Graph Rewriting¶
In this document we will explain how graph rewriting works and how graph optimizations can be constructed using graph rewriting.
Todo
The old “optimization” nomenclature is still in use throughout these documents and the codebase; however, this is being changed to more accurately distinguish between general graph rewriting for any purpose and the kind that is explicitly intended to “optimize” a graph in some way.
Global and Local Optimizations¶
First, let’s lay out the way optimizations work in Aesara. There are
two types of optimizations: global optimizations and local
optimizations. A global optimization takes a FunctionGraph
object (see its
documentation for more details) and navigates through it
in a suitable way, replacing some Variable
s by others in the process. A
local optimization, on the other hand, is defined as a function on a
single Apply node and must return either False
(to mean that
nothing is to be done) or a list of new Variable
s that we would like to
replace the node’s outputs with. A NavigatorOptimizer is a special kind
of global optimization which navigates the computation graph in some
fashion (e.g. in topological order, reversetopological 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.
Global optimization¶

class
GlobalOptimizer
¶ 
apply
(fgraph)¶ This method takes a
FunctionGraph
object which contains the computation graph and does modifications in line with what the optimization is meant to do. This is one of the main methods of the optimizer.

add_requirements
(fgraph)¶ This method takes a
FunctionGraph
object and adds features to it. These features are “plugins” that are needed for theGlobalOptimizer.apply()
method to do its job properly.

optimize
(fgraph)¶ This is the interface function called by Aesara. It calls
GlobalOptimizer.apply()
by default.

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

class
LocalOptimizer
¶ 
transform
(fgraph, node)¶ This method takes a
FunctionGraph
and anApply
node and returns eitherFalse
to signify that no changes are to be done or a list ofVariable
s which matches the length of the node’soutputs
list. When theLocalOptimizer
is applied by aNavigatorOptimizer
, the outputs of the node passed as argument to theLocalOptimizer
will be replaced by the list returned.

A simplification rule¶
For starters, let’s define the following simplification:
We will implement it in three ways: using a global optimization, a
local optimization with a NavigatorOptimizer
and then using the PatternSub
facility.
Global optimization¶
Here is the code for a global optimization implementing the simplification described above:
import aesara
from aesara.graph.opt import GlobalOptimizer
from aesara.graph.features import ReplaceValidate
class Simplify(GlobalOptimizer):
def add_requirements(self, fgraph):
fgraph.attach_feature(ReplaceValidate())
def apply(self, fgraph):
for node in fgraph.toposort():
if node.op == true_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:
fgraph.replace_validate(z, b)
elif y == b:
fgraph.replace_validate(z, a)
simplify = Simplify()
Here’s how it works: first, in add_requirements()
, we add the
ReplaceValidate
Feature
located in
features – [doc TODO]. This feature adds the replace_validate()
method to fgraph
, which is an enhanced version of FunctionGraph.replace()
that
does additional checks to ensure that we are not messing up the
computation graph.
In a nutshell, ReplaceValidate
grants access to fgraph.replace_validate()
,
and fgraph.replace_validate()
allows us to replace a Variable
with
another while respecting certain validation constraints. As an
exercise, try to rewrite Simplify
using NodeFinder
. (Hint: you
want to use the method it publishes instead of the call to toposort)
Then, in GlobalOptimizer.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 FunctionGraph.replace_validate()
; otherwise, we do
nothing.
Now, we test the optimization:
>>> from aesara.scalar import float64, add, mul, true_div
>>> x = float64('x')
>>> y = float64('y')
>>> z = float64('z')
>>> a = add(z, mul(true_div(mul(y, x), y), true_div(z, x)))
>>> e = aesara.graph.fg.FunctionGraph([x, y, z], [a])
>>> e
FunctionGraph(add(z, mul(true_div(mul(y, x), y), true_div(z, x))))
>>> simplify.optimize(e)
>>> e
FunctionGraph(add(z, mul(x, true_div(z, x))))
You can check what happens if you put many instances of 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 = float64('x')
>>> y = float64('y')
>>> z = float64('z')
>>> a = true_div(mul(add(y, z), x), add(y, z))
>>> e = aesara.graph.fg.FunctionGraph([x, y, z], [a])
>>> e
FunctionGraph(true_div(mul(add(y, z), x), add(y, z)))
>>> simplify.optimize(e)
>>> e
FunctionGraph(true_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 MergeOptimizer
defined in
aesara.graph.opt
.
>>> from aesara.graph.opt import MergeOptimizer
>>> MergeOptimizer().optimize(e)
(0, ..., None, None, {}, 1, 0)
>>> e
FunctionGraph(true_div(mul(*1 > add(y, z), x), *1))
>>> simplify.optimize(e)
>>> e
FunctionGraph(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 Aesara 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
arithmetic that your Ops implement. Aesara might provide facilities
for this somewhere in the future.
Note
FunctionGraph
is an Aesara structure intended for the optimization
phase. It is used internally by aesara.function()
and is rarely
exposed to the end user.
Local Optimization¶
The local version of the above code would be the following:
from aesara.graph.opt import LocalOptimizer
class LocalSimplify(LocalOptimizer):
def transform(self, fgraph, node):
if node.op == true_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 tells certain navigators to only apply this `LocalOptimizer`
# on these kinds of `Op`s
return [true_div]
local_simplify = LocalSimplify()
In this case, the transformation is defined in the
LocalOptimizer.transform()
method, which is given an explicit
Apply
node on which to work. The entire graph–as a fgraph
–is
also provided, in case global information is needed.
If no changes are to be made, False
must be returned; otherwise, a list of replacements for the node’s
outputs are returned. This list must have the same length as
node.outputs
. If one of node.outputs
doesn’t have clients
(e.g. available via fgraph.clients
), then it is not used elsewhere in the graph and
you can put None
in the returned list to remove it.
In order to apply the local optimizer we can use it in conjunction
with a NavigatorOptimizer
. Basically, a NavigatorOptimizer
is
a global optimizer that loops through all nodes in the graph (or a welldefined
subset of them) and applies one or several local optimizers.
>>> x = float64('x')
>>> y = float64('y')
>>> z = float64('z')
>>> a = add(z, mul(true_div(mul(y, x), y), true_div(z, x)))
>>> e = aesara.graph.fg.FunctionGraph([x, y, z], [a])
>>> e
FunctionGraph(add(z, mul(true_div(mul(y, x), y), true_div(z, x))))
>>> simplify = aesara.graph.opt.TopoOptimizer(local_simplify)
>>> simplify.optimize(e)
(<aesara.graph.opt.TopoOptimizer object at 0x...>, 1, 5, 3, ..., ..., ...)
>>> e
FunctionGraph(add(z, mul(x, true_div(z, x))))
OpSub
, OpRemove
, PatternSub
¶
Aesara defines some shortcuts to make LocalOptimizer
s:

OpSub
(op1, op2)¶ Replaces all uses of
op1
byop2
. In other words, the outputs of allApply
nodes usingop1
by the outputs ofApply
nodes involvingop2
, where their inputs are the same.

OpRemove
(op)¶ Removes all uses of
op
in the following way: ify = op(x)
theny
is replaced byx
.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.

PatternSub
(pattern1, pattern2)¶ Replaces all occurrences of the first pattern by the second pattern. See
PatternSub
.
from aesara.scalar import identity
from aesara.graph.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((true_div, (mul, 'x', 'y'), 'y'), 'x')
local_simplify_2 = PatternSub((true_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 (e.g. they need to be wrapped in a NavigatorOptimizer
, etc.)
When an optimization can be naturally expressed using OpSub
, OpRemove
or PatternSub
, it is highly recommended to use them.
Unification and reification¶
The PatternSub
class uses unification and reification to implement a
more succinct and reusable form of “pattern matching and replacement”.
In general, use of the unification and reification tools is preferable when
a rewrite’s matching and replacement are nontrivial, so we will briefly explain
them in the following.
Aesara’s unification and reification tools are provided by the
logicalunification package.
The basic tools are unify()
, reify()
, and var
. The class var
construct logic variables, which represent the elements to be unified/matched, unify()
performs the “matching”, and reify()
performs the “replacements”.
See unification
’s documentation for an introduction to unification and reification.
In order to use unify()
and reify()
with Aesara graphs, we need an intermediate
structure that will allow us to represent Aesara graphs that contain var
s, because
Aesara Op
s and Apply
nodes will not accept these foreign objects as inputs.
PatternSub
uses Python tuple
s to effectively represent Apply
nodes and
str
s to represent logic variables (i.e. var
s in the unification
library).
Behind the scenes, these tuple
s are converted to a tuple
subclass called ExpressionTuple
s,
which behave just like normal tuple
s except for some special caching features that allow for easy
evaluation and caching. These ExpressionTuple
s are provided by the
etuples library.
Here is an illustration of all the above components used together:
>>> from unification import unify, reify, var
>>> from etuples import etuple
>>> y_lv = var() # Create a logic variable
>>> y_lv
~_1
>>> s = unify(add(x, y), etuple(add, x, y_lv))
>>> s
{~_1: y}
In this example, unify()
matched the Aesara graph in the first argument with the “pattern”
given by the etuple()
in the second. The result is a dict
mapping logic variables to
the objects to which they were successfully unified. When a unify()
doesn’t succeed, it will
return False
.
reify()
uses dict
s like the kind produced by unify()
to replace
logic variables within structures:
>>> res = reify(etuple(add, y_lv, y_lv), s)
>>> res
e(<aesara.scalar.basic.Add at 0x7f54dfa5a350>, y, y)
Since ExpressionTuple
s can be evaluated, we can produce a complete Aesara graph from these
results as follows:
>>> res.evaled_obj
add.0
>>> aesara.dprint(res.evaled_obj)
add [id A] ''
y [id B]
y [id B]
Because ExpressionTuple
s effectively model Sexpressions, they can be used with the cons package to unify and reify
graphs structurally.
Let’s say we want to match graphs that use the add
Op
but could have a
varying number of arguments:
>>> from cons import cons
>>> op_lv = var()
>>> args_lv = var()
>>> s = unify(cons(op_lv, args_lv), add(x, y))
>>> s
{~_2: <aesara.scalar.basic.Add at 0x7f54dfa5a350>, ~_3: e(x, y)}
>>> s = unify(cons(op_lv, args_lv), add(x, y, z))
>>> s
{~_2: <aesara.scalar.basic.Add at 0x7f54dfa5a350>, ~_3: e(x, y, z)}
From here, we can check s[op_lv] == add
to confirm that we have the correct Op
and
proceed with our rewrite.
>>> res = reify(cons(mul, args_lv), s)
>>> res
e(<aesara.scalar.basic.Mul at 0x7f54dfa5ae10>, x, y, z)
>>> aesara.dprint(res.evaled_obj)
mul [id A] ''
x [id B]
y [id C]
z [id D]
miniKanren¶
Given that unification and reification are fully implemented for Aesara objects via the unificiation
package,
the kanren package can be used with Aesara graphs, as well.
kanren
implements the miniKanren domainspecific language for relational programming.
Refer to the links above for a proper introduction to miniKanren, but suffice it to say that miniKanren orchestrates the unification and reification operations described in Unification and reification, and it does so in the context of relational operators (e.g. equations like ). This means that a relation that–say–represents can be utilized in both directions.
Currently, the local optimizer KanrenRelationSub
provides a means of
turning kanren
relations into LocalOptimizer
s; however,
kanren
can always be used directly from within a custom Rewriter
, so
KanrenRelationSub
is not necessary.
The following is an example that distributes dot products across additions.
import aesara
import aesara.tensor as at
from aesara.graph.kanren import KanrenRelationSub
from aesara.graph.opt import EquilibriumOptimizer
from aesara.graph.opt_utils import optimize_graph
from aesara.tensor.math import _dot
from etuples import etuple
from kanren import conso, eq, fact, heado, tailo
from kanren.assoccomm import assoc_flatten, associative
from kanren.core import lall
from kanren.graph import mapo
from unification import vars as lvars
# Make the graph pretty printing results a little more readable
aesara.pprint.assign(
_dot, aesara.printing.OperatorPrinter("@", 1, "left")
)
# Tell `kanren` that `add` is associative
fact(associative, at.add)
def dot_distributeo(in_lv, out_lv):
"""A `kanren` goal constructor relation for the relation ``A.dot(a + b ...) == A.dot(a) + A.dot(b) ...``."""
A_lv, add_term_lv, add_cdr_lv, dot_cdr_lv, add_flat_lv = lvars(5)
return lall(
# Make sure the input is a `_dot`
eq(in_lv, etuple(_dot, A_lv, add_term_lv)),
# Make sure the term being `_dot`ed is an `add`
heado(at.add, add_term_lv),
# Flatten the associative pairings of `add` operations
assoc_flatten(add_term_lv, add_flat_lv),
# Get the flattened `add` arguments
tailo(add_cdr_lv, add_flat_lv),
# Add all the `_dot`ed arguments and set the output
conso(at.add, dot_cdr_lv, out_lv),
# Apply the `_dot` to all the flattened `add` arguments
mapo(lambda x, y: conso(_dot, etuple(A_lv, x), y), add_cdr_lv, dot_cdr_lv),
)
dot_distribute_opt = EquilibriumOptimizer([KanrenRelationSub(dot_distributeo)], max_use_ratio=10)
Below, we apply dot_distribute_opt
to a few example graphs. First we create simple test graph:
>>> x_at = at.vector("x")
>>> y_at = at.vector("y")
>>> A_at = at.matrix("A")
>>> test_at = A_at.dot(x_at + y_at)
>>> print(aesara.pprint(test_at))
(A @ (x + y))
Next we apply the rewrite to the graph:
>>> res = optimize_graph(test_at, include=[], custom_opt=dot_distribute_opt, clone=False)
>>> print(aesara.pprint(res))
((A @ x) + (A @ y))
We see that the dot product has been distributed, as desired. Now, let’s try a few more test cases:
>>> z_at = at.vector("z")
>>> w_at = at.vector("w")
>>> test_at = A_at.dot((x_at + y_at) + (z_at + w_at))
>>> print(aesara.pprint(test_at))
(A @ ((x + y) + (z + w)))
>>> res = optimize_graph(test_at, include=[], custom_opt=dot_distribute_opt, clone=False)
>>> print(aesara.pprint(res))
(((A @ x) + (A @ y)) + ((A @ z) + (A @ w)))
>>> B_at = at.matrix("B")
>>> w_at = at.vector("w")
>>> test_at = A_at.dot(x_at + (y_at + B_at.dot(z_at + w_at)))
>>> print(aesara.pprint(test_at))
(A @ (x + (y + ((B @ z) + (B @ w)))))
>>> res = optimize_graph(test_at, include=[], custom_opt=dot_distribute_opt, clone=False)
>>> print(aesara.pprint(res))
((A @ x) + ((A @ y) + ((A @ (B @ z)) + (A @ (B @ w)))))
This example demonstrates how nontrivial matching and replacement logic can be neatly expressed in miniKanren’s DSL, but it doesn’t quite demonstrate miniKanren’s relational properties.
To do that, we will create another Rewriter
that simply reverses the arguments
to the relation dot_distributeo()
and apply it to the distributed result in res
:
>>> dot_gather_opt = EquilibriumOptimizer([KanrenRelationSub(lambda x, y: dot_distributeo(y, x))], max_use_ratio=10)
>>> rev_res = optimize_graph(res, include=[], custom_opt=dot_gather_opt, clone=False)
>>> print(aesara.pprint(rev_res))
(A @ (x + (y + (B @ (z + w)))))
As we can see, the kanren
relation works both ways, just like the underlying
mathematical relation does.
miniKanren relations can be used to explore rewrites of graphs in sophisticated
ways. It also provides a framework that more directly maps to the mathematical
identities that drive graph rewrites. For some simple examples of relational graph rewriting
in kanren
see here. For a
highlevel overview of miniKanren’s use as a tool for symbolic computation see
“miniKanren as a Tool for Symbolic Computation in Python”.
The optimization database (optdb
)¶
Aesara 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 LocalOptimizer
s 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 CPUintensive 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.
For instance, the optimization tag cxx_only
is used for optimizations that
insert Op
s that have no Python implementation (i.e. they only have C
implementations). Optimizations with this tag can be skipped when the C backend
is not being used.
Definition of optdb
¶
optdb
is an object which is an instance of
SequenceDB
,
itself a subclass of OptimizationDatabase
.
There exist (for now) two types of OptimizationDatabase
, SequenceDB
and EquilibriumDB
.
When given an appropriate OptimizationQuery
, OptimizationDatabase
objects build an Optimizer
matching
the query.
A SequenceDB
contains Optimizer
or OptimizationDatabase
objects. Each of them
has a name, an arbitrary number of tags and an integer representing their order
in the sequence. When a OptimizationQuery
is applied to a SequenceDB
, all Optimizer
s whose
tags match the query are inserted in proper order in a SequenceOptimizer
, which
is returned. If the SequenceDB
contains OptimizationDatabase
instances, the OptimizationQuery
will be passed to them as well and the
optimizers they return will be put in their places.
An EquilibriumDB
contains LocalOptimizer
or OptimizationDatabase
objects. Each of them
has a name and an arbitrary number of tags. When a OptimizationQuery
is applied to
an EquilibriumDB
, all LocalOptimizer
s that match the query are
inserted into an EquilibriumOptimizer
, which is returned. If the
SequenceDB
contains OptimizationDatabase
instances, the
OptimizationQuery
will be passed to them as well and the
LocalOptimizer
s they return will be put in their places
(note that as of yet no OptimizationDatabase
can produce LocalOptimizer
objects, so this
is a moot point).
Aesara contains one principal OptimizationDatabase
object, optdb
, which
contains all of Aesara’s optimizers with proper tags. It is
recommended to insert new Optimizer
s in it. As mentioned previously,
optdb is a SequenceDB
, so, at the top level, Aesara applies a sequence
of global optimizations to the computation graphs.
OptimizationQuery
¶
A OptimizationQuery
is built by the following call:
aesara.graph.optdb.OptimizationQuery(include, require=None, exclude=None, subquery=None)

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

require
¶ A set of tags such that every optimization obtained through this
OptimizationQuery
must have all of these tags.

exclude
¶ A set of tags such that every optimization obtained through this
OptimizationQuery
must have none of these tags.

subquery
¶ optdb
can contain subdatabases; subquery is a dictionary mapping the name of a subdatabase to a specialOptimizationQuery
. If no subquery is given for a subdatabase, the originalOptimizationQuery
will be used again.

Furthermore, a OptimizationQuery
object includes three methods, including()
,
requiring()
and excluding()
, which each produce a new OptimizationQuery
object
with the include, require, and exclude sets refined to contain the new entries.
Examples¶
Here are a few examples of how to use a OptimizationQuery
on optdb
to produce an
Optimizer
:
from aesara.graph.optdb import OptimizationQuery
from aesara.compile import optdb
# This is how the optimizer for the fast_run mode is defined
fast_run = optdb.query(OptimizationQuery(include=['fast_run']))
# This is how the optimizer for the fast_compile mode is defined
fast_compile = optdb.query(OptimizationQuery(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(OptimizationQuery(include=['fast_run'],
exclude=['inplace']))
Registering an Optimizer
¶
Let’s say we have a global optimizer called simplify
. We can add
it to optdb
as follows:
# optdb.register(name, optimizer, order, *tags)
optdb.register('simplify', simplify, 'fast_run', position=0.5)
Once this is done, the FAST_RUN
mode will automatically include your
optimization (since you gave it the 'fast_run'
tag). Of course,
alreadycompiled functions will see no change. The ‘order’ parameter
(what it means and how to choose it) will be explained in
optdb structure below.
Registering a LocalOptimizer
¶
LocalOptimizer
s may be registered in two ways:
 Wrap them in a
NavigatorOptimizer
and insert them like a global optimizer (see previous section).  Put them in an
EquilibriumDB
.
Aesara defines two EquilibriumDB
s in which one can put local
optimizations:

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 rewrittenx**2
; etc.)  Fold constants (
Constant(2)*Constant(2)
becomesConstant(4)
)

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 OptimizationQuery
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
contains the following Optimizer
s and subDBs, 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 differentlooking 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.
Profiling Aesara function compilation¶
You find that compiling an Aesara function is taking too much time? You can get profiling information about Aesara optimization. The normal Aesara profiler will provide you with very highlevel information. The indentation shows the included in/subset relationship between sections. The top of its output look like this:
Function profiling
==================
Message: PATH_TO_A_FILE:23
Time in 0 calls to Function.__call__: 0.000000e+00s
Total compile time: 1.131874e+01s
Number of Apply nodes: 50
Aesara Optimizer time: 1.152431e+00s
Aesara validate time: 2.790451e02s
Aesara Linker time (includes C, CUDA code generation/compiling): 7.893991e02s
Import time 1.153541e02s
Time in all call to aesara.grad() 4.732513e02s
Explanations:
Total compile time: 1.131874e+01s
gives the total time spent insideaesara.function
.Number of Apply nodes: 50
means that after optimization, there are 50 apply node in the graph.Aesara Optimizer time: 1.152431e+00s
means that we spend 1.15s in theaesara.function
phase where we optimize (modify) the graph to make it faster / more stable numerically /…Aesara validate time: 2.790451e02s
means that we spent 2.8e2s in the validate subset of the optimization phase.Aesara Linker time (includes C code generation/compiling): 7.893991e02s
means that we spent 7.9e2s in linker phase ofaesara.function
.Import time 1.153541e02s
is a subset of the linker time where we import the compiled module.Time in all call to aesara.grad() 4.732513e02s
tells that we spent a total of 4.7e2s in all calls toaesara.grad
. This is outside of the calls toaesara.function
.
The linker phase includes the generation of the C code, the time spent by g++ to compile and the time needed by Aesara to build the object we return. The C code generation and compilation is cached, so the first time you compile a function and the following ones could take different amount of execution time.
Detailed profiling of Aesara optimizations¶
You can get more detailed profiling information about the Aesara
optimizer phase by setting to True
the Aesara flags
config.profile_optimizer
(this requires config.profile
to be True
as well).
This will output something like this:
Optimizer Profile

SeqOptimizer OPT_FAST_RUN time 1.152s for 123/50 nodes before/after optimization
0.028s for fgraph.validate()
0.131s for callback
time  (name, class, index)  validate time
0.751816s  ('canonicalize', 'EquilibriumOptimizer', 4)  0.004s
EquilibriumOptimizer canonicalize
time 0.751s for 14 passes
nb nodes (start, end, max) 108 81 117
time io_toposort 0.029s
time in local optimizers 0.687s
time in global optimizers 0.010s
0  0.050s 27 (0.000s in global opts, 0.002s io_toposort)  108 nodes  ('local_dimshuffle_lift', 9) ('local_upcast_elemwise_constant_inputs', 5) ('local_shape_to_shape_i', 3) ('local_fill_sink', 3) ('local_fill_to_alloc', 2) ...
1  0.288s 26 (0.002s in global opts, 0.002s io_toposort)  117 nodes  ('local_dimshuffle_lift', 8) ('local_fill_sink', 4) ('constant_folding', 4) ('local_useless_elemwise', 3) ('local_subtensor_make_vector', 3) ...
2  0.044s 13 (0.002s in global opts, 0.003s io_toposort)  96 nodes  ('constant_folding', 4) ('local_dimshuffle_lift', 3) ('local_fill_sink', 3) ('local_useless_elemwise', 1) ('local_fill_to_alloc', 1) ...
3  0.045s 11 (0.000s in global opts, 0.002s io_toposort)  91 nodes  ('constant_folding', 3) ('local_fill_to_alloc', 2) ('local_dimshuffle_lift', 2) ('local_mul_canonizer', 2) ('MergeOptimizer', 1) ...
4  0.035s 8 (0.002s in global opts, 0.002s io_toposort)  93 nodes  ('local_fill_sink', 3) ('local_dimshuffle_lift', 2) ('local_fill_to_alloc', 1) ('MergeOptimizer', 1) ('constant_folding', 1)
5  0.035s 6 (0.000s in global opts, 0.002s io_toposort)  88 nodes  ('local_fill_sink', 2) ('local_dimshuffle_lift', 2) ('local_fill_to_alloc', 1) ('local_mul_canonizer', 1)
6  0.038s 10 (0.001s in global opts, 0.002s io_toposort)  95 nodes  ('local_fill_sink', 3) ('local_dimshuffle_lift', 3) ('constant_folding', 2) ('local_fill_to_alloc', 1) ('MergeOptimizer', 1)
7  0.032s 5 (0.001s in global opts, 0.002s io_toposort)  91 nodes  ('local_fill_sink', 3) ('MergeOptimizer', 1) ('local_dimshuffle_lift', 1)
8  0.034s 5 (0.000s in global opts, 0.002s io_toposort)  92 nodes  ('local_fill_sink', 3) ('MergeOptimizer', 1) ('local_greedy_distributor', 1)
9  0.031s 6 (0.001s in global opts, 0.002s io_toposort)  90 nodes  ('local_fill_sink', 2) ('local_fill_to_alloc', 1) ('MergeOptimizer', 1) ('local_dimshuffle_lift', 1) ('local_greedy_distributor', 1)
10  0.032s 5 (0.000s in global opts, 0.002s io_toposort)  89 nodes  ('local_dimshuffle_lift', 2) ('local_fill_to_alloc', 1) ('MergeOptimizer', 1) ('local_fill_sink', 1)
11  0.030s 5 (0.000s in global opts, 0.002s io_toposort)  88 nodes  ('local_dimshuffle_lift', 2) ('local_fill_to_alloc', 1) ('MergeOptimizer', 1) ('constant_folding', 1)
12  0.026s 1 (0.000s in global opts, 0.003s io_toposort)  81 nodes  ('MergeOptimizer', 1)
13  0.031s 0 (0.000s in global opts, 0.003s io_toposort)  81 nodes 
times  times applied  nb node created  name:
0.263s  15  0  constant_folding
0.096s  2  14  local_greedy_distributor
0.066s  4  19  local_mul_canonizer
0.046s  28  57  local_fill_sink
0.042s  35  78  local_dimshuffle_lift
0.018s  5  15  local_upcast_elemwise_constant_inputs
0.010s  11  4  MergeOptimizer
0.009s  4  0  local_useless_elemwise
0.005s  11  2  local_fill_to_alloc
0.004s  3  6  local_neg_to_mul
0.002s  1  3  local_lift_transpose_through_dot
0.002s  3  4  local_shape_to_shape_i
0.002s  2  4  local_subtensor_lift
0.001s  3  0  local_subtensor_make_vector
0.001s  1  1  local_sum_all_to_none
0.131s  in 62 optimization that where not used (display only those with a runtime > 0)
0.050s  local_add_canonizer
0.018s  local_mul_zero
0.016s  local_one_minus_erf
0.010s  local_func_inv
0.006s  local_0_dot_x
0.005s  local_track_shape_i
0.004s  local_mul_switch_sink
0.004s  local_fill_cut
0.004s  local_one_minus_erf2
0.003s  local_remove_switch_const_cond
0.003s  local_cast_cast
0.002s  local_IncSubtensor_serialize
0.001s  local_sum_div_dimshuffle
0.001s  local_div_switch_sink
0.001s  local_dimshuffle_no_inplace_at_canonicalize
0.001s  local_cut_useless_reduce
0.001s  local_reduce_join
0.000s  local_sum_sum
0.000s  local_useless_alloc
0.000s  local_reshape_chain
0.000s  local_useless_subtensor
0.000s  local_reshape_lift
0.000s  local_flatten_lift
0.000s  local_useless_slice
0.000s  local_subtensor_of_alloc
0.000s  local_subtensor_of_dot
0.000s  local_subtensor_merge
0.101733s  ('elemwise_fusion', 'SeqOptimizer', 13)  0.000s
SeqOptimizer elemwise_fusion time 0.102s for 78/50 nodes before/after optimization
0.000s for fgraph.validate()
0.004s for callback
0.095307s  ('composite_elemwise_fusion', 'FusionOptimizer', 1)  0.000s
FusionOptimizer
nb_iter 3
nb_replacement 10
nb_inconsistency_replace 0
validate_time 0.000249624252319
callback_time 0.00316381454468
time_toposort 0.00375390052795
0.006412s  ('local_add_mul_fusion', 'FusionOptimizer', 0)  0.000s
FusionOptimizer
nb_iter 2
nb_replacement 3
nb_inconsistency_replace 0
validate_time 6.43730163574e05
callback_time 0.000783205032349
time_toposort 0.0035240650177
0.090089s  ('inplace_elemwise_optimizer', 'FromFunctionOptimizer', 30)  0.019s
0.048993s  ('BlasOpt', 'SeqOptimizer', 8)  0.000s
SeqOptimizer BlasOpt time 0.049s for 81/80 nodes before/after optimization
0.000s for fgraph.validate()
0.003s for callback
0.035997s  ('gemm_optimizer', 'GemmOptimizer', 1)  0.000s
GemmOptimizer
nb_iter 2
nb_replacement 2
nb_replacement_didn_t_remove 0
nb_inconsistency_make 0
nb_inconsistency_replace 0
time_canonicalize 0.00720071792603
time_factor_can 9.05990600586e06
time_factor_list 0.00128507614136
time_toposort 0.00311398506165
validate_time 4.60147857666e05
callback_time 0.00174236297607
0.004569s  ('local_dot_to_dot22', 'TopoOptimizer', 0)  0.000s
TopoOptimizer
nb_node (start, end, changed) (81, 81, 5)
init io_toposort 0.00139284133911
loop time 0.00312399864197
callback_time 0.00172805786133
0.002283s  ('local_dot22_to_dot22scalar', 'TopoOptimizer', 2)  0.000s
TopoOptimizer
nb_node (start, end, changed) (80, 80, 0)
init io_toposort 0.00171804428101
loop time 0.000502109527588
callback_time 0.0
0.002257s  ('local_gemm_to_gemv', 'EquilibriumOptimizer', 3)  0.000s
EquilibriumOptimizer local_gemm_to_gemv
time 0.002s for 1 passes
nb nodes (start, end, max) 80 80 80
time io_toposort 0.001s
time in local optimizers 0.000s
time in global optimizers 0.000s
0  0.002s 0 (0.000s in global opts, 0.001s io_toposort)  80 nodes 
0.002227s  ('use_c_blas', 'TopoOptimizer', 4)  0.000s
TopoOptimizer
nb_node (start, end, changed) (80, 80, 0)
init io_toposort 0.0014750957489
loop time 0.00068998336792
callback_time 0.0
0.001632s  ('use_scipy_ger', 'TopoOptimizer', 5)  0.000s
TopoOptimizer
nb_node (start, end, changed) (80, 80, 0)
init io_toposort 0.00138401985168
loop time 0.000202178955078
callback_time 0.0
0.031740s  ('specialize', 'EquilibriumOptimizer', 9)  0.000s
EquilibriumOptimizer specialize
time 0.031s for 2 passes
nb nodes (start, end, max) 80 78 80
time io_toposort 0.003s
time in local optimizers 0.022s
time in global optimizers 0.004s
0  0.017s 6 (0.002s in global opts, 0.001s io_toposort)  80 nodes  ('constant_folding', 2) ('local_mul_to_sqr', 1) ('local_elemwise_alloc', 1) ('local_div_to_inv', 1) ('local_mul_specialize', 1)
1  0.014s 0 (0.002s in global opts, 0.001s io_toposort)  78 nodes 
times  times applied  nb node created  name:
0.003s  1  1  local_mul_specialize
0.002s  1  2  local_elemwise_alloc
0.002s  2  0  constant_folding
0.001s  1  1  local_div_to_inv
0.001s  1  1  local_mul_to_sqr
0.016s  in 69 optimization that where not used (display only those with a runtime > 0)
0.004s  crossentropy_to_crossentropy_with_softmax_with_bias
0.002s  local_one_minus_erf
0.002s  Elemwise{sub,no_inplace}(z, Elemwise{mul,no_inplace}(alpha subject to <function <lambda> at 0x7f475e4da050>, SparseDot(x, y))) > Usmm{no_inplace}(Elemwise{neg,no_inplace}(alpha), x, y, z)
0.002s  local_add_specialize
0.001s  local_func_inv
0.001s  local_useless_elemwise
0.001s  local_abs_merge
0.001s  local_track_shape_i
0.000s  local_one_minus_erf2
0.000s  local_sum_mul_by_scalar
0.000s  local_elemwise_sub_zeros
0.000s  local_cast_cast
0.000s  local_alloc_unary
0.000s  Elemwise{log,no_inplace}(Softmax(x)) > <function make_out_pattern at 0x7f47619a8410>(x)
0.000s  local_sum_div_dimshuffle
0.000s  local_sum_alloc
0.000s  local_dimshuffle_lift
0.000s  local_reduce_broadcastable
0.000s  local_grad_log_erfc_neg
0.000s  local_advanced_indexing_crossentropy_onehot
0.000s  local_log_erfc
0.000s  local_log1p
0.000s  local_log_add
0.000s  local_useless_alloc
0.000s  local_neg_neg
0.000s  local_neg_div_neg
...
To understand this profile here is some explanation of how optimizations work:
Optimizations are organized in an hierarchy. At the top level, there is a
SeqOptimizer
. It contains other optimizers, and applies them in the order they were specified. Those suboptimizers can be of other types, but are all global optimizers.Each
Optimizer
in the hierarchy will print some stats about itself. The information that it prints depends of the type of the optimizer.The
SeqOptimizer
will print some stats at the start:Optimizer Profile  SeqOptimizer OPT_FAST_RUN time 1.152s for 123/50 nodes before/after optimization 0.028s for fgraph.validate() 0.131s for callback time  (name, class, index)  validate time
Then it will print, with some additional indentation, each suboptimizer’s profile information. These subprofiles are ordered by the time they took to execute, not by their execution order.
OPT_FAST_RUN
is the name of the optimizer 1.152s is the total time spent in that optimizer
 123/50 means that before this optimization, there were 123 apply node in the function graph, and after only 50.
 0.028s means it spent that time calls to
fgraph.validate()
 0.131s means it spent that time for callbacks. This is a mechanism that can trigger other execution when there is a change to the FunctionGraph.
time  (name, class, index)  validate time
tells how the information for each suboptimizer get printed. All other instances of
SeqOptimizer
are described like this. In particular, some suboptimizer fromOPT_FAST_RUN
that are alsoSeqOptimizer
.
The
SeqOptimizer
will print some stats at the start:0.751816s  ('canonicalize', 'EquilibriumOptimizer', 4)  0.004s EquilibriumOptimizer canonicalize time 0.751s for 14 passes nb nodes (start, end, max) 108 81 117 time io_toposort 0.029s time in local optimizers 0.687s time in global optimizers 0.010s 0  0.050s 27 (0.000s in global opts, 0.002s io_toposort)  108 nodes  ('local_dimshuffle_lift', 9) ('local_upcast_elemwise_constant_inputs', 5) ('local_shape_to_shape_i', 3) ('local_fill_sink', 3) ('local_fill_to_alloc', 2) ... 1  0.288s 26 (0.002s in global opts, 0.002s io_toposort)  117 nodes  ('local_dimshuffle_lift', 8) ('local_fill_sink', 4) ('constant_folding', 4) ('local_useless_elemwise', 3) ('local_subtensor_make_vector', 3) ... 2  0.044s 13 (0.002s in global opts, 0.003s io_toposort)  96 nodes  ('constant_folding', 4) ('local_dimshuffle_lift', 3) ('local_fill_sink', 3) ('local_useless_elemwise', 1) ('local_fill_to_alloc', 1) ... 3  0.045s 11 (0.000s in global opts, 0.002s io_toposort)  91 nodes  ('constant_folding', 3) ('local_fill_to_alloc', 2) ('local_dimshuffle_lift', 2) ('local_mul_canonizer', 2) ('MergeOptimizer', 1) ... 4  0.035s 8 (0.002s in global opts, 0.002s io_toposort)  93 nodes  ('local_fill_sink', 3) ('local_dimshuffle_lift', 2) ('local_fill_to_alloc', 1) ('MergeOptimizer', 1) ('constant_folding', 1) 5  0.035s 6 (0.000s in global opts, 0.002s io_toposort)  88 nodes  ('local_fill_sink', 2) ('local_dimshuffle_lift', 2) ('local_fill_to_alloc', 1) ('local_mul_canonizer', 1) 6  0.038s 10 (0.001s in global opts, 0.002s io_toposort)  95 nodes  ('local_fill_sink', 3) ('local_dimshuffle_lift', 3) ('constant_folding', 2) ('local_fill_to_alloc', 1) ('MergeOptimizer', 1) 7  0.032s 5 (0.001s in global opts, 0.002s io_toposort)  91 nodes  ('local_fill_sink', 3) ('MergeOptimizer', 1) ('local_dimshuffle_lift', 1) 8  0.034s 5 (0.000s in global opts, 0.002s io_toposort)  92 nodes  ('local_fill_sink', 3) ('MergeOptimizer', 1) ('local_greedy_distributor', 1) 9  0.031s 6 (0.001s in global opts, 0.002s io_toposort)  90 nodes  ('local_fill_sink', 2) ('local_fill_to_alloc', 1) ('MergeOptimizer', 1) ('local_dimshuffle_lift', 1) ('local_greedy_distributor', 1) 10  0.032s 5 (0.000s in global opts, 0.002s io_toposort)  89 nodes  ('local_dimshuffle_lift', 2) ('local_fill_to_alloc', 1) ('MergeOptimizer', 1) ('local_fill_sink', 1) 11  0.030s 5 (0.000s in global opts, 0.002s io_toposort)  88 nodes  ('local_dimshuffle_lift', 2) ('local_fill_to_alloc', 1) ('MergeOptimizer', 1) ('constant_folding', 1) 12  0.026s 1 (0.000s in global opts, 0.003s io_toposort)  81 nodes  ('MergeOptimizer', 1) 13  0.031s 0 (0.000s in global opts, 0.003s io_toposort)  81 nodes  times  times applied  nb node created  name: 0.263s  15  0  constant_folding 0.096s  2  14  local_greedy_distributor 0.066s  4  19  local_mul_canonizer 0.046s  28  57  local_fill_sink 0.042s  35  78  local_dimshuffle_lift 0.018s  5  15  local_upcast_elemwise_constant_inputs 0.010s  11  4  MergeOptimizer 0.009s  4  0  local_useless_elemwise 0.005s  11  2  local_fill_to_alloc 0.004s  3  6  local_neg_to_mul 0.002s  1  3  local_lift_transpose_through_dot 0.002s  3  4  local_shape_to_shape_i 0.002s  2  4  local_subtensor_lift 0.001s  3  0  local_subtensor_make_vector 0.001s  1  1  local_sum_all_to_none 0.131s  in 62 optimization that where not used (display only those with a runtime > 0) 0.050s  local_add_canonizer 0.018s  local_mul_zero 0.016s  local_one_minus_erf 0.010s  local_func_inv 0.006s  local_0_dot_x 0.005s  local_track_shape_i 0.004s  local_mul_switch_sink 0.004s  local_fill_cut 0.004s  local_one_minus_erf2 0.003s  local_remove_switch_const_cond 0.003s  local_cast_cast 0.002s  local_IncSubtensor_serialize 0.001s  local_sum_div_dimshuffle 0.001s  local_div_switch_sink 0.001s  local_dimshuffle_no_inplace_at_canonicalize 0.001s  local_cut_useless_reduce 0.001s  local_reduce_join 0.000s  local_sum_sum 0.000s  local_useless_alloc 0.000s  local_reshape_chain 0.000s  local_useless_subtensor 0.000s  local_reshape_lift 0.000s  local_flatten_lift 0.000s  local_useless_slice 0.000s  local_subtensor_of_alloc 0.000s  local_subtensor_of_dot 0.000s  local_subtensor_merge
0.751816s  ('canonicalize', 'EquilibriumOptimizer', 4)  0.004s
This line is fromSeqOptimizer
, and indicates information related to a suboptimizer. It means that this suboptimizer took a total of .7s. Its name is'canonicalize'
. It is anEquilibriumOptimizer
. It was executed at index 4 by theSeqOptimizer
. It spent 0.004s in the validate phase.All other lines are from the profiler of the
EquilibriumOptimizer
.An
EquilibriumOptimizer
does multiple passes on the Apply nodes from the graph, trying to apply local and global optimizations. Conceptually, it tries to execute all global optimizations, and to apply all local optimizations on all nodes in the graph. If no optimization got applied during a pass, it stops. So it tries to find an equilibrium state where none of the optimizations get applied. This is useful when we do not know a fixed order for the execution of the optimization.time 0.751s for 14 passes
means that it took .7s and did 14 passes over the graph.nb nodes (start, end, max) 108 81 117
means that at the start, the graph had 108 node, at the end, it had 81 and the maximum size was 117.Then it prints some global timing information: it spent 0.029s in
io_toposort()
, all local optimizers took 0.687s together for all passes, and global optimizers took a total of 0.010s.Then we print the timing for each pass, the optimization that got applied, and the number of time they got applied. For example, in pass 0, the
local_dimshuffle_lift()
optimizer changed the graph 9 time.Then we print the time spent in each optimizer, the number of times they changed the graph and the number of nodes they introduced in the graph.
Optimizations with that pattern
local_op_lift()
means that a node with that op will be replaced by another node, with the same op, but will do computation closer to the inputs of the graph. For instance,local_op(f(x))
getting replaced byf(local_op(x))
.Optimization with that pattern
local_op_sink()
is the opposite of “lift”. For instancef(local_op(x))
getting replaced bylocal_op(f(x))
.Local optimizers can replace any arbitrary node in the graph, not only the node it received as input. For this, it must return a
dict
. The keys being nodes to replace and the values being the corresponding replacement.This is useful to replace a client of the node received as parameter.