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doc/tutorial/symbolic_graphs.txt 0 → 100644
浏览文件 @ ef0085ff
.. _symbolic_graph:
================
Graph Structures
================
In order to be able to take advantage of Theano, you need to understand
how Theano works. Theano represents mathematical computations as graphs
( for a detailed rendering see :ref:`graphstructures` - parts of this
are directly taken from there). Graphs are composed of itnerconnected
:ref:`apply` and :ref:`variable` nodes. They are associated to *function
application* and *data*, respectively. An operation is represented by
an :ref:`op` and data types are represented by :ref:`type` instances.
Here is a piece of code and a diagram showing the structure built by
that piece of code. This should help you understand how these pieces fit
together:
-----------------------
**Code**
.. code-block:: python
x = dmatrix('x')
y = dmatrix('y')
z = x + y
**Diagram**
.. image:: apply.png
-----------------------
Arrows represent references to the Python objects pointed at. The blue
box is an :ref:`apply` node. Red boxes are :ref:`variable` nodes. Green
circles are :ref:`Ops <op>`. Purple boxes are :ref:`Types <type>`.
When we create :ref:`Variables <variable>` and then :ref:`apply`
:ref:`Ops <op>` to them to make more Variables, we build a
bi-partite, directed, acyclic graph. Variables point to the Apply nodes
representing the function application producing them via their
``owner`` field. These Apply nodes point in turn to their input and
output Variables via their ``inputs`` and ``outputs`` fields.
(Apply instances also contain a list of references to their ``outputs``, but
those pointers don't count in this graph.)
The ``owner`` field of both ``x`` and ``y`` point to ``None`` because
they are not the result of another computation. If one of them was the
result of another computation, it's ``owner`` field would point to another
blue box like ``z`` does, and so on.
Note that the ``Apply`` instance's outputs points to
``z``, and ``z.owner`` points back to the ``Apply`` instance.
The graph structure is needed for *Optimizations* and *Automatic
Differentiation*.
Automatic Differentiation
=========================
Having the graph structure, computing automatic differentiation is
simple. The only thing :func:`tensor.grad` has to do is to traverse the
graph from the outputs back towards the inputs through all :ref:`apply`
nodes ( :ref:`apply` nodes are those who define what computations the
graph does). For each such :ref:`apply` node, its :ref:`op` defines
how to compute the gradient of the node's outputs with respect to its
inputs. Note that if an :ref:`op` does not define how to compute the
gradient, then any expression containing this :ref:`op` is not
differentiable. Using the `chain rule <http://en.wikipedia.org/wiki/Chain_rile>`_
these gradients can be composed in order to obtain the expression of the
gradient of the graph's output with respect to the graph's inputs .
Optimizations
=============
When compiling a Theano function, what you give to the
:ref:`theano.function <libdoc_compile_function>` is actually a graph
(starting from the outputs variables you can traverse the graph up to
the input variables). While this graph structure shows how to compute
the output from the input, it also offers the posibility to improve the
the way this computation is carried out. The way optimizations work in
Theano is by indentifying and replacing certain patterns in the graph
with other specialized patterns that produce the same results but are either
faster or more stable. Optimizations can also detect
identical subgraphs and ensure that the same values are not computed
twice or reformulate parts of the graph to a GPU specific version.
For example, one (simple) optimization that Theano uses is to replace
the pattern :math:`\frac{xy}{y}` by :math:`x`.
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