
.. _tutorial_graphstructures:

================
Graph Structures
================


Theano Graphs
=============

Debugging or profiling code written in Theano is not that simple if you
do not know what goes on under the hood. This chapter is meant to
introduce you to a required minimum of the inner workings of Theano.  
For more detail see :ref:`extending`.

The first step in writing Theano code is to write down all mathematical 
relations using symbolic placeholders (**variables**). When writing down 
these expressions you use operations like ``+``, ``-``, ``**``,
``sum()``, ``tanh()``. All these are represented internally as **ops**. 
An *op* represents a certain computation on some type of inputs
producing some type of output. You can see it as a *function definition*
in most programming languages. 

Theano builds internally a graph structure composed of interconnected 
**variable** nodes, **op** nodes and **apply** nodes. An 
*apply* node represents the application of an *op* to some 
*variables*. It is important to draw the difference between the
definition of a computation represented by an *op* and its application
to some actual data which is represented by the *apply* node. For more
detail about these building blocks refer to :ref:`variable`, :ref:`op`, 
:ref:`apply`. Here is an example of a graph:


**Code**

.. code-block:: python

    x = T.dmatrix('x')
    y = T.dmatrix('y')
    z = x + y

**Diagram**

.. _tutorial-graphfigure: 

.. figure:: apply.png 
    :align: center

    Interaction between instances of Apply (blue), Variable (red), Op (green),
    and Type (purple).

.. # COMMENT
    WARNING: hyper-links and ref's seem to break the PDF build when placed
    into this figure caption.

Arrows in this figure 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>`.


The graph can be traversed starting from outputs (the result of some
computation) down to its inputs using the owner field.
Take for example the following code:

>>> import theano
>>> x = theano.tensor.dmatrix('x')
>>> y = x * 2.

If you enter ``type(y.owner)`` you get ``<class 'theano.gof.graph.Apply'>``, 
which is the apply node that connects the op and the inputs to get this
output. You can now print the name of the op that is applied to get 
*y*:

>>> y.owner.op.name
'Elemwise{mul,no_inplace}'

Hence, an elementwise multiplication is used to compute *y*. This
multiplication is done between the inputs:

>>> len(y.owner.inputs)
2
>>> y.owner.inputs[0]
x
>>> y.owner.inputs[1]
DimShuffle{x,x}.0

Note that the second input is not 2 as we would have expected. This is 
because 2 was first :term:`broadcasted <broadcasting>` to a matrix of 
same shape as *x*. This is done by using the op ``DimShuffle`` :

>>> type(y.owner.inputs[1])
<class 'theano.tensor.var.TensorVariable'>
>>> type(y.owner.inputs[1].owner)
<class 'theano.gof.graph.Apply'>
>>> y.owner.inputs[1].owner.op # doctest: +SKIP
<theano.tensor.elemwise.DimShuffle object at 0x106fcaf10>
>>> y.owner.inputs[1].owner.inputs
[TensorConstant{2.0}]


Starting from this graph structure it is easier to understand how 
*automatic differentiation* proceeds and how the symbolic relations
can be *optimized* for performance or stability.  


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 *apply*
nodes (*apply* nodes are those that define which computations the
graph does). For each such *apply* node, its *op* defines 
how to compute the *gradient* of the node's outputs with respect to its
inputs. Note that if an *op* does not provide this information, 
it is assumed that the *gradient* is not defined.
Using the 
`chain rule <http://en.wikipedia.org/wiki/Chain_rule>`_ 
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 .

A following section of this tutorial will examine the topic of :ref:`differentiation<tutcomputinggrads>`
in greater detail.


Optimizations
=============

When compiling a Theano function, what you give to the
:func:`theano.function <function.function>` is actually a graph
(starting from the output 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 possibility to improve the  
way this computation is carried out. The way optimizations work in 
Theano is by identifying 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 *x.*

Further information regarding the optimization
:ref:`process<optimization>` and the specific :ref:`optimizations<optimizations>` that are applicable
is respectively available in the library and on the entrance page of the documentation.  


**Example**

Symbolic programming involves a change of paradigm: it will become clearer
as we apply it. Consider the following example of optimization:

>>> import theano
>>> a = theano.tensor.vector("a")      # declare symbolic variable
>>> b = a + a ** 10                    # build symbolic expression
>>> f = theano.function([a], b)        # compile function
>>> print f([0, 1, 2])                 # prints `array([0,2,1026])`
[    0.     2.  1026.]
>>> theano.printing.pydotprint(b, outfile="./pics/symbolic_graph_unopt.png", var_with_name_simple=True)
The output file is available at ./pics/symbolic_graph_unopt.png
>>> theano.printing.pydotprint(f, outfile="./pics/symbolic_graph_opt.png", var_with_name_simple=True)
The output file is available at ./pics/symbolic_graph_opt.png


.. |g1| image:: ./pics/symbolic_graph_unopt.png
    :width: 500 px

.. |g2| image:: ./pics/symbolic_graph_opt.png
    :width: 500 px

We used :func:`theano.printing.pydotprint` to visualize the optimized graph
(right), which is much more compact than the unoptimized graph (left).

======================================================  =====================================================
        Unoptimized graph                                    Optimized graph
======================================================  =====================================================
|g1|                                                    |g2|
======================================================  =====================================================

