Graph Structures#
Aesara works by modeling mathematical operations and their outputs using
symbolic placeholders, or variables, which inherit from the class
Variable
. When writing expressions in Aesara one uses operations like
+
, 
, **
, sum()
, tanh()
. These are represented
internally as Ops. An Op
represents a computation that is
performed on a set of symbolic inputs and produces a set of symbolic outputs.
These symbolic input and output Variable
s carry information about
their types, like their data type (e.g. float, int), the number of dimensions,
etc.
Aesara graphs are composed of interconnected Apply, Variable and
Op
nodes. An Apply
node represents the application of an
Op
to specific Variable
s. It is important to draw the
difference between the definition of a computation represented by an Op
and its application to specific inputs, which is represented by the
Apply
node.
The following illustrates these elements:
Code
import aesara.tensor as at
x = at.dmatrix('x')
y = at.dmatrix('y')
z = x + y
Diagram
The blue box is an Apply
node. Red boxes are Variable
s. Green
circles are Op
s. Purple boxes are Type
s.
When we create Variable
s and then Apply
Op
s to them to make more Variable
s, we build a
bipartite, directed, acyclic graph. Variable
s point to the Apply
nodes
representing the function application producing them via their
Variable.owner
field. These Apply
nodes point in turn to their input and
output Variable
s via their Apply.inputs
and Apply.outputs
fields.
The Variable.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, its Variable.owner
field would point to another
blue box like z
does, and so on.
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 broadcasted 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
<aesara.tensor.elemwise.DimShuffle object at 0x106fcaf10>
>>> y.owner.inputs[1].owner.inputs
[TensorConstant{2.0}]
All of the above can be succinctly summarized with the aesara.dprint()
function:
>>> aesara.dprint(y)
Elemwise{mul,no_inplace} [id A] ''
x [id B]
InplaceDimShuffle{x,x} [id C] ''
TensorConstant{2.0} [id D]
Starting from this graph structure it is easier to understand how automatic differentiation proceeds and how the symbolic relations can be rewritten for performance or stability.
Graph Structures#
The following section outlines each type of structure that may be used in an Aesarabuilt computation graph.
Apply
#
An Apply
node is a type of internal node used to represent a
computation graph in Aesara. Unlike
Variable
, Apply
nodes are usually not
manipulated directly by the end user. They may be accessed via
the Variable.owner
field.
An Apply
node is typically an instance of the Apply
class. It represents the application
of an Op
on one or more inputs, where each input is a
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 Apply
node is
Aesara’s version of a function call whereas an Op
is
Aesara’s version of a function definition.
An Apply
instance has three important fields:
 op
An
Op
that determines the function/transformation being applied here. inputs
A list of
Variable
s that represent the arguments of the function. outputs
A list of
Variable
s that represent the return values of the function.
An Apply
instance can be created by calling graph.basic.Apply(op, inputs, outputs)
.
Op
#
An 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 Variables and an Op
, you can build an 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
.
Type
#
A Type
in Aesara provides static information (or constraints) about
data objects in a graph. The information provided by Type
s allows
Aesara to perform rewrites and produce more efficient compiled code.
Every symbolic Variable
in an Aesara graph has an associated
Type
instance, and Type
s also serve as a means of
constructing Variable
instances. In other words, Type
s and
Variable
s go handinhand.
For example, aesara.tensor.irow is an instance of a
Type
and it can be used to construct variables as follows:
>>> from aesara.tensor import irow
>>> irow()
<TensorType(int32, (1, ?))>
As the string printout shows, irow
specifies the following information about
the Variable
s it constructs:
They represent tensors that are backed by
numpy.ndarray
s. This comes from the fact thatirow
is an instance ofTensorType
, which is the baseType
for symbolicnumpy.ndarray
s.They represent arrays of 32bit integers (i.e. from the
int32
).They represent arrays with shapes of , or, in code,
(1, None)
, whereNone
represents any shape value.
Note that Aesara Type
s are not necessarily equivalent to Python types or
classes. Aesara’s TensorType
’s, like irow
, use numpy.ndarray
as the underlying Python type for performing computations and storing data, but
numpy.ndarray
s model a much wider class of arrays than most TensorType
s.
In other words, Aesara Type
’s try to be more specific.
For more information see Types.
Variable
#
A Variable
is the main data structure you work with when using
Aesara. The symbolic inputs that you operate on are Variable
s and what
you get from applying various Op
s to these inputs are also
Variable
s. For example, when one inputs
>>> import aesara
>>> x = aesara.tensor.ivector()
>>> y = x
x
and y
are both Variable
s. The 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 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 Variable
or
one of its subclasses.
A Variable
r
contains four important fields:
 type
a
Type
defining the kind of value thisVariable
can hold in computation. owner
this is either
None
or anApply
node of which theVariable
is an output. index
the integer such that
owner.outputs[index] is r
(ignored ifVariable.owner
isNone
) name
a string to use in prettyprinting and debugging.
Variable
has an important subclass: Constant.
Constant
#
A Constant
is a Variable
with one extra, immutable field:
Constant.data
.
When used in a computation graph as the input of an
Op
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 Op
will not under
any circumstances modify the input. This means that a Constant
is
eligible to participate in numerous rewrites: constant inlining
in C code, constant folding, etc.
Automatic Differentiation#
Having the graph structure, computing automatic differentiation is
simple. The only thing aesara.grad()
has to do is to traverse the
graph from the outputs back towards the inputs through all Apply
nodes. 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, 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 differentiation in greater detail.
Rewrites#
When compiling an Aesara graph using aesara.function()
, a graph is
necessarily provided. 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 rewrites 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. Rewrites can also detect identical subgraphs and ensure that the
same values are not computed twice.
For example, one simple rewrite that Aesara uses is to replace the pattern by .
See Graph Rewriting and Optimizations for more information.
Example
Consider the following example of rewrites:
>>> 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_no_rewrite.png", var_with_name_simple=True)
The output file is available at ./pics/symbolic_graph_no_rewrite.png
>>> aesara.printing.pydotprint(f, outfile="./pics/symbolic_graph_rewite.png", var_with_name_simple=True)
The output file is available at ./pics/symbolic_graph_rewrite.png
We used aesara.printing.pydotprint()
to visualize the rewritten graph
(right), which is much more compact than the unrewritten graph (left).
Unrewritten graph 
Rewritten graph 

