
.. _graphstructures:

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

Debugging or profiling code written in Aesara 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 Aesara.

The first step in writing Aesara 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.

Aesara represents symbolic mathematical computations as graphs. These
graphs are composed of interconnected :ref:`apply`, :ref:`variable` and
:ref:`op` nodes. *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.
Furthermore, 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**

.. testcode::

   import aesara.tensor as aet

   x = aet.dmatrix('x')
   y = aet.dmatrix('y')
   z = x + y

**Diagram**

.. _tutorial-graphfigure:

.. image:: apply.png
    :align: center


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>`.

.. TODO
    Clarify the 'acyclic' graph and the 'back' pointers or references that
    'don't count'.

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.


Traversing the graph
====================

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

If you enter ``type(y.owner)`` you get ``<class 'aesara.graph.basic.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]
InplaceDimShuffle{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 'aesara.tensor.var.TensorVariable'>
>>> type(y.owner.inputs[1].owner)
<class 'aesara.graph.basic.Apply'>
>>> y.owner.inputs[1].owner.op # doctest: +SKIP
<aesara.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.


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

The following section outlines each type of structure that may be used
in an Aesara-built computation graph. The following structures are
explained: :ref:`apply`, :ref:`constant`, :ref:`op`, :ref:`variable` and
:ref:`type`.


.. index::
   single: Apply
   single: graph construct; Apply

.. _apply:

Apply
-----

An *Apply node* is a type of internal node used to represent a
:term:`computation graph <graph>` in Aesara. Unlike
:ref:`Variable nodes <variable>`, Apply nodes are usually not
manipulated directly by the end user. They may be accessed via
a Variable's ``owner`` field.

An Apply node is typically an instance of the :class:`Apply`
class. It represents the application
of an :ref:`op` on one or more inputs, where each input is a
:ref:`variable`. By convention, each Op is responsible for
knowing how to build an Apply node from a list of
inputs. Therefore, an Apply node may be obtained from an Op
and a list of inputs by calling ``Op.make_node(*inputs)``.

Comparing with the Python language, an :ref:`apply` node is
Aesara's version of a function call whereas an :ref:`op` is
Aesara's version of a function definition.

An Apply instance has three important fields:

**op**
  An :ref:`op` that determines the function/transformation being
  applied here.

**inputs**
  A list of :ref:`Variables <variable>` that represent the arguments of
  the function.

**outputs**
  A list of :ref:`Variables <variable>` that represent the return values
  of the function.

An Apply instance can be created by calling ``graph.basic.Apply(op, inputs, outputs)``.



.. index::
   single: Op
   single: graph construct; Op

.. _op:


Op
--

An :ref:`op` in Aesara defines a certain computation on some types of
inputs, producing some types of outputs. It is equivalent to a
function definition in most programming languages. From a list of
input :ref:`Variables <variable>` and an Op, you can build an :ref:`apply`
node representing the application of the Op to the inputs.

It is important to understand the distinction between an Op (the
definition of a function) and an Apply node (the application of a
function). If you were to interpret the Python language using Aesara's
structures, code going like ``def f(x): ...`` would produce an Op for
``f`` whereas code like ``a = f(x)`` or ``g(f(4), 5)`` would produce an
Apply node involving the ``f`` Op.


.. index::
   single: Type
   single: graph construct; Type

.. _type:


Type
----

A :ref:`type` in Aesara represents a set of constraints on potential
data objects. These constraints allow Aesara to tailor C code to handle
them and to statically optimize the computation graph. For instance,
the :ref:`irow <libdoc_tensor_creation>` type in the ``aesara.tensor`` package
gives the following constraints on the data the Variables of type ``irow``
may contain:

#. Must be an instance of ``numpy.ndarray``: ``isinstance(x, numpy.ndarray)``
#. Must be an array of 32-bit integers: ``str(x.dtype) == 'int32'``
#. Must have a shape of 1xN: ``len(x.shape) == 2 and x.shape[0] == 1``

Knowing these restrictions, Aesara may generate C code for addition, etc.
that declares the right data types and that contains the right number
of loops over the dimensions.

Note that an Aesara :ref:`type` is not equivalent to a Python type or
class. Indeed, in Aesara, :ref:`irow <libdoc_tensor_creation>` and :ref:`dmatrix
<libdoc_tensor_creation>` both use ``numpy.ndarray`` as the underlying type
for doing computations and storing data, yet they are different Aesara
Types. Indeed, the constraints set by ``dmatrix`` are:

#. Must be an instance of ``numpy.ndarray``: ``isinstance(x, numpy.ndarray)``
#. Must be an array of 64-bit floating point numbers: ``str(x.dtype) == 'float64'``
#. Must have a shape of MxN, no restriction on M or N: ``len(x.shape) == 2``

These restrictions are different from those of ``irow`` which are listed above.

There are cases in which a Type can fully correspond to a Python type,
such as the ``double`` Type we will define here, which corresponds to
Python's ``float``. But, it's good to know that this is not necessarily
the case. Unless specified otherwise, when we say "Type" we mean a
Aesara Type.


.. index::
   single: Variable
   single: graph construct; Variable

.. _variable:



Variable
--------

A :ref:`variable` is the main data structure you work with when using
Aesara. The symbolic inputs that you operate on are Variables and what
you get from applying various Ops to these inputs are also
Variables. For example, when I type

>>> import aesara
>>> x = aesara.tensor.ivector()
>>> y = -x

``x`` and ``y`` are both Variables, i.e. instances of the :class:`Variable` class. The :ref:`type` of both ``x`` and
``y`` is ``aesara.tensor.ivector``.

Unlike ``x``, ``y`` is a Variable produced by a computation (in this
case, it is the negation of ``x``). ``y`` is the Variable corresponding to
the output of the computation, while ``x`` is the Variable
corresponding to its input. The computation itself is represented by
another type of node, an :ref:`apply` node, and may be accessed
through ``y.owner``.

More specifically, a Variable is a basic structure in Aesara that
represents a datum at a certain point in computation. It is typically
an instance of the class :class:`Variable` or
one of its subclasses.

A Variable ``r`` contains four important fields:

**type**
  a :ref:`type` defining the kind of value this Variable can hold in
  computation.

**owner**
  this is either None or an :ref:`apply` node of which the Variable is
  an output.

**index**
  the integer such that ``owner.outputs[index] is r`` (ignored if
  ``owner`` is None)

**name**
  a string to use in pretty-printing and debugging.

Variable has one special subclass: :ref:`Constant <constant>`.

.. index::
   single: Constant
   single: graph construct; Constant

.. _constant:


Constant
^^^^^^^^

A Constant is a :ref:`Variable` with one extra field, *data* (only
settable once). When used in a computation graph as the input of an
:ref:`Op` :ref:`application <Apply>`, it is assumed that said input
will *always* take the value contained in the constant's data
field. Furthermore, it is assumed that the :ref:`Op` will not under
any circumstances modify the input. This means that a constant is
eligible to participate in numerous optimizations: constant inlining
in C code, constant folding, etc.

A constant does not need to be specified in a :func:`function
<function.function>`'s list
of inputs.  In fact, doing so will raise an exception.



Graph Structures Extension
==========================

When we start the compilation of an Aesara function, we compute some
extra information. This section describes a portion of the information
that is made available.

The graph gets cloned at the start of compilation, so modifications done
during compilation won't affect the user graph.

Each variable receives a new field called clients. It is a list with
references to every place in the graph where this variable is used. If
its length is 0, it means the variable isn't used. Each place where it
is used is described by a tuple of 2 elements. There are two types of
pairs:

- The first element is an Apply node.
- The first element is the string "output". It means the
  function outputs this variable.

In both types of pairs, the second element of the tuple is an index,
such that: ``fgraph.clients[var][*][0].inputs[index]`` or
``fgraph.outputs[index]`` is that variable.


>>> import aesara
>>> v = aesara.tensor.vector()
>>> f = aesara.function([v], (v+1).sum())
>>> aesara.printing.debugprint(f)
Sum{acc_dtype=float64} [id A] ''   1
 |Elemwise{add,no_inplace} [id B] ''   0
   |TensorConstant{(1,) of 1.0} [id C]
   |<TensorType(float64, vector)> [id D]
>>> # Sorted list of all nodes in the compiled graph.
>>> fgraph = f.maker.fgraph
>>> topo = fgraph.toposort()
>>> fgraph.clients[topo[0].outputs[0]]
[(Sum{acc_dtype=float64}(Elemwise{add,no_inplace}.0), 0)]
>>> fgraph.clients[topo[1].outputs[0]]
[('output', 0)]

>>> # An internal variable
>>> var = topo[0].outputs[0]
>>> client = fgraph.clients[var][0]
>>> client
(Sum{acc_dtype=float64}(Elemwise{add,no_inplace}.0), 0)
>>> type(client[0])
<class 'aesara.graph.basic.Apply'>
>>> assert client[0].inputs[client[1]] is var

>>> # An output of the graph
>>> var = topo[1].outputs[0]
>>> client = fgraph.clients[var][0]
>>> client
('output', 0)
>>> assert fgraph.outputs[client[1]] is var


Automatic Differentiation
=========================

Having the graph structure, computing automatic differentiation is
simple. The only thing :func:`aesara.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 an Aesara function, what you give to the
:func:`aesara.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
Aesara 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 Aesara 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 aesara
>>> a = aesara.tensor.vector("a")      # declare symbolic variable
>>> b = a + a ** 10                    # build symbolic expression
>>> f = aesara.function([a], b)        # compile function
>>> print(f([0, 1, 2]))                # prints `array([0,2,1026])`
[    0.     2.  1026.]
>>> aesara.printing.pydotprint(b, outfile="./pics/symbolic_graph_unopt.png", var_with_name_simple=True)  # doctest: +SKIP
The output file is available at ./pics/symbolic_graph_unopt.png
>>> aesara.printing.pydotprint(f, outfile="./pics/symbolic_graph_opt.png", var_with_name_simple=True)  # doctest: +SKIP
The output file is available at ./pics/symbolic_graph_opt.png

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

.. |g1| image:: ./pics/symbolic_graph_unopt.png
        :width: 500 px
.. |g2| image:: ./pics/symbolic_graph_opt.png
        :width: 500 px

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