
.. _optimization:

==================
Graph optimization
==================

In this section we will define a couple optimizations on doubles.

.. todo::

   This tutorial goes way too far under the hood, for someone who just wants
   to add yet another pattern to the libraries in tensor.opt for example.

   We need another tutorial that covers the decorator syntax, and explains how
   to register your optimization right away.  That's what you need to get
   going.

   Later, the rest is more useful for when that decorator syntax type thing
   doesn't work. (There are optimizations that don't fit that model).



Global and local optimizations
==============================

First, let's lay out the way optimizations work in Theano. There are
two types of optimizations: *global* optimizations and *local*
optimizations. A global optimization takes an ``Env`` object (an
Env is a wrapper around a whole computation graph, you can see its
:class:`documentation <Env>` for more details) and navigates through it
in a suitable way, replacing some Variables by others in the process. A
local optimization, on the other hand, is defined as a function on a
*single* :ref:`apply` node and must return either ``False`` (to mean that
nothing is to be done) or a list of new Variables that we would like to
replace the node's outputs with. A :ref:`navigator` is a special kind
of global optimization which navigates the computation graph in some
fashion (in topological order, reverse-topological order, random
order, etc.) and applies one or more local optimizations at each step.

Optimizations which are holistic, meaning that they must take into
account dependencies that might be all over the graph, should be
global. Optimizations that can be done with a narrow perspective are
better defined as local optimizations. The majority of optimizations
we want to define are local.

.. optimizer:

Global optimization
-------------------

A global optimization (or optimizer) is an object which defines the following
methods:

.. class:: Optimizer

    .. method:: apply(env)

      This method takes an Env object which contains the computation graph
      and does modifications in line with what the optimization is meant
      to do. This is of the main method of the optimizer.

    .. method:: add_requirements(env)

      This method takes an Env object and adds :ref:`features
      <libdoc_gof_envfeature>` to it. These features are "plugins" that are needed
      for the ``apply`` method to do its job properly.

    .. method:: optimize(env)

      This is the interface function called by Theano.

      *Default:* this is defined by Optimizer as ``add_requirement(env);
      apply(env)``.

See the section about :class:`Env` to understand how to define these
methods.


Local optimization
------------------

A local optimization is an object which defines the following methods:

.. class:: LocalOptimizer

    .. method:: transform(node)

      This method takes an :ref:`apply` node and returns either ``False`` to
      signify that no changes are to be done or a list of Variables which
      matches the length of the node's ``outputs`` list. When the
      LocalOptimizer is applied by a Navigator, the outputs of the node
      passed as argument to the LocalOptimizer will be replaced by the
      list returned.



One simplification rule
=======================

For starters, let's define the following simplification:

.. math::

   \frac{xy}{y} = x

We will implement it in three ways: using a global optimization, a
local optimization with a Navigator and then using the PatternSub
facility.


Global optimization
-------------------

Here is the code for a global optimization implementing the
simplification described above:

.. code-block:: python

   from theano.gof import toolbox

   class Simplify(gof.Optimizer):
       def add_requirements(self, env):
           env.extend(toolbox.ReplaceValidate())
       def apply(self, env):
           for node in env.toposort():
               if node.op == div:
                   x, y = node.inputs
                   z = node.outputs[0]
                   if x.owner and x.owner.op == mul:
                       a, b = x.owner.inputs
                       if y == a:
                           env.replace_validate(z, b)
                       elif y == b:
                           env.replace_validate(z, a)

   simplify = Simplify()

.. todo::

   What is add_requirements? Why would we know to do this? Are there other
   requirements we might want to  know about?

Here's how it works: first, in ``add_requirements``, we add the
``ReplaceValidate`` :ref:`libdoc_gof_envfeature` located in
:ref:`libdoc_gof_toolbox`. This feature adds the ``replace_validate``
method to ``env``, which is an enhanced version of ``replace`` that
does additional checks to ensure that we are not messing up the
computation graph (note: if ``ReplaceValidate`` was already added by
another optimizer, ``extend`` will do nothing). In a nutshell,
``toolbox.ReplaceValidate`` grants access to ``env.replace_validate``,
and ``env.replace_validate`` allows us to replace a Variable with
another while respecting certain validation constraints. You can
browse the list of :ref:`libdoc_gof_envfeaturelist` and see if some of
them might be useful to write optimizations with. For example, as an
exercise, try to rewrite Simplify using :class:`NodeFinder`. (Hint: you
want to use the method it publishes instead of the call to toposort!)

Then, in ``apply`` we do the actual job of simplification. We start by
iterating through the graph in topological order. For each node
encountered, we check if it's a ``div`` node. If not, we have nothing
to do here. If so, we put in ``x``, ``y`` and ``z`` the numerator,
denominator and quotient (output) of the division.
The simplification only occurs when the numerator is a multiplication,
so we check for that. If the numerator is a multiplication we put the
two operands in ``a`` and ``b``, so
we can now say that ``z == (a*b)/y``. If ``y==a`` then ``z==b`` and if
``y==b`` then ``z==a``. When either case happens then we can replace
``z`` by either ``a`` or ``b`` using ``env.replace_validate`` - else we do
nothing. You might want to check the documentation about :ref:`variable`
and :ref:`apply` to get a better understanding of the
pointer-following game you need to get ahold of the nodes of interest
for the simplification (``x``, ``y``, ``z``, ``a``, ``b``, etc.).


Test time:

>>> x = double('x')
>>> y = double('y')
>>> z = double('z')
>>> a = add(z, mul(div(mul(y, x), y), div(z, x)))
>>> e = gof.Env([x, y, z], [a])
>>> e
[add(z, mul(div(mul(y, x), y), div(z, x)))]
>>> simplify.optimize(e)
>>> e
[add(z, mul(x, div(z, x)))]

Cool! It seems to work. You can check what happens if you put many
instances of :math:`\frac{xy}{y}` in the graph. Note that it sometimes
won't work for reasons that have nothing to do with the quality of the
optimization you wrote. For example, consider the following:

>>> x = double('x')
>>> y = double('y')
>>> z = double('z')
>>> a = div(mul(add(y, z), x), add(y, z))
>>> e = gof.Env([x, y, z], [a])
>>> e
[div(mul(add(y, z), x), add(y, z))]
>>> simplify.optimize(e)
>>> e
[div(mul(add(y, z), x), add(y, z))]

Nothing happened here. The reason is: ``add(y, z) != add(y,
z)``. That is the case for efficiency reasons. To fix this problem we
first need to merge the parts of the graph that represent the same
computation, using the ``merge_optimizer`` defined in
``theano.gof.opt``.

>>> from theano.gof.opt import merge_optimizer
>>> merge_optimizer.optimize(e)
>>> e
[div(mul(*1 -> add(y, z), x), *1)]
>>> simplify.optimize(e)
>>> e
[x]

Once the merge is done, both occurrences of ``add(y, z)`` are
collapsed into a single one and is used as an input in two
places. Note that ``add(x, y)`` and ``add(y, x)`` are still considered
to be different because Theano has no clue that ``add`` is
commutative. You may write your own global optimizer to identify
computations that are identical with full knowledge of the rules of
arithmetics that your Ops implement. Theano might provide facilities
for this somewhere in the future.

.. note::
   :class:`Env` is a Theano structure intended for the optimization
   phase. It is used internally by function and Module and is rarely
   exposed to the end user. You can use it to test out optimizations,
   etc. if you are comfortable with it, but it is recommended to use
   the function/Module frontends and to interface optimizations with
   :class:`optdb` (we'll see how to do that soon).


Local optimization
------------------

The local version of the above code would be the following:


.. code-block:: python

   class LocalSimplify(gof.LocalOptimizer):
       def transform(self, node):
           if node.op == div:
               x, y = node.inputs
               if x.owner and x.owner.op == mul:
                   a, b = x.owner.inputs
                   if y == a:
                       return [b]
                   elif y == b:
                       return [a]
           return False
       def tracks(self):
           # This should be needed for the EquilibriumOptimizer
           # but it isn't now
           # TODO: do this and explain it
           return [] # that's not what you should do

   local_simplify = LocalSimplify()

.. todo::

    Fix up previous example... it's bad and incomplete.

The definition of transform is the inner loop of the global optimizer,
where the node is given as argument. If no changes are to be made,
``False`` must be returned. Else, a list of what to replace the node's
outputs with must be returned.

In order to apply the local optimizer we must use it in conjunction
with a :ref:`navigator`. Basically, a :ref:`navigator` is a global
optimizer that loops through all nodes in the graph (or a well-defined
subset of them) and applies one or several local optimizers on them.

>>> x = double('x')
>>> y = double('y')
>>> z = double('z')
>>> a = add(z, mul(div(mul(y, x), y), div(z, x)))
>>> e = gof.Env([x, y, z], [a])
>>> e
[add(z, mul(div(mul(y, x), y), div(z, x)))]
>>> simplify = gof.TopoOptimizer(local_simplify)
>>> simplify.optimize(e)
>>> e
[add(z, mul(x, div(z, x)))]


OpSub, OpRemove, PatternSub
+++++++++++++++++++++++++++

Theano defines some shortcuts to make LocalOptimizers:

.. function:: OpSub(op1, op2)

  Replaces all uses of *op1* by *op2*. In other
  words, the outputs of all :ref:`apply` involving *op1* by the outputs
  of Apply nodes involving *op2*, where their inputs are the same.

.. function:: OpRemove(op)

  Removes all uses of *op* in the following way:
  if ``y = op(x)`` then ``y`` is replaced by ``x``. *op* must have as many
  outputs as it has inputs. The first output becomes the first input,
  the second output becomes the second input, and so on.

.. function:: PatternSub(pattern1, pattern2)

  Replaces all occurrences of the first pattern by the second pattern.
  See :class:`PatternSub`.


.. code-block:: python

   from theano.gof.opt import OpSub, OpRemove, PatternSub

   # Replacing add by mul (this is not recommended for primarily
   # mathematical reasons):
   add_to_mul = OpSub(add, mul)

   # Removing identity
   remove_identity = OpRemove(identity)

   # The "simplify" operation we've been defining in the past few
   # sections. Note that we need two patterns to account for the
   # permutations of the arguments to mul.
   local_simplify_1 = PatternSub((div, (mul, 'x', 'y'), 'y'),
                                 'x')
   local_simplify_2 = PatternSub((div, (mul, 'x', 'y'), 'x'),
                                 'y')

.. note::

   ``OpSub``, ``OpRemove`` and ``PatternSub`` produce local optimizers, which
   means that everything we said previously about local optimizers
   apply: they need to be wrapped in a Navigator, etc.

.. todo::

   wtf is a navigator?

When an optimization can be naturally expressed using ``OpSub``, ``OpRemove``
or ``PatternSub``, it is highly recommended to use them.

WRITEME: more about using PatternSub (syntax for the patterns, how to
use constraints, etc. - there's some decent doc at
:class:`PatternSub` for those interested)



.. _optdb:

The optimization database (optdb)
=================================

Theano exports a symbol called ``optdb`` which acts as a sort of
ordered database of optimizations. When you make a new optimization,
you must insert it at the proper place in the database. Furthermore,
you can give each optimization in the database a set of tags that can
serve as a basis for filtering.

The point of optdb is that you might want to apply many optimizations
to a computation graph in many unique patterns. For example, you might
want to do optimization X, then optimization Y, then optimization
Z. And then maybe optimization Y is an EquilibriumOptimizer containing
LocalOptimizers A, B and C which are applied on every node of the
graph until they all fail to change it. If some optimizations act up,
we want an easy way to turn them off. Ditto if some optimizations are
very CPU-intensive and we don't want to take the time to apply them.

The optdb system allows us to tag each optimization with a unique name
as well as informative tags such as 'stable', 'buggy' or
'cpu_intensive', all this without compromising the structure of the
optimizations.


Definition of optdb
-------------------

optdb is an object which is an instance of
:class:`SequenceDB <optdb.SequenceDB>`,
itself a subclass of :class:`DB <optdb.DB>`.
There exist (for now) two types of DB, SequenceDB and EquilibriumDB.
When given an appropriate Query, DB objects build an Optimizer matching
the query.

A SequenceDB contains Optimizer or DB objects. Each of them has a
name, an arbitrary number of tags and an integer representing their
order in the sequence. When a Query is applied to a SequenceDB, all
Optimizers whose tags match the query are inserted in proper order in
a SequenceOptimizer, which is returned. If the SequenceDB contains DB
instances, the Query will be passed to them as well and the optimizers
they return will be put in their places.

An EquilibriumDB contains LocalOptimizer or DB objects. Each of them
has a name and an arbitrary number of tags. When a Query is applied to
an EquilibriumDB, all LocalOptimizers that match the query are
inserted into an EquilibriumOptimizer, which is returned. If the
SequenceDB contains DB instances, the Query will be passed to them as
well and the LocalOptimizers they return will be put in their places
(note that as of yet no DB can produce LocalOptimizer objects, so this
is a moot point).

Theano contains one principal DB object, :class:`optdb`, which
contains all of Theano's optimizers with proper tags. It is
recommended to insert new Optimizers in it. As mentioned previously,
optdb is a SequenceDB, so, at the top level, Theano applies a sequence
of global optimizations to the computation graphs.


Query
-----

A Query is built by the following call:

.. code-block:: python

   theano.gof.Query(include, require = None, exclude = None, subquery = None)

.. class:: Query

    .. attribute:: include

       A set of tags (a tag being a string) such that every
       optimization obtained through this Query must have **one** of the tags
       listed. This field is required and basically acts as a starting point
       for the search.

    .. attribute:: require

       A set of tags such that every optimization obtained
       through this Query must have **all** of these tags.

    .. attribute:: exclude

       A set of tags such that every optimization obtained
       through this Query must have **none** of these tags.

    .. attribute:: subquery

       optdb can contain sub-databases; subquery is a
       dictionary mapping the name of a sub-database to a special Query.
       If no subquery is given for a sub-database, the original Query will be
       used again.

Furthermore, a Query object includes three methods, ``including``,
``requiring`` and ``excluding`` which each produce a new Query object
with include, require and exclude sets refined to contain the new [WRITEME]


Examples
--------

Here are a few examples of how to use a Query on optdb to produce an
Optimizer:

.. code-block:: python

   # This is how the optimizer for the fast_run mode is defined
   fast_run = optdb.query(Query(include = ['fast_run']))

   # This is how the optimizer for the fast_compile mode is defined
   fast_compile = optdb.query(Query(include = ['fast_compile']))

   # This is the same as fast_run but no optimizations will replace
   # any operation by an inplace version. This assumes, of course,
   # that all inplace operations are tagged as 'inplace' (as they
   # should!)
   fast_run_no_inplace = optdb.query(Query(include = ['fast_run'], exclude = ['inplace']))
   fast_run_no_inplace = fast_run.excluding('inplace')


Registering an Optimizer
------------------------

Let's say we have a global optimizer called ``simplify``. We can add
it to ``optdb`` as follows:

.. code-block:: python

   # optdb.register(name, optimizer, order, *tags)
   optdb.register('simplify', simplify, 0.5, 'fast_run')

Once this is done, the FAST_RUN mode will automatically include your
optimization (since you gave it the 'fast_run' tag). Of course,
already-compiled functions will see no change. The 'order' parameter
(what it means and how to choose it) will be explained in
:ref:`optdb-structure` below.



Registering a LocalOptimizer
----------------------------

LocalOptimizers may be registered in two ways:

* Wrap them in a Navigator and insert them like a global optimizer
  (see previous section).
* Put them in an EquilibriumDB.

Theano defines two EquilibriumDBs where you can put local
optimizations:


.. function:: canonicalize

  This contains optimizations that aim to *simplify* the graph:

  * Replace rare or esoterical operations with their equivalents using
    elementary operations.

  * Order operations in a canonical way (any sequence of
    multiplications and divisions can be rewritten to contain at most
    one division, for example; ``x*x`` can be rewritten ``x**2``; etc.)

  * Fold constants (``Constant(2)*Constant(2)`` becomes ``Constant(4)``)


.. function:: specialize

  This contains optimizations that aim to *specialize* the graph:

  * Replace a combination of operations with a special operation that
    does the same thing (but better).


For each group, all optimizations of the group that are selected by
the Query will be applied on the graph over and over again until none
of them is applicable, so keep that in mind when designing it: check
carefully that your optimization leads to a fixpoint (a point where it
cannot apply anymore) at which point it returns ``False`` to indicate its
job is done. Also be careful not to undo the work of another local
optimizer in the group, because then the graph will oscillate between
two or more states and nothing will get done.


.. _optdb-structure:

optdb structure
---------------

optdb contains the following Optimizers and sub-DBs, with the given
priorities and tags:

+-------+---------------------+------------------------------+
| Order | Name                | Description                  |
+=======+=====================+==============================+
| 0     | merge1              | First merge operation        |
+-------+---------------------+------------------------------+
| 1     | canonicalize        | Simplify the graph           |
+-------+---------------------+------------------------------+
| 2     | specialize          | Add specialized operations   |
+-------+---------------------+------------------------------+
| 49    | merge2              | Second merge operation       |
+-------+---------------------+------------------------------+
| 49.5  | add_destroy_handler | Enable inplace optimizations |
+-------+---------------------+------------------------------+
| 100   | merge3              | Third merge operation        |
+-------+---------------------+------------------------------+

The merge operations are meant to put together parts of the graph that
represent the same computation. Since optimizations can modify the
graph in such a way that two previously different-looking parts of the
graph become similar, we merge at the beginning, in the middle and at
the very end. Technically, we only really need to do it at the end,
but doing it in previous steps reduces the size of the graph and
therefore increases the efficiency of the process.

See previous section for more information about the canonicalize and
specialize steps.

The ``add_destroy_handler`` step is not really an optimization. It is
a marker. Basically:

.. warning::

   Any optimization which inserts inplace operations in the
   computation graph must appear **after** the ``add_destroy_handler``
   "optimizer". In other words, the priority of any such optimization
   must be **>= 50**. Failure to comply by this restriction can lead
   to the creation of incorrect computation graphs.

The reason the destroy handler is not inserted at the beginning is
that it is costly to run. It is cheaper to run most optimizations
under the assumption there are no inplace operations.


.. _navigator:

Navigator
-------------------

WRITEME
