From dce92332de3e8cb8326f7f80409af7f362396f83 Mon Sep 17 00:00:00 2001 From: Travis CI Date: Fri, 11 Aug 2017 22:55:37 +0000 Subject: [PATCH] Deploy to GitHub Pages: 0dc75ab3ef6bcf07bc93e08f8fa4bcd02503cbc4 --- .../design/auto_gradient_check.md.txt | 146 ++++++ develop/doc/design/auto_gradient_check.html | 439 +++++++++++++++++ develop/doc/objects.inv | Bin 2379 -> 2411 bytes develop/doc/searchindex.js | 2 +- .../design/auto_gradient_check.md.txt | 146 ++++++ .../doc_cn/design/auto_gradient_check.html | 445 ++++++++++++++++++ develop/doc_cn/objects.inv | Bin 2870 -> 2900 bytes develop/doc_cn/searchindex.js | 2 +- 8 files changed, 1178 insertions(+), 2 deletions(-) create mode 100644 develop/doc/_sources/design/auto_gradient_check.md.txt create mode 100644 develop/doc/design/auto_gradient_check.html create mode 100644 develop/doc_cn/_sources/design/auto_gradient_check.md.txt create mode 100644 develop/doc_cn/design/auto_gradient_check.html diff --git a/develop/doc/_sources/design/auto_gradient_check.md.txt b/develop/doc/_sources/design/auto_gradient_check.md.txt new file mode 100644 index 00000000000..1f4d4ec16f7 --- /dev/null +++ b/develop/doc/_sources/design/auto_gradient_check.md.txt @@ -0,0 +1,146 @@ +## Auto Gradient Checker Design + +## Backgraound： +- Operator forward computing is easy to check if the result is right because it has a clear definition. **But** backpropagation is a notoriously difficult algorithm to debug and get right: + - 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. + +- Auto gradient check gets a numeric gradient by forward Operator and use it as a reference of the backward Operator's result. It has several advantages: + - 1. numeric gradient checker only need forward operator. + - 2. user only need to prepare the input data for forward Operator. + +## Mathematical Theory +The following two document from stanford has a detailed explanation of how to get numeric gradient and why it's useful. + +- [Gradient checking and advanced optimization(en)](http://deeplearning.stanford.edu/wiki/index.php/Gradient_checking_and_advanced_optimization) +- [Gradient checking and advanced optimization(cn)](http://ufldl.stanford.edu/wiki/index.php/%E6%A2%AF%E5%BA%A6%E6%A3%80%E9%AA%8C%E4%B8%8E%E9%AB%98%E7%BA%A7%E4%BC%98%E5%8C%96) + + +## Numeric Gradient Implementation +### Python Interface +```python +def get_numeric_gradient(op, + input_values, + output_name, + input_to_check, + delta=0.005, + local_scope=None): + """ + Get Numeric Gradient for an operator's input. + + :param op: C++ operator instance, could be an network + :param input_values: The input variables. Should be an dictionary, key is + variable name. Value is numpy array. + :param output_name: The final output variable name. + :param input_to_check: The input variable need to get gradient. + :param delta: The perturbation value for numeric gradient method. The + smaller delta is, the more accurate result will get. But if that delta is + too small, it could occur numerical stability problem. + :param local_scope: The local scope used for get_numeric_gradient. + :return: The gradient array in numpy format. + """ +``` + +### Explaination: + +- Why need `output_name` + - One Operator may have multiple Output, you can get independent gradient from each Output. So user should set one output to calculate. + +- Why need `input_to_check` + - One operator may have multiple inputs. Gradient Op can calculate the gradient of these Inputs at the same time. But Numeric Gradient needs to calculate them one by one. So `get_numeric_gradient` is designed to calculate the gradient for one input. If you need to compute multiple inputs, you can call `get_numeric_gradient` multiple times. + + +### Core Algorithm Implementation + + +```python + # we only compute gradient of one element each time. + # we use a for loop to compute the gradient of every element. + for i in xrange(tensor_size): + # get one input element throw it's index i. + origin = tensor_to_check.get_float_element(i) + + # add delta to it, run op and then get the sum of the result tensor. + x_pos = origin + delta + tensor_to_check.set_float_element(i, x_pos) + y_pos = get_output() + + # plus delta to this element, run op and get the sum of the result tensor. + x_neg = origin - delta + tensor_to_check.set_float_element(i, x_neg) + y_neg = get_output() + + # restore old value + tensor_to_check.set_float_element(i, origin) + + # compute the gradient of this element and store it into a numpy array. + gradient_flat[i] = (y_pos - y_neg) / delta / 2 + + # reshape the gradient result to the shape of the source tensor. + return gradient_flat.reshape(tensor_to_check.get_dims()) +``` + +## Auto Graident Checker Framework + +Each Operator Kernel has three kinds of Gradient: + +- 1. Numeric Gradient +- 2. CPU Operator Gradient +- 3. GPU Operator Gradient(if supported) + +Numeric Gradient Only relies on forward Operator. So we use Numeric Gradient as the reference value. + +- 1. calculate the numeric gradient. +- 2. calculate CPU kernel Gradient with the backward Operator and compare it with the numeric gradient. +- 3. calculate GPU kernel Gradient with the backward Operator and compare it with the numeric gradient.(if support GPU) + +#### Python Interface + +```python + def check_grad(self, + forward_op, + input_vars, + inputs_to_check, + output_name, + no_grad_set=None, + only_cpu=False, + max_relative_error=0.005): + """ + :param forward_op: used to create backward_op + :param input_vars: numpy value of input variable. The following + computation will use these variables. + :param inputs_to_check: inputs var names that should check gradient. + :param output_name: output name that used to + :param max_relative_error: The relative tolerance parameter. + :param no_grad_set: used when create backward ops + :param only_cpu: only compute and check gradient on cpu kernel. + :return: + """ +``` + +### How to check if two numpy array is close enough? +if `abs_numeric_grad` is nearly zero, then use abs error for numeric_grad, not relative + +```python +numeric_grad = ... +operator_grad = numpy.array(scope.find_var(grad_var_name(name)).get_tensor()) + +abs_numeric_grad = numpy.abs(numeric_grad) +# if abs_numeric_grad is nearly zero, then use abs error for numeric_grad, not relative +# error. +abs_numeric_grad[abs_numeric_grad < 1e-3] = 1 + +diff_mat = numpy.abs(abs_numeric_grad - operator_grad) / abs_numeric_grad +max_diff = numpy.max(diff_mat) +``` + + +#### Notes： +1，The Input data for auto gradient checker should be reasonable to avoid numeric problem. + + +#### Refs: + +- [Gradient checking and advanced optimization(en)](http://deeplearning.stanford.edu/wiki/index.php/Gradient_checking_and_advanced_optimization) +- [Gradient checking and advanced optimization(cn)](http://ufldl.stanford.edu/wiki/index.php/%E6%A2%AF%E5%BA%A6%E6%A3%80%E9%AA%8C%E4%B8%8E%E9%AB%98%E7%BA%A7%E4%BC%98%E5%8C%96) diff --git a/develop/doc/design/auto_gradient_check.html b/develop/doc/design/auto_gradient_check.html new file mode 100644 index 00000000000..67249ff6b9e --- /dev/null +++ b/develop/doc/design/auto_gradient_check.html @@ -0,0 +1,439 @@ + + + + + + + + + + + Auto Gradient Checker Design — PaddlePaddle documentation + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +
+ +
+
+ Fork me on Github +
+ + English + + + + +
+ +
+
+ + + + +
+
+ + + +
+
+
+
+
+ +
+ + + + +
+ + + + + + + + + + + +
+
    + +
  • Auto Gradient Checker Design
    • +
+
+ +
+
+
+
+ +
+

Auto Gradient Checker Design

+
+
+

Backgraound：

+
    +
  • Operator forward computing is easy to check if the result is right because it has a clear definition. But backpropagation is a notoriously difficult algorithm to debug and get right:
      +
      • +
      1. you should get the right backpropagation formula according to the forward computation.
        2. +
      +
      • +
      • +
      1. you should implement it right in CPP.
        2. +
      +
      • +
      • +
      1. it’s difficult to prepare test data.
        2. +
      +
      • +
    +
    • +
  • Auto gradient check gets a numeric gradient by forward Operator and use it as a reference of the backward Operator’s result. It has several advantages:
      +
      • +
      1. numeric gradient checker only need forward operator.
        2. +
      +
      • +
      • +
      1. user only need to prepare the input data for forward Operator.
        2. +
      +
      • +
    +
    • +
+
+
+

Mathematical Theory

+

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

+
+ +
+
+
+

Numeric Gradient Implementation

+
+

Python Interface

+
def get_numeric_gradient(op,
+                         input_values,
+                         output_name,
+                         input_to_check,
+                         delta=0.005,
+                         local_scope=None):
+    """
+    Get Numeric Gradient for an operator's input.
+
+    :param op: C++ operator instance, could be an network
+    :param input_values: The input variables. Should be an dictionary, key is
+    variable name. Value is numpy array.
+    :param output_name: The final output variable name.
+    :param input_to_check: The input variable need to get gradient.
+    :param delta: The perturbation value for numeric gradient method. The
+    smaller delta is, the more accurate result will get. But if that delta is
+     too small, it could occur numerical stability problem.
+    :param local_scope: The local scope used for get_numeric_gradient.
+    :return: The gradient array in numpy format.
+    """
+
+
+
+
+

Explaination:

+
    +
  • Why need output_name
      +
    • One Operator may have multiple Output, you can get independent gradient from each Output. So user should set one output to calculate.
      • +
    +
    • +
  • Why need input_to_check
      +
    • One operator may have multiple inputs. Gradient Op can calculate the gradient of these Inputs at the same time. But Numeric Gradient needs to calculate them one by one. So get_numeric_gradient is designed to calculate the gradient for one input. If you need to compute multiple inputs, you can call get_numeric_gradient multiple times.
      • +
    +
    • +
+
+
+

Core Algorithm Implementation

+
    # we only compute gradient of one element each time.
+    # we use a for loop to compute the gradient of every element.
+    for i in xrange(tensor_size):
+        # get one input element throw it's index i.
+        origin = tensor_to_check.get_float_element(i)
+
+        # add delta to it, run op and then get the sum of the result tensor.
+        x_pos = origin + delta
+        tensor_to_check.set_float_element(i, x_pos)
+        y_pos = get_output()
+
+        # plus delta to this element, run op and get the sum of the result tensor.
+        x_neg = origin - delta
+        tensor_to_check.set_float_element(i, x_neg)
+        y_neg = get_output()
+
+        # restore old value
+        tensor_to_check.set_float_element(i, origin)
+
+        # compute the gradient of this element and store it into a numpy array.
+        gradient_flat[i] = (y_pos - y_neg) / delta / 2
+
+    # reshape the gradient result to the shape of the source tensor.
+    return gradient_flat.reshape(tensor_to_check.get_dims())
+
+
+
+
+
+

Auto Graident Checker Framework

+

Each Operator Kernel has three kinds of Gradient:

+
    +
    • +
    1. Numeric Gradient
      2. +
    +
    • +
    • +
    1. CPU Operator Gradient
      2. +
    +
    • +
    • +
    1. GPU Operator Gradient(if supported)
      2. +
    +
    • +
+

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

+
    +
    • +
    1. calculate the numeric gradient.
      2. +
    +
    • +
    • +
    1. calculate CPU kernel Gradient with the backward Operator and compare it with the numeric gradient.
      2. +
    +
    • +
    • +
    1. calculate GPU kernel Gradient with the backward Operator and compare it with the numeric gradient.(if support GPU)
      2. +
    +
    • +
+
+

Python Interface

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

How to check if two numpy array is close enough?

+

if abs_numeric_grad is nearly zero, then use abs error for numeric_grad, not relative

+
numeric_grad = ...
+operator_grad = numpy.array(scope.find_var(grad_var_name(name)).get_tensor())
+
+abs_numeric_grad = numpy.abs(numeric_grad)
+# if abs_numeric_grad is nearly zero, then use abs error for numeric_grad, not relative
+# error.
+abs_numeric_grad[abs_numeric_grad < 1e-3] = 1
+
+diff_mat = numpy.abs(abs_numeric_grad - operator_grad) / abs_numeric_grad
+max_diff = numpy.max(diff_mat)
+
+
+
+

Notes：

+

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

+
+ +
+
+ + +
+
+
+ + +
+ +
+

+ © Copyright 2016, PaddlePaddle developers. + +

+
+ Built with Sphinx using a theme provided by Read the Docs. + +
+ +
+
+ +
+ +
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\ No newline at end of file diff --git a/develop/doc_cn/_sources/design/auto_gradient_check.md.txt b/develop/doc_cn/_sources/design/auto_gradient_check.md.txt new file mode 100644 index 00000000000..1f4d4ec16f7 --- /dev/null +++ b/develop/doc_cn/_sources/design/auto_gradient_check.md.txt @@ -0,0 +1,146 @@ +## Auto Gradient Checker Design + +## Backgraound： +- Operator forward computing is easy to check if the result is right because it has a clear definition. **But** backpropagation is a notoriously difficult algorithm to debug and get right: + - 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. + +- Auto gradient check gets a numeric gradient by forward Operator and use it as a reference of the backward Operator's result. It has several advantages: + - 1. numeric gradient checker only need forward operator. + - 2. user only need to prepare the input data for forward Operator. + +## Mathematical Theory +The following two document from stanford has a detailed explanation of how to get numeric gradient and why it's useful. + +- [Gradient checking and advanced optimization(en)](http://deeplearning.stanford.edu/wiki/index.php/Gradient_checking_and_advanced_optimization) +- [Gradient checking and advanced optimization(cn)](http://ufldl.stanford.edu/wiki/index.php/%E6%A2%AF%E5%BA%A6%E6%A3%80%E9%AA%8C%E4%B8%8E%E9%AB%98%E7%BA%A7%E4%BC%98%E5%8C%96) + + +## Numeric Gradient Implementation +### Python Interface +```python +def get_numeric_gradient(op, + input_values, + output_name, + input_to_check, + delta=0.005, + local_scope=None): + """ + Get Numeric Gradient for an operator's input. + + :param op: C++ operator instance, could be an network + :param input_values: The input variables. Should be an dictionary, key is + variable name. Value is numpy array. + :param output_name: The final output variable name. + :param input_to_check: The input variable need to get gradient. + :param delta: The perturbation value for numeric gradient method. The + smaller delta is, the more accurate result will get. But if that delta is + too small, it could occur numerical stability problem. + :param local_scope: The local scope used for get_numeric_gradient. + :return: The gradient array in numpy format. + """ +``` + +### Explaination: + +- Why need `output_name` + - One Operator may have multiple Output, you can get independent gradient from each Output. So user should set one output to calculate. + +- Why need `input_to_check` + - One operator may have multiple inputs. Gradient Op can calculate the gradient of these Inputs at the same time. But Numeric Gradient needs to calculate them one by one. So `get_numeric_gradient` is designed to calculate the gradient for one input. If you need to compute multiple inputs, you can call `get_numeric_gradient` multiple times. + + +### Core Algorithm Implementation + + +```python + # we only compute gradient of one element each time. + # we use a for loop to compute the gradient of every element. + for i in xrange(tensor_size): + # get one input element throw it's index i. + origin = tensor_to_check.get_float_element(i) + + # add delta to it, run op and then get the sum of the result tensor. + x_pos = origin + delta + tensor_to_check.set_float_element(i, x_pos) + y_pos = get_output() + + # plus delta to this element, run op and get the sum of the result tensor. + x_neg = origin - delta + tensor_to_check.set_float_element(i, x_neg) + y_neg = get_output() + + # restore old value + tensor_to_check.set_float_element(i, origin) + + # compute the gradient of this element and store it into a numpy array. + gradient_flat[i] = (y_pos - y_neg) / delta / 2 + + # reshape the gradient result to the shape of the source tensor. + return gradient_flat.reshape(tensor_to_check.get_dims()) +``` + +## Auto Graident Checker Framework + +Each Operator Kernel has three kinds of Gradient: + +- 1. Numeric Gradient +- 2. CPU Operator Gradient +- 3. GPU Operator Gradient(if supported) + +Numeric Gradient Only relies on forward Operator. So we use Numeric Gradient as the reference value. + +- 1. calculate the numeric gradient. +- 2. calculate CPU kernel Gradient with the backward Operator and compare it with the numeric gradient. +- 3. calculate GPU kernel Gradient with the backward Operator and compare it with the numeric gradient.(if support GPU) + +#### Python Interface + +```python + def check_grad(self, + forward_op, + input_vars, + inputs_to_check, + output_name, + no_grad_set=None, + only_cpu=False, + max_relative_error=0.005): + """ + :param forward_op: used to create backward_op + :param input_vars: numpy value of input variable. The following + computation will use these variables. + :param inputs_to_check: inputs var names that should check gradient. + :param output_name: output name that used to + :param max_relative_error: The relative tolerance parameter. + :param no_grad_set: used when create backward ops + :param only_cpu: only compute and check gradient on cpu kernel. + :return: + """ +``` + +### How to check if two numpy array is close enough? +if `abs_numeric_grad` is nearly zero, then use abs error for numeric_grad, not relative + +```python +numeric_grad = ... +operator_grad = numpy.array(scope.find_var(grad_var_name(name)).get_tensor()) + +abs_numeric_grad = numpy.abs(numeric_grad) +# if abs_numeric_grad is nearly zero, then use abs error for numeric_grad, not relative +# error. +abs_numeric_grad[abs_numeric_grad < 1e-3] = 1 + +diff_mat = numpy.abs(abs_numeric_grad - operator_grad) / abs_numeric_grad +max_diff = numpy.max(diff_mat) +``` + + +#### Notes： +1，The Input data for auto gradient checker should be reasonable to avoid numeric problem. + + +#### Refs: + +- [Gradient checking and advanced optimization(en)](http://deeplearning.stanford.edu/wiki/index.php/Gradient_checking_and_advanced_optimization) +- [Gradient checking and advanced optimization(cn)](http://ufldl.stanford.edu/wiki/index.php/%E6%A2%AF%E5%BA%A6%E6%A3%80%E9%AA%8C%E4%B8%8E%E9%AB%98%E7%BA%A7%E4%BC%98%E5%8C%96) diff --git a/develop/doc_cn/design/auto_gradient_check.html b/develop/doc_cn/design/auto_gradient_check.html new file mode 100644 index 00000000000..ea9e4b29d25 --- /dev/null +++ b/develop/doc_cn/design/auto_gradient_check.html @@ -0,0 +1,445 @@ + + + + + + + + + + + Auto Gradient Checker Design — PaddlePaddle 文档 + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +
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+ Fork me on Github +
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  • Auto Gradient Checker Design
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Auto Gradient Checker Design

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Backgraound：

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  • Operator forward computing is easy to check if the result is right because it has a clear definition. But backpropagation is a notoriously difficult algorithm to debug and get right:
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      1. you should get the right backpropagation formula according to the forward computation.
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      1. numeric gradient checker only need forward operator.
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Mathematical Theory

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

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Numeric Gradient Implementation

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Python Interface

+
def get_numeric_gradient(op,
+                         input_values,
+                         output_name,
+                         input_to_check,
+                         delta=0.005,
+                         local_scope=None):
+    """
+    Get Numeric Gradient for an operator's input.
+
+    :param op: C++ operator instance, could be an network
+    :param input_values: The input variables. Should be an dictionary, key is
+    variable name. Value is numpy array.
+    :param output_name: The final output variable name.
+    :param input_to_check: The input variable need to get gradient.
+    :param delta: The perturbation value for numeric gradient method. The
+    smaller delta is, the more accurate result will get. But if that delta is
+     too small, it could occur numerical stability problem.
+    :param local_scope: The local scope used for get_numeric_gradient.
+    :return: The gradient array in numpy format.
+    """
+
+
+
+
+

Explaination:

+
    +
  • Why need output_name
      +
    • One Operator may have multiple Output, you can get independent gradient from each Output. So user should set one output to calculate.
      • +
    +
    • +
  • Why need input_to_check
      +
    • One operator may have multiple inputs. Gradient Op can calculate the gradient of these Inputs at the same time. But Numeric Gradient needs to calculate them one by one. So get_numeric_gradient is designed to calculate the gradient for one input. If you need to compute multiple inputs, you can call get_numeric_gradient multiple times.
      • +
    +
    • +
+
+
+

Core Algorithm Implementation

+
    # we only compute gradient of one element each time.
+    # we use a for loop to compute the gradient of every element.
+    for i in xrange(tensor_size):
+        # get one input element throw it's index i.
+        origin = tensor_to_check.get_float_element(i)
+
+        # add delta to it, run op and then get the sum of the result tensor.
+        x_pos = origin + delta
+        tensor_to_check.set_float_element(i, x_pos)
+        y_pos = get_output()
+
+        # plus delta to this element, run op and get the sum of the result tensor.
+        x_neg = origin - delta
+        tensor_to_check.set_float_element(i, x_neg)
+        y_neg = get_output()
+
+        # restore old value
+        tensor_to_check.set_float_element(i, origin)
+
+        # compute the gradient of this element and store it into a numpy array.
+        gradient_flat[i] = (y_pos - y_neg) / delta / 2
+
+    # reshape the gradient result to the shape of the source tensor.
+    return gradient_flat.reshape(tensor_to_check.get_dims())
+
+
+
+
+
+

Auto Graident Checker Framework

+

Each Operator Kernel has three kinds of Gradient:

+
    +
    • +
    1. Numeric Gradient
      2. +
    +
    • +
    • +
    1. CPU Operator Gradient
      2. +
    +
    • +
    • +
    1. GPU Operator Gradient(if supported)
      2. +
    +
    • +
+

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

+
    +
    • +
    1. calculate the numeric gradient.
      2. +
    +
    • +
    • +
    1. calculate CPU kernel Gradient with the backward Operator and compare it with the numeric gradient.
      2. +
    +
    • +
    • +
    1. calculate GPU kernel Gradient with the backward Operator and compare it with the numeric gradient.(if support GPU)
      2. +
    +
    • +
+
+

Python Interface

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

How to check if two numpy array is close enough?

+

if abs_numeric_grad is nearly zero, then use abs error for numeric_grad, not relative

+
numeric_grad = ...
+operator_grad = numpy.array(scope.find_var(grad_var_name(name)).get_tensor())
+
+abs_numeric_grad = numpy.abs(numeric_grad)
+# if abs_numeric_grad is nearly zero, then use abs error for numeric_grad, not relative
+# error.
+abs_numeric_grad[abs_numeric_grad < 1e-3] = 1
+
+diff_mat = numpy.abs(abs_numeric_grad - operator_grad) / abs_numeric_grad
+max_diff = numpy.max(diff_mat)
+
+
+
+

Notes：

+

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

+
+ +
+
+ + +
+
+
+ + +
+ +
+

+ © Copyright 2016, PaddlePaddle developers. + +

+
+ Built with Sphinx using a theme provided by Read the Docs. + +
+ +
+
+ +
+ +
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