the new value of the result tensor.
x_neg = original - delta
tensor_to_check.set_float_element(i, x_neg) tensor_to_check.set_float_element(i, x_neg)
y_neg = get_output() y_neg = get_output()
# restore old value # 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 gradient_flat[i] = (y_pos - y_neg) / delta / 2
# reshape the gradient result to the shape of the source tensor. # reshape the gradient result to the shape of the source tensor.
return gradient_flat.reshape(tensor_to_check.get_dims()) 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: Each Operator Kernel has three kinds of Gradient:
1. Numerical gradient 1. Numerical gradient
2. CPU kernel 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 1. Calculate the numerical gradient
2. calculate CPU kernel gradient with the backward Operator and compare it with 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) 3. Calculate GPU kernel gradient with the backward Operator and compare it with the numeric gradient. (if supported)
#### Python Interface #### Python Interface
...@@ -110,25 +113,26 @@ The numerical gradient only relies on forward Operator. So we use the numerical ...@@ -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 forward_op: used to create backward_op
:param input_vars: numpy value of input variable. The following :param input_vars: numpy value of input variable. The following
computation will use these variables. 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 output_name: The final output variable name.
:param max_relative_error: The relative tolerance parameter. :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. :param only_cpu: only compute and check gradient on cpu kernel.
:return: :return:
""" """
``` ```
### How to check if two numpy array is close enough? ### How to check if two numpy arrays are close enough?
if `abs_numerical_grad` is nearly zero, then use abs error for numerical_grad if `abs_numerical_grad` is nearly zero, then use absolute error for numerical_grad.
```python ```python
numerical_grad = ... numerical_grad = ...
operator_grad = numpy.array(scope.find_var(grad_var_name(name)).get_tensor()) operator_grad = numpy.array(scope.find_var(grad_var_name(name)).get_tensor())
abs_numerical_grad = numpy.abs(numerical_grad) abs_numerical_grad = numpy.abs(numerical_grad)
# if abs_numerical_grad is nearly zero, then use abs error for numeric_grad, not relative # if abs_numerical_grad is nearly zero, then use abs error for
# error. # numeric_grad, instead of relative error.
abs_numerical_grad[abs_numerical_grad < 1e-3] = 1 abs_numerical_grad[abs_numerical_grad < 1e-3] = 1
diff_mat = numpy.abs(abs_numerical_grad - operator_grad) / abs_numerical_grad diff_mat = numpy.abs(abs_numerical_grad - operator_grad) / abs_numerical_grad
...@@ -140,7 +144,7 @@ max_diff = numpy.max(diff_mat) ...@@ -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. 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(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) - [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)
develop/doc/design/auto_gradient_check.html
浏览文件 @ d451b098
...@@ -8,7 +8,7 @@ ...@@ -8,7 +8,7 @@
<meta name="viewport" content="width=device-width, initial-scale=1.0"> <meta name="viewport" content="width=device-width, initial-scale=1.0">
<title>Auto Gradient Checker Design &mdash; PaddlePaddle documentation</title> <title>Auto Gradient Check Design &mdash; PaddlePaddle documentation</title>
...@@ -193,7 +193,7 @@ ...@@ -193,7 +193,7 @@
<div role="navigation" aria-label="breadcrumbs navigation"> <div role="navigation" aria-label="breadcrumbs navigation">
<ul class="wy-breadcrumbs"> <ul class="wy-breadcrumbs">
<li>Auto Gradient Checker Design</li> <li>Auto Gradient Check Design</li>
</ul> </ul>
</div> </div>
...@@ -202,28 +202,28 @@ ...@@ -202,28 +202,28 @@
<div role="main" class="document" itemscope="itemscope" itemtype="http://schema.org/Article"> <div role="main" class="document" itemscope="itemscope" itemtype="http://schema.org/Article">
<div itemprop="articleBody"> <div itemprop="articleBody">
<div class="section" id="auto-gradient-checker-design"> <div class="section" id="auto-gradient-check-design">
<span id="auto-gradient-checker-design"></span><h1>Auto Gradient Checker Design<a class="headerlink" href="#auto-gradient-checker-design" title="Permalink to this headline"></a></h1> <span id="auto-gradient-check-design"></span><h1>Auto Gradient Check Design<a class="headerlink" href="#auto-gradient-check-design" title="Permalink to this headline"></a></h1>
</div> </div>
<div class="section" id="backgraound"> <div class="section" id="background">
<span id="backgraound"></span><h1>Backgraound：<a class="headerlink" href="#backgraound" title="Permalink to this headline"></a></h1> <span id="background"></span><h1>Background：<a class="headerlink" href="#background" title="Permalink to this headline"></a></h1>
<ul class="simple"> <ul class="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>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>you should get the right backpropagation formula according to the forward computation.</li> <li>The formula for backpropagation formula should be correct according to the forward computation.</li>
<li>you should implement it right in CPP.</li> <li>The Implementation of the above shoule be correct in CPP.</li>
<li>it&#8217;s difficult to prepare test data.</li> <li>It is difficult to prepare an unbiased test data.</li>
</ol> </ol>
</li> </li>
<li>Auto gradient checking gets a numerical gradient by forward Operator and use it as a reference of the backward Operator&#8217;s result. It has several advantages:<ol> <li>Auto gradient checking gets a numerical gradient using forward Operator and uses it as a reference for the backward Operator&#8217;s result. It has several advantages:<ol>
<li>numerical gradient checker only need forward operator.</li> <li>Numerical gradient checker only needs the forward operator.</li>
<li>user only need to prepare the input data for 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> </ol>
</li> </li>
</ul> </ul>
</div> </div>
<div class="section" id="mathematical-theory"> <div class="section" id="mathematical-theory">
<span id="mathematical-theory"></span><h1>Mathematical Theory<a class="headerlink" href="#mathematical-theory" title="Permalink to this headline"></a></h1> <span id="mathematical-theory"></span><h1>Mathematical Theory<a class="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&#8217;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>
<div class="toctree-wrapper compound"> <div class="toctree-wrapper compound">
<ul> <ul>
<li class="toctree-l1"><a class="reference external" href="http://deeplearning.stanford.edu/wiki/index.php/Gradient_checking_and_advanced_optimization">Gradient checking and advanced optimization(en)</a></li> <li class="toctree-l1"><a class="reference external" href="http://deeplearning.stanford.edu/wiki/index.php/Gradient_checking_and_advanced_optimization">Gradient checking and advanced optimization(en)</a></li>
...@@ -231,8 +231,8 @@ ...@@ -231,8 +231,8 @@
</ul> </ul>
</div> </div>
</div> </div>
<div class="section" id="numeric-gradient-implementation"> <div class="section" id="numerical-gradient-implementation">
<span id="numeric-gradient-implementation"></span><h1>Numeric Gradient Implementation<a class="headerlink" href="#numeric-gradient-implementation" title="Permalink to this headline"></a></h1> <span id="numerical-gradient-implementation"></span><h1>Numerical Gradient Implementation<a class="headerlink" href="#numerical-gradient-implementation" title="Permalink to this headline"></a></h1>
<div class="section" id="python-interface"> <div class="section" id="python-interface">
<span id="python-interface"></span><h2>Python Interface<a class="headerlink" href="#python-interface" title="Permalink to this headline"></a></h2> <span id="python-interface"></span><h2>Python Interface<a class="headerlink" href="#python-interface" title="Permalink to this headline"></a></h2>
<div class="highlight-python"><div class="highlight"><pre><span></span><span class="k">def</span> <span class="nf">get_numerical_gradient</span><span class="p">(</span><span class="n">op</span><span class="p">,</span> <div class="highlight-python"><div class="highlight"><pre><span></span><span class="k">def</span> <span class="nf">get_numerical_gradient</span><span class="p">(</span><span class="n">op</span><span class="p">,</span>
...@@ -242,57 +242,60 @@ ...@@ -242,57 +242,60 @@
<span class="n">delta</span><span class="o">=</span><span class="mf">0.005</span><span class="p">,</span> <span class="n">delta</span><span class="o">=</span><span class="mf">0.005</span><span class="p">,</span>
<span class="n">local_scope</span><span class="o">=</span><span class="bp">None</span><span class="p">):</span> <span class="n">local_scope</span><span class="o">=</span><span class="bp">None</span><span class="p">):</span>
<span class="sd">&quot;&quot;&quot;</span> <span class="sd">&quot;&quot;&quot;</span>
<span class="sd"> Get Numeric Gradient for an operator&#39;s input.</span> <span class="sd"> Get Numerical Gradient for the input of an operator.</span>
<span class="sd"> :param op: C++ operator instance, could be an network</span> <span class="sd"> :param op: C++ operator instance, could be an network.</span>
<span class="sd"> :param input_values: The input variables. Should be an dictionary, whose key is</span> <span class="sd"> :param input_values: The input variables. Should be an dictionary, whose key is</span>
<span class="sd"> variable name, and value is numpy array.</span> <span class="sd"> variable name, and value is a numpy array.</span>
<span class="sd"> :param output_name: The final output variable name.</span> <span class="sd"> :param output_name: The final output variable name.</span>
<span class="sd"> :param input_to_check: The input variable with respect to which to compute the gradient.</span> <span class="sd"> :param input_to_check: The input variable with respect to which the gradient has to be computed.</span>
<span class="sd"> :param delta: The perturbation value for numeric gradient method. The</span> <span class="sd"> :param delta: The perturbation value for numerical gradient method. The</span>
<span class="sd"> smaller delta is, the more accurate result will get. But if that delta is</span> <span class="sd"> smaller the delta, the more accurate the result. But if the delta is too</span>
<span class="sd"> too small, it will suffer from numerical stability problem.</span> <span class="sd"> small, it will suffer from the numerical stability problem.</span>
<span class="sd"> :param local_scope: The local scope used for get_numeric_gradient.</span> <span class="sd"> :param local_scope: The local scope used for get_numeric_gradient.</span>
<span class="sd"> :return: The gradient array in numpy format.</span> <span class="sd"> :return: The gradient array in numpy format.</span>
<span class="sd"> &quot;&quot;&quot;</span> <span class="sd"> &quot;&quot;&quot;</span>
</pre></div> </pre></div>
</div> </div>
</div> </div>
<div class="section" id="explaination"> <div class="section" id="explanation">
<span id="explaination"></span><h2>Explaination:<a class="headerlink" href="#explaination" title="Permalink to this headline"></a></h2> <span id="explanation"></span><h2>Explanation:<a class="headerlink" href="#explanation" title="Permalink to this headline"></a></h2>
<ul class="simple"> <ul class="simple">
<li>Why need <code class="docutils literal"><span class="pre">output_name</span></code><ul> <li>Why do we need an <code class="docutils literal"><span class="pre">output_name</span></code><ul>
<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>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> </ul>
</li> </li>
<li>Why need <code class="docutils literal"><span class="pre">input_to_check</span></code><ul> <li>Why do we need <code class="docutils literal"><span class="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 <code class="docutils literal"><span class="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 <code class="docutils literal"><span class="pre">get_numeric_gradient</span></code> multiple times.</li> <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 <code class="docutils literal"><span class="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 <code class="docutils literal"><span class="pre">get_numeric_gradient</span></code> multiple times each with a different input.</li>
</ul> </ul>
</li> </li>
</ul> </ul>
</div> </div>
<div class="section" id="core-algorithm-implementation"> <div class="section" id="core-algorithm-implementation">
<span id="core-algorithm-implementation"></span><h2>Core Algorithm Implementation<a class="headerlink" href="#core-algorithm-implementation" title="Permalink to this headline"></a></h2> <span id="core-algorithm-implementation"></span><h2>Core Algorithm Implementation<a class="headerlink" href="#core-algorithm-implementation" title="Permalink to this headline"></a></h2>
<div class="highlight-python"><div class="highlight"><pre><span></span> <span class="c1"># we only compute gradient of one element a time.</span> <div class="highlight-python"><div class="highlight"><pre><span></span> <span class="c1"># we only compute the gradient of one element a time.</span>
<span class="c1"># we use a for loop to compute the gradient of each element.</span> <span class="c1"># we use a for loop to compute the gradient of each element.</span>
<span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">xrange</span><span class="p">(</span><span class="n">tensor_size</span><span class="p">):</span> <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">xrange</span><span class="p">(</span><span class="n">tensor_size</span><span class="p">):</span>
<span class="c1"># get one input element by its index i.</span> <span class="c1"># get one input element using the index i.</span>
<span class="n">origin</span> <span class="o">=</span> <span class="n">tensor_to_check</span><span class="o">.</span><span class="n">get_float_element</span><span class="p">(</span><span class="n">i</span><span class="p">)</span> <span class="n">original</span> <span class="o">=</span> <span class="n">tensor_to_check</span><span class="o">.</span><span class="n">get_float_element</span><span class="p">(</span><span class="n">i</span><span class="p">)</span>
<span class="c1"># add delta to it, run op and then get the new value of the result tensor.</span> <span class="c1"># add delta to it, run the forward op and then</span>
<span class="n">x_pos</span> <span class="o">=</span> <span class="n">origin</span> <span class="o">+</span> <span class="n">delta</span> <span class="c1"># get the new value of the result tensor.</span>
<span class="n">x_pos</span> <span class="o">=</span> <span class="n">original</span> <span class="o">+</span> <span class="n">delta</span>
<span class="n">tensor_to_check</span><span class="o">.</span><span class="n">set_float_element</span><span class="p">(</span><span class="n">i</span><span class="p">,</span> <span class="n">x_pos</span><span class="p">)</span> <span class="n">tensor_to_check</span><span class="o">.</span><span class="n">set_float_element</span><span class="p">(</span><span class="n">i</span><span class="p">,</span> <span class="n">x_pos</span><span class="p">)</span>
<span class="n">y_pos</span> <span class="o">=</span> <span class="n">get_output</span><span class="p">()</span> <span class="n">y_pos</span> <span class="o">=</span> <span class="n">get_output</span><span class="p">()</span>
<span class="c1"># plus delta to this element, run op and get the new value of the result tensor.</span> <span class="c1"># Subtract delta from this element, run the op again</span>
<span class="n">x_neg</span> <span class="o">=</span> <span class="n">origin</span> <span class="o">-</span> <span class="n">delta</span> <span class="c1"># and get the new value of the result tensor.</span>
<span class="n">x_neg</span> <span class="o">=</span> <span class="n">original</span> <span class="o">-</span> <span class="n">delta</span>
<span class="n">tensor_to_check</span><span class="o">.</span><span class="n">set_float_element</span><span class="p">(</span><span class="n">i</span><span class="p">,</span> <span class="n">x_neg</span><span class="p">)</span> <span class="n">tensor_to_check</span><span class="o">.</span><span class="n">set_float_element</span><span class="p">(</span><span class="n">i</span><span class="p">,</span> <span class="n">x_neg</span><span class="p">)</span>
<span class="n">y_neg</span> <span class="o">=</span> <span class="n">get_output</span><span class="p">()</span> <span class="n">y_neg</span> <span class="o">=</span> <span class="n">get_output</span><span class="p">()</span>
<span class="c1"># restore old value</span> <span class="c1"># restore old value</span>
<span class="n">tensor_to_check</span><span class="o">.</span><span class="n">set_float_element</span><span class="p">(</span><span class="n">i</span><span class="p">,</span> <span class="n">origin</span><span class="p">)</span> <span class="n">tensor_to_check</span><span class="o">.</span><span class="n">set_float_element</span><span class="p">(</span><span class="n">i</span><span class="p">,</span> <span class="n">original</span><span class="p">)</span>
<span class="c1"># compute the gradient of this element and store it into a numpy array.</span> <span class="c1"># compute the gradient of this element and store</span>
<span class="c1"># it into a numpy array.</span>
<span class="n">gradient_flat</span><span class="p">[</span><span class="n">i</span><span class="p">]</span> <span class="o">=</span> <span class="p">(</span><span class="n">y_pos</span> <span class="o">-</span> <span class="n">y_neg</span><span class="p">)</span> <span class="o">/</span> <span class="n">delta</span> <span class="o">/</span> <span class="mi">2</span> <span class="n">gradient_flat</span><span class="p">[</span><span class="n">i</span><span class="p">]</span> <span class="o">=</span> <span class="p">(</span><span class="n">y_pos</span> <span class="o">-</span> <span class="n">y_neg</span><span class="p">)</span> <span class="o">/</span> <span class="n">delta</span> <span class="o">/</span> <span class="mi">2</span>
<span class="c1"># reshape the gradient result to the shape of the source tensor.</span> <span class="c1"># reshape the gradient result to the shape of the source tensor.</span>
...@@ -301,19 +304,19 @@ ...@@ -301,19 +304,19 @@
</div> </div>
</div> </div>
</div> </div>
<div class="section" id="auto-graident-checker-framework"> <div class="section" id="auto-gradient-check-framework">
<span id="auto-graident-checker-framework"></span><h1>Auto Graident Checker Framework<a class="headerlink" href="#auto-graident-checker-framework" title="Permalink to this headline"></a></h1> <span id="auto-gradient-check-framework"></span><h1>Auto Gradient Check Framework<a class="headerlink" href="#auto-gradient-check-framework" title="Permalink to this headline"></a></h1>
<p>Each Operator Kernel has three kinds of Gradient:</p> <p>Each Operator Kernel has three kinds of Gradient:</p>
<ol class="simple"> <ol class="simple">
<li>Numerical gradient</li> <li>Numerical gradient</li>
<li>CPU kernel 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> </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>
<ol class="simple"> <ol class="simple">
<li>calculate the numerical gradient</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 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 GPU kernel gradient with the backward Operator and compare it with the numeric gradient. (if supported)</li>
</ol> </ol>
<div class="section" id="python-interface"> <div class="section" id="python-interface">
<span id="id1"></span><h2>Python Interface<a class="headerlink" href="#python-interface" title="Permalink to this headline"></a></h2> <span id="id1"></span><h2>Python Interface<a class="headerlink" href="#python-interface" title="Permalink to this headline"></a></h2>
...@@ -329,25 +332,26 @@ ...@@ -329,25 +332,26 @@
<span class="sd"> :param forward_op: used to create backward_op</span> <span class="sd"> :param forward_op: used to create backward_op</span>
<span class="sd"> :param input_vars: numpy value of input variable. The following</span> <span class="sd"> :param input_vars: numpy value of input variable. The following</span>
<span class="sd"> computation will use these variables.</span> <span class="sd"> computation will use these variables.</span>
<span class="sd"> :param inputs_to_check: the input variable with respect to which to compute the gradient.</span> <span class="sd"> :param inputs_to_check: the input variable with respect to which the</span>
<span class="sd"> gradient will be computed.</span>
<span class="sd"> :param output_name: The final output variable name.</span> <span class="sd"> :param output_name: The final output variable name.</span>
<span class="sd"> :param max_relative_error: The relative tolerance parameter.</span> <span class="sd"> :param max_relative_error: The relative tolerance parameter.</span>
<span class="sd"> :param no_grad_set: used when create backward ops</span> <span class="sd"> :param no_grad_set: used to create backward ops</span>
<span class="sd"> :param only_cpu: only compute and check gradient on cpu kernel.</span> <span class="sd"> :param only_cpu: only compute and check gradient on cpu kernel.</span>
<span class="sd"> :return:</span> <span class="sd"> :return:</span>
<span class="sd"> &quot;&quot;&quot;</span> <span class="sd"> &quot;&quot;&quot;</span>
</pre></div> </pre></div>
</div> </div>
</div> </div>
<div class="section" id="how-to-check-if-two-numpy-array-is-close-enough"> <div class="section" id="how-to-check-if-two-numpy-arrays-are-close-enough">
<span id="how-to-check-if-two-numpy-array-is-close-enough"></span><h2>How to check if two numpy array is close enough?<a class="headerlink" href="#how-to-check-if-two-numpy-array-is-close-enough" title="Permalink to this headline"></a></h2> <span id="how-to-check-if-two-numpy-arrays-are-close-enough"></span><h2>How to check if two numpy arrays are close enough?<a class="headerlink" href="#how-to-check-if-two-numpy-arrays-are-close-enough" title="Permalink to this headline"></a></h2>
<p>if <code class="docutils literal"><span class="pre">abs_numerical_grad</span></code> is nearly zero, then use abs error for numerical_grad</p> <p>if <code class="docutils literal"><span class="pre">abs_numerical_grad</span></code> is nearly zero, then use absolute error for numerical_grad.</p>
<div class="highlight-python"><div class="highlight"><pre><span></span><span class="n">numerical_grad</span> <span class="o">=</span> <span class="o">...</span> <div class="highlight-python"><div class="highlight"><pre><span></span><span class="n">numerical_grad</span> <span class="o">=</span> <span class="o">...</span>
<span class="n">operator_grad</span> <span class="o">=</span> <span class="n">numpy</span><span class="o">.</span><span class="n">array</span><span class="p">(</span><span class="n">scope</span><span class="o">.</span><span class="n">find_var</span><span class="p">(</span><span class="n">grad_var_name</span><span class="p">(</span><span class="n">name</span><span class="p">))</span><span class="o">.</span><span class="n">get_tensor</span><span class="p">())</span> <span class="n">operator_grad</span> <span class="o">=</span> <span class="n">numpy</span><span class="o">.</span><span class="n">array</span><span class="p">(</span><span class="n">scope</span><span class="o">.</span><span class="n">find_var</span><span class="p">(</span><span class="n">grad_var_name</span><span class="p">(</span><span class="n">name</span><span class="p">))</span><span class="o">.</span><span class="n">get_tensor</span><span class="p">())</span>
<span class="n">abs_numerical_grad</span> <span class="o">=</span> <span class="n">numpy</span><span class="o">.</span><span class="n">abs</span><span class="p">(</span><span class="n">numerical_grad</span><span class="p">)</span> <span class="n">abs_numerical_grad</span> <span class="o">=</span> <span class="n">numpy</span><span class="o">.</span><span class="n">abs</span><span class="p">(</span><span class="n">numerical_grad</span><span class="p">)</span>
<span class="c1"># if abs_numerical_grad is nearly zero, then use abs error for numeric_grad, not relative</span> <span class="c1"># if abs_numerical_grad is nearly zero, then use abs error for</span>
<span class="c1"># error.</span> <span class="c1"># numeric_grad, instead of relative error.</span>
<span class="n">abs_numerical_grad</span><span class="p">[</span><span class="n">abs_numerical_grad</span> <span class="o">&lt;</span> <span class="mf">1e-3</span><span class="p">]</span> <span class="o">=</span> <span class="mi">1</span> <span class="n">abs_numerical_grad</span><span class="p">[</span><span class="n">abs_numerical_grad</span> <span class="o">&lt;</span> <span class="mf">1e-3</span><span class="p">]</span> <span class="o">=</span> <span class="mi">1</span>
<span class="n">diff_mat</span> <span class="o">=</span> <span class="n">numpy</span><span class="o">.</span><span class="n">abs</span><span class="p">(</span><span class="n">abs_numerical_grad</span> <span class="o">-</span> <span class="n">operator_grad</span><span class="p">)</span> <span class="o">/</span> <span class="n">abs_numerical_grad</span> <span class="n">diff_mat</span> <span class="o">=</span> <span class="n">numpy</span><span class="o">.</span><span class="n">abs</span><span class="p">(</span><span class="n">abs_numerical_grad</span> <span class="o">-</span> <span class="n">operator_grad</span><span class="p">)</span> <span class="o">/</span> <span class="n">abs_numerical_grad</span>
...@@ -358,8 +362,8 @@ ...@@ -358,8 +362,8 @@
<span id="notes"></span><h3>Notes：<a class="headerlink" href="#notes" title="Permalink to this headline"></a></h3> <span id="notes"></span><h3>Notes：<a class="headerlink" href="#notes" title="Permalink to this headline"></a></h3>
<p>The Input data for auto gradient checker should be reasonable to avoid numerical stability problem.</p> <p>The Input data for auto gradient checker should be reasonable to avoid numerical stability problem.</p>
</div> </div>
<div class="section" id="refs"> <div class="section" id="references">
<span id="refs"></span><h3>Refs:<a class="headerlink" href="#refs" title="Permalink to this headline"></a></h3> <span id="references"></span><h3>References:<a class="headerlink" href="#references" title="Permalink to this headline"></a></h3>
<div class="toctree-wrapper compound"> <div class="toctree-wrapper compound">
<ul> <ul>
<li class="toctree-l1"><a class="reference external" href="http://deeplearning.stanford.edu/wiki/index.php/Gradient_checking_and_advanced_optimization">Gradient checking and advanced optimization(en)</a></li> <li class="toctree-l1"><a class="reference external" href="http://deeplearning.stanford.edu/wiki/index.php/Gradient_checking_and_advanced_optimization">Gradient checking and advanced optimization(en)</a></li>
......
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## Auto Gradient Checker Design ## Auto Gradient Check Design
## Backgraound： ## 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: - 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. you should get the right backpropagation formula according to the forward computation. 1. The formula for backpropagation formula should be correct according to the forward computation.
2. you should implement it right in CPP. 2. The Implementation of the above shoule be correct in CPP.
3. it's difficult to prepare test data. 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: - 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 need forward operator. 1. Numerical gradient checker only needs the forward operator.
2. user only need to prepare the input data for forward Operator. 2. The user only needs to prepare the input data for forward Operator and not worry about the backward Operator.
## Mathematical Theory ## 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(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) - [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 Interface
```python ```python
def get_numerical_gradient(op, def get_numerical_gradient(op,
...@@ -27,73 +27,76 @@ def get_numerical_gradient(op, ...@@ -27,73 +27,76 @@ def get_numerical_gradient(op,
delta=0.005, delta=0.005,
local_scope=None): 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 :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 output_name: The final output variable name.
:param input_to_check: The input variable with respect to which to compute the gradient. :param input_to_check: The input variable with respect to which the gradient has to be computed.
:param delta: The perturbation value for numeric gradient method. The :param delta: The perturbation value for numerical gradient method. The
smaller delta is, the more accurate result will get. But if that delta is smaller the delta, the more accurate the result. But if the delta is too
too small, it will suffer from numerical stability problem. small, it will suffer from the numerical stability problem.
:param local_scope: The local scope used for get_numeric_gradient. :param local_scope: The local scope used for get_numeric_gradient.
:return: The gradient array in numpy format. :return: The gradient array in numpy format.
""" """
``` ```
### Explaination: ### Explanation:
- Why need `output_name` - Why do we need an `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. - 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` - Why do we 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. - 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 ### Core Algorithm Implementation
```python ```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. # we use a for loop to compute the gradient of each element.
for i in xrange(tensor_size): for i in xrange(tensor_size):
# get one input element by its index i. # get one input element using the index i.
origin = tensor_to_check.get_float_element(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. # add delta to it, run the forward op and then
x_pos = origin + delta # get the new value of the result tensor.
x_pos = original + delta
tensor_to_check.set_float_element(i, x_pos) tensor_to_check.set_float_element(i, x_pos)
y_pos = get_output() y_pos = get_output()
# plus delta to this element, run op and get the new value of the result tensor. # Subtract delta from this element, run the op again
x_neg = origin - delta # and get the new value of the result tensor.
x_neg = original - delta
tensor_to_check.set_float_element(i, x_neg) tensor_to_check.set_float_element(i, x_neg)
y_neg = get_output() y_neg = get_output()
# restore old value # 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 gradient_flat[i] = (y_pos - y_neg) / delta / 2
# reshape the gradient result to the shape of the source tensor. # reshape the gradient result to the shape of the source tensor.
return gradient_flat.reshape(tensor_to_check.get_dims()) 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: Each Operator Kernel has three kinds of Gradient:
1. Numerical gradient 1. Numerical gradient
2. CPU kernel 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 1. Calculate the numerical gradient
2. calculate CPU kernel gradient with the backward Operator and compare it with 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) 3. Calculate GPU kernel gradient with the backward Operator and compare it with the numeric gradient. (if supported)
#### Python Interface #### Python Interface
...@@ -110,25 +113,26 @@ The numerical gradient only relies on forward Operator. So we use the numerical ...@@ -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 forward_op: used to create backward_op
:param input_vars: numpy value of input variable. The following :param input_vars: numpy value of input variable. The following
computation will use these variables. 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 output_name: The final output variable name.
:param max_relative_error: The relative tolerance parameter. :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. :param only_cpu: only compute and check gradient on cpu kernel.
:return: :return:
""" """
``` ```
### How to check if two numpy array is close enough? ### How to check if two numpy arrays are close enough?
if `abs_numerical_grad` is nearly zero, then use abs error for numerical_grad if `abs_numerical_grad` is nearly zero, then use absolute error for numerical_grad.
```python ```python
numerical_grad = ... numerical_grad = ...
operator_grad = numpy.array(scope.find_var(grad_var_name(name)).get_tensor()) operator_grad = numpy.array(scope.find_var(grad_var_name(name)).get_tensor())
abs_numerical_grad = numpy.abs(numerical_grad) abs_numerical_grad = numpy.abs(numerical_grad)
# if abs_numerical_grad is nearly zero, then use abs error for numeric_grad, not relative # if abs_numerical_grad is nearly zero, then use abs error for
# error. # numeric_grad, instead of relative error.
abs_numerical_grad[abs_numerical_grad < 1e-3] = 1 abs_numerical_grad[abs_numerical_grad < 1e-3] = 1
diff_mat = numpy.abs(abs_numerical_grad - operator_grad) / abs_numerical_grad diff_mat = numpy.abs(abs_numerical_grad - operator_grad) / abs_numerical_grad
...@@ -140,7 +144,7 @@ max_diff = numpy.max(diff_mat) ...@@ -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. 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(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) - [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)
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...@@ -8,7 +8,7 @@ ...@@ -8,7 +8,7 @@
<meta name="viewport" content="width=device-width, initial-scale=1.0"> <meta name="viewport" content="width=device-width, initial-scale=1.0">
<title>Auto Gradient Checker Design &mdash; PaddlePaddle 文档</title> <title>Auto Gradient Check Design &mdash; PaddlePaddle 文档</title>
...@@ -212,7 +212,7 @@ ...@@ -212,7 +212,7 @@
<div role="navigation" aria-label="breadcrumbs navigation"> <div role="navigation" aria-label="breadcrumbs navigation">
<ul class="wy-breadcrumbs"> <ul class="wy-breadcrumbs">
<li>Auto Gradient Checker Design</li> <li>Auto Gradient Check Design</li>
</ul> </ul>
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...@@ -221,28 +221,28 @@ ...@@ -221,28 +221,28 @@
<div role="main" class="document" itemscope="itemscope" itemtype="http://schema.org/Article"> <div role="main" class="document" itemscope="itemscope" itemtype="http://schema.org/Article">
<div itemprop="articleBody"> <div itemprop="articleBody">
<div class="section" id="auto-gradient-checker-design"> <div class="section" id="auto-gradient-check-design">
<span id="auto-gradient-checker-design"></span><h1>Auto Gradient Checker Design<a class="headerlink" href="#auto-gradient-checker-design" title="永久链接至标题"></a></h1> <span id="auto-gradient-check-design"></span><h1>Auto Gradient Check Design<a class="headerlink" href="#auto-gradient-check-design" title="永久链接至标题"></a></h1>
</div> </div>
<div class="section" id="backgraound"> <div class="section" id="background">
<span id="backgraound"></span><h1>Backgraound：<a class="headerlink" href="#backgraound" title="永久链接至标题"></a></h1> <span id="background"></span><h1>Background：<a class="headerlink" href="#background" title="永久链接至标题"></a></h1>
<ul class="simple"> <ul class="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>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>you should get the right backpropagation formula according to the forward computation.</li> <li>The formula for backpropagation formula should be correct according to the forward computation.</li>
<li>you should implement it right in CPP.</li> <li>The Implementation of the above shoule be correct in CPP.</li>
<li>it&#8217;s difficult to prepare test data.</li> <li>It is difficult to prepare an unbiased test data.</li>
</ol> </ol>
</li> </li>
<li>Auto gradient checking gets a numerical gradient by forward Operator and use it as a reference of the backward Operator&#8217;s result. It has several advantages:<ol> <li>Auto gradient checking gets a numerical gradient using forward Operator and uses it as a reference for the backward Operator&#8217;s result. It has several advantages:<ol>
<li>numerical gradient checker only need forward operator.</li> <li>Numerical gradient checker only needs the forward operator.</li>
<li>user only need to prepare the input data for 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> </ol>
</li> </li>
</ul> </ul>
</div> </div>
<div class="section" id="mathematical-theory"> <div class="section" id="mathematical-theory">
<span id="mathematical-theory"></span><h1>Mathematical Theory<a class="headerlink" href="#mathematical-theory" title="永久链接至标题"></a></h1> <span id="mathematical-theory"></span><h1>Mathematical Theory<a class="headerlink" href="#mathematical-theory" title="永久链接至标题"></a></h1>
<p>The following two document from Stanford has a detailed explanation of how to get numerical gradient and why it&#8217;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>
<div class="toctree-wrapper compound"> <div class="toctree-wrapper compound">
<ul> <ul>
<li class="toctree-l1"><a class="reference external" href="http://deeplearning.stanford.edu/wiki/index.php/Gradient_checking_and_advanced_optimization">Gradient checking and advanced optimization(en)</a></li> <li class="toctree-l1"><a class="reference external" href="http://deeplearning.stanford.edu/wiki/index.php/Gradient_checking_and_advanced_optimization">Gradient checking and advanced optimization(en)</a></li>
...@@ -250,8 +250,8 @@ ...@@ -250,8 +250,8 @@
</ul> </ul>
</div> </div>
</div> </div>
<div class="section" id="numeric-gradient-implementation"> <div class="section" id="numerical-gradient-implementation">
<span id="numeric-gradient-implementation"></span><h1>Numeric Gradient Implementation<a class="headerlink" href="#numeric-gradient-implementation" title="永久链接至标题"></a></h1> <span id="numerical-gradient-implementation"></span><h1>Numerical Gradient Implementation<a class="headerlink" href="#numerical-gradient-implementation" title="永久链接至标题"></a></h1>
<div class="section" id="python-interface"> <div class="section" id="python-interface">
<span id="python-interface"></span><h2>Python Interface<a class="headerlink" href="#python-interface" title="永久链接至标题"></a></h2> <span id="python-interface"></span><h2>Python Interface<a class="headerlink" href="#python-interface" title="永久链接至标题"></a></h2>
<div class="highlight-python"><div class="highlight"><pre><span></span><span class="k">def</span> <span class="nf">get_numerical_gradient</span><span class="p">(</span><span class="n">op</span><span class="p">,</span> <div class="highlight-python"><div class="highlight"><pre><span></span><span class="k">def</span> <span class="nf">get_numerical_gradient</span><span class="p">(</span><span class="n">op</span><span class="p">,</span>
...@@ -261,57 +261,60 @@ ...@@ -261,57 +261,60 @@
<span class="n">delta</span><span class="o">=</span><span class="mf">0.005</span><span class="p">,</span> <span class="n">delta</span><span class="o">=</span><span class="mf">0.005</span><span class="p">,</span>
<span class="n">local_scope</span><span class="o">=</span><span class="bp">None</span><span class="p">):</span> <span class="n">local_scope</span><span class="o">=</span><span class="bp">None</span><span class="p">):</span>
<span class="sd">&quot;&quot;&quot;</span> <span class="sd">&quot;&quot;&quot;</span>
<span class="sd"> Get Numeric Gradient for an operator&#39;s input.</span> <span class="sd"> Get Numerical Gradient for the input of an operator.</span>
<span class="sd"> :param op: C++ operator instance, could be an network</span> <span class="sd"> :param op: C++ operator instance, could be an network.</span>
<span class="sd"> :param input_values: The input variables. Should be an dictionary, whose key is</span> <span class="sd"> :param input_values: The input variables. Should be an dictionary, whose key is</span>
<span class="sd"> variable name, and value is numpy array.</span> <span class="sd"> variable name, and value is a numpy array.</span>
<span class="sd"> :param output_name: The final output variable name.</span> <span class="sd"> :param output_name: The final output variable name.</span>
<span class="sd"> :param input_to_check: The input variable with respect to which to compute the gradient.</span> <span class="sd"> :param input_to_check: The input variable with respect to which the gradient has to be computed.</span>
<span class="sd"> :param delta: The perturbation value for numeric gradient method. The</span> <span class="sd"> :param delta: The perturbation value for numerical gradient method. The</span>
<span class="sd"> smaller delta is, the more accurate result will get. But if that delta is</span> <span class="sd"> smaller the delta, the more accurate the result. But if the delta is too</span>
<span class="sd"> too small, it will suffer from numerical stability problem.</span> <span class="sd"> small, it will suffer from the numerical stability problem.</span>
<span class="sd"> :param local_scope: The local scope used for get_numeric_gradient.</span> <span class="sd"> :param local_scope: The local scope used for get_numeric_gradient.</span>
<span class="sd"> :return: The gradient array in numpy format.</span> <span class="sd"> :return: The gradient array in numpy format.</span>
<span class="sd"> &quot;&quot;&quot;</span> <span class="sd"> &quot;&quot;&quot;</span>
</pre></div> </pre></div>
</div> </div>
</div> </div>
<div class="section" id="explaination"> <div class="section" id="explanation">
<span id="explaination"></span><h2>Explaination:<a class="headerlink" href="#explaination" title="永久链接至标题"></a></h2> <span id="explanation"></span><h2>Explanation:<a class="headerlink" href="#explanation" title="永久链接至标题"></a></h2>
<ul class="simple"> <ul class="simple">
<li>Why need <code class="docutils literal"><span class="pre">output_name</span></code><ul> <li>Why do we need an <code class="docutils literal"><span class="pre">output_name</span></code><ul>
<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>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> </ul>
</li> </li>
<li>Why need <code class="docutils literal"><span class="pre">input_to_check</span></code><ul> <li>Why do we need <code class="docutils literal"><span class="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 <code class="docutils literal"><span class="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 <code class="docutils literal"><span class="pre">get_numeric_gradient</span></code> multiple times.</li> <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 <code class="docutils literal"><span class="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 <code class="docutils literal"><span class="pre">get_numeric_gradient</span></code> multiple times each with a different input.</li>
</ul> </ul>
</li> </li>
</ul> </ul>
</div> </div>
<div class="section" id="core-algorithm-implementation"> <div class="section" id="core-algorithm-implementation">
<span id="core-algorithm-implementation"></span><h2>Core Algorithm Implementation<a class="headerlink" href="#core-algorithm-implementation" title="永久链接至标题"></a></h2> <span id="core-algorithm-implementation"></span><h2>Core Algorithm Implementation<a class="headerlink" href="#core-algorithm-implementation" title="永久链接至标题"></a></h2>
<div class="highlight-python"><div class="highlight"><pre><span></span> <span class="c1"># we only compute gradient of one element a time.</span> <div class="highlight-python"><div class="highlight"><pre><span></span> <span class="c1"># we only compute the gradient of one element a time.</span>
<span class="c1"># we use a for loop to compute the gradient of each element.</span> <span class="c1"># we use a for loop to compute the gradient of each element.</span>
<span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">xrange</span><span class="p">(</span><span class="n">tensor_size</span><span class="p">):</span> <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">xrange</span><span class="p">(</span><span class="n">tensor_size</span><span class="p">):</span>
<span class="c1"># get one input element by its index i.</span> <span class="c1"># get one input element using the index i.</span>
<span class="n">origin</span> <span class="o">=</span> <span class="n">tensor_to_check</span><span class="o">.</span><span class="n">get_float_element</span><span class="p">(</span><span class="n">i</span><span class="p">)</span> <span class="n">original</span> <span class="o">=</span> <span class="n">tensor_to_check</span><span class="o">.</span><span class="n">get_float_element</span><span class="p">(</span><span class="n">i</span><span class="p">)</span>
<span class="c1"># add delta to it, run op and then get the new value of the result tensor.</span> <span class="c1"># add delta to it, run the forward op and then</span>
<span class="n">x_pos</span> <span class="o">=</span> <span class="n">origin</span> <span class="o">+</span> <span class="n">delta</span> <span class="c1"># get the new value of the result tensor.</span>
<span class="n">x_pos</span> <span class="o">=</span> <span class="n">original</span> <span class="o">+</span> <span class="n">delta</span>
<span class="n">tensor_to_check</span><span class="o">.</span><span class="n">set_float_element</span><span class="p">(</span><span class="n">i</span><span class="p">,</span> <span class="n">x_pos</span><span class="p">)</span> <span class="n">tensor_to_check</span><span class="o">.</span><span class="n">set_float_element</span><span class="p">(</span><span class="n">i</span><span class="p">,</span> <span class="n">x_pos</span><span class="p">)</span>
<span class="n">y_pos</span> <span class="o">=</span> <span class="n">get_output</span><span class="p">()</span> <span class="n">y_pos</span> <span class="o">=</span> <span class="n">get_output</span><span class="p">()</span>
<span class="c1"># plus delta to this element, run op and get the new value of the result tensor.</span> <span class="c1"># Subtract delta from this element, run the op again</span>
<span class="n">x_neg</span> <span class="o">=</span> <span class="n">origin</span> <span class="o">-</span> <span class="n">delta</span> <span class="c1"># and get the new value of the result tensor.</span>
<span class="n">x_neg</span> <span class="o">=</span> <span class="n">original</span> <span class="o">-</span> <span class="n">delta</span>
<span class="n">tensor_to_check</span><span class="o">.</span><span class="n">set_float_element</span><span class="p">(</span><span class="n">i</span><span class="p">,</span> <span class="n">x_neg</span><span class="p">)</span> <span class="n">tensor_to_check</span><span class="o">.</span><span class="n">set_float_element</span><span class="p">(</span><span class="n">i</span><span class="p">,</span> <span class="n">x_neg</span><span class="p">)</span>
<span class="n">y_neg</span> <span class="o">=</span> <span class="n">get_output</span><span class="p">()</span> <span class="n">y_neg</span> <span class="o">=</span> <span class="n">get_output</span><span class="p">()</span>
<span class="c1"># restore old value</span> <span class="c1"># restore old value</span>
<span class="n">tensor_to_check</span><span class="o">.</span><span class="n">set_float_element</span><span class="p">(</span><span class="n">i</span><span class="p">,</span> <span class="n">origin</span><span class="p">)</span> <span class="n">tensor_to_check</span><span class="o">.</span><span class="n">set_float_element</span><span class="p">(</span><span class="n">i</span><span class="p">,</span> <span class="n">original</span><span class="p">)</span>
<span class="c1"># compute the gradient of this element and store it into a numpy array.</span> <span class="c1"># compute the gradient of this element and store</span>
<span class="c1"># it into a numpy array.</span>
<span class="n">gradient_flat</span><span class="p">[</span><span class="n">i</span><span class="p">]</span> <span class="o">=</span> <span class="p">(</span><span class="n">y_pos</span> <span class="o">-</span> <span class="n">y_neg</span><span class="p">)</span> <span class="o">/</span> <span class="n">delta</span> <span class="o">/</span> <span class="mi">2</span> <span class="n">gradient_flat</span><span class="p">[</span><span class="n">i</span><span class="p">]</span> <span class="o">=</span> <span class="p">(</span><span class="n">y_pos</span> <span class="o">-</span> <span class="n">y_neg</span><span class="p">)</span> <span class="o">/</span> <span class="n">delta</span> <span class="o">/</span> <span class="mi">2</span>
<span class="c1"># reshape the gradient result to the shape of the source tensor.</span> <span class="c1"># reshape the gradient result to the shape of the source tensor.</span>
...@@ -320,19 +323,19 @@ ...@@ -320,19 +323,19 @@
</div> </div>
</div> </div>
</div> </div>
<div class="section" id="auto-graident-checker-framework"> <div class="section" id="auto-gradient-check-framework">
<span id="auto-graident-checker-framework"></span><h1>Auto Graident Checker Framework<a class="headerlink" href="#auto-graident-checker-framework" title="永久链接至标题"></a></h1> <span id="auto-gradient-check-framework"></span><h1>Auto Gradient Check Framework<a class="headerlink" href="#auto-gradient-check-framework" title="永久链接至标题"></a></h1>
<p>Each Operator Kernel has three kinds of Gradient:</p> <p>Each Operator Kernel has three kinds of Gradient:</p>
<ol class="simple"> <ol class="simple">
<li>Numerical gradient</li> <li>Numerical gradient</li>
<li>CPU kernel 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> </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>
<ol class="simple"> <ol class="simple">
<li>calculate the numerical gradient</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 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 GPU kernel gradient with the backward Operator and compare it with the numeric gradient. (if supported)</li>
</ol> </ol>
<div class="section" id="python-interface"> <div class="section" id="python-interface">
<span id="id1"></span><h2>Python Interface<a class="headerlink" href="#python-interface" title="永久链接至标题"></a></h2> <span id="id1"></span><h2>Python Interface<a class="headerlink" href="#python-interface" title="永久链接至标题"></a></h2>
...@@ -348,25 +351,26 @@ ...@@ -348,25 +351,26 @@
<span class="sd"> :param forward_op: used to create backward_op</span> <span class="sd"> :param forward_op: used to create backward_op</span>
<span class="sd"> :param input_vars: numpy value of input variable. The following</span> <span class="sd"> :param input_vars: numpy value of input variable. The following</span>
<span class="sd"> computation will use these variables.</span> <span class="sd"> computation will use these variables.</span>
<span class="sd"> :param inputs_to_check: the input variable with respect to which to compute the gradient.</span> <span class="sd"> :param inputs_to_check: the input variable with respect to which the</span>
<span class="sd"> gradient will be computed.</span>
<span class="sd"> :param output_name: The final output variable name.</span> <span class="sd"> :param output_name: The final output variable name.</span>
<span class="sd"> :param max_relative_error: The relative tolerance parameter.</span> <span class="sd"> :param max_relative_error: The relative tolerance parameter.</span>
<span class="sd"> :param no_grad_set: used when create backward ops</span> <span class="sd"> :param no_grad_set: used to create backward ops</span>
<span class="sd"> :param only_cpu: only compute and check gradient on cpu kernel.</span> <span class="sd"> :param only_cpu: only compute and check gradient on cpu kernel.</span>
<span class="sd"> :return:</span> <span class="sd"> :return:</span>
<span class="sd"> &quot;&quot;&quot;</span> <span class="sd"> &quot;&quot;&quot;</span>
</pre></div> </pre></div>
</div> </div>
</div> </div>
<div class="section" id="how-to-check-if-two-numpy-array-is-close-enough"> <div class="section" id="how-to-check-if-two-numpy-arrays-are-close-enough">
<span id="how-to-check-if-two-numpy-array-is-close-enough"></span><h2>How to check if two numpy array is close enough?<a class="headerlink" href="#how-to-check-if-two-numpy-array-is-close-enough" title="永久链接至标题"></a></h2> <span id="how-to-check-if-two-numpy-arrays-are-close-enough"></span><h2>How to check if two numpy arrays are close enough?<a class="headerlink" href="#how-to-check-if-two-numpy-arrays-are-close-enough" title="永久链接至标题"></a></h2>
<p>if <code class="docutils literal"><span class="pre">abs_numerical_grad</span></code> is nearly zero, then use abs error for numerical_grad</p> <p>if <code class="docutils literal"><span class="pre">abs_numerical_grad</span></code> is nearly zero, then use absolute error for numerical_grad.</p>
<div class="highlight-python"><div class="highlight"><pre><span></span><span class="n">numerical_grad</span> <span class="o">=</span> <span class="o">...</span> <div class="highlight-python"><div class="highlight"><pre><span></span><span class="n">numerical_grad</span> <span class="o">=</span> <span class="o">...</span>
<span class="n">operator_grad</span> <span class="o">=</span> <span class="n">numpy</span><span class="o">.</span><span class="n">array</span><span class="p">(</span><span class="n">scope</span><span class="o">.</span><span class="n">find_var</span><span class="p">(</span><span class="n">grad_var_name</span><span class="p">(</span><span class="n">name</span><span class="p">))</span><span class="o">.</span><span class="n">get_tensor</span><span class="p">())</span> <span class="n">operator_grad</span> <span class="o">=</span> <span class="n">numpy</span><span class="o">.</span><span class="n">array</span><span class="p">(</span><span class="n">scope</span><span class="o">.</span><span class="n">find_var</span><span class="p">(</span><span class="n">grad_var_name</span><span class="p">(</span><span class="n">name</span><span class="p">))</span><span class="o">.</span><span class="n">get_tensor</span><span class="p">())</span>
<span class="n">abs_numerical_grad</span> <span class="o">=</span> <span class="n">numpy</span><span class="o">.</span><span class="n">abs</span><span class="p">(</span><span class="n">numerical_grad</span><span class="p">)</span> <span class="n">abs_numerical_grad</span> <span class="o">=</span> <span class="n">numpy</span><span class="o">.</span><span class="n">abs</span><span class="p">(</span><span class="n">numerical_grad</span><span class="p">)</span>
<span class="c1"># if abs_numerical_grad is nearly zero, then use abs error for numeric_grad, not relative</span> <span class="c1"># if abs_numerical_grad is nearly zero, then use abs error for</span>
<span class="c1"># error.</span> <span class="c1"># numeric_grad, instead of relative error.</span>
<span class="n">abs_numerical_grad</span><span class="p">[</span><span class="n">abs_numerical_grad</span> <span class="o">&lt;</span> <span class="mf">1e-3</span><span class="p">]</span> <span class="o">=</span> <span class="mi">1</span> <span class="n">abs_numerical_grad</span><span class="p">[</span><span class="n">abs_numerical_grad</span> <span class="o">&lt;</span> <span class="mf">1e-3</span><span class="p">]</span> <span class="o">=</span> <span class="mi">1</span>
<span class="n">diff_mat</span> <span class="o">=</span> <span class="n">numpy</span><span class="o">.</span><span class="n">abs</span><span class="p">(</span><span class="n">abs_numerical_grad</span> <span class="o">-</span> <span class="n">operator_grad</span><span class="p">)</span> <span class="o">/</span> <span class="n">abs_numerical_grad</span> <span class="n">diff_mat</span> <span class="o">=</span> <span class="n">numpy</span><span class="o">.</span><span class="n">abs</span><span class="p">(</span><span class="n">abs_numerical_grad</span> <span class="o">-</span> <span class="n">operator_grad</span><span class="p">)</span> <span class="o">/</span> <span class="n">abs_numerical_grad</span>
...@@ -377,8 +381,8 @@ ...@@ -377,8 +381,8 @@
<span id="notes"></span><h3>Notes：<a class="headerlink" href="#notes" title="永久链接至标题"></a></h3> <span id="notes"></span><h3>Notes：<a class="headerlink" href="#notes" title="永久链接至标题"></a></h3>
<p>The Input data for auto gradient checker should be reasonable to avoid numerical stability problem.</p> <p>The Input data for auto gradient checker should be reasonable to avoid numerical stability problem.</p>
</div> </div>
<div class="section" id="refs"> <div class="section" id="references">
<span id="refs"></span><h3>Refs:<a class="headerlink" href="#refs" title="永久链接至标题"></a></h3> <span id="references"></span><h3>References:<a class="headerlink" href="#references" title="永久链接至标题"></a></h3>
<div class="toctree-wrapper compound"> <div class="toctree-wrapper compound">
<ul> <ul>
<li class="toctree-l1"><a class="reference external" href="http://deeplearning.stanford.edu/wiki/index.php/Gradient_checking_and_advanced_optimization">Gradient checking and advanced optimization(en)</a></li> <li class="toctree-l1"><a class="reference external" href="http://deeplearning.stanford.edu/wiki/index.php/Gradient_checking_and_advanced_optimization">Gradient checking and advanced optimization(en)</a></li>
......
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