Op
s#
An Op
is a graph object that defines and performs computations in a graph.
It has to define the following methods.
- make_node(*inputs)#
This method is responsible for creating output
Variable
s of a suitable symbolicType
to serve as the outputs of thisOp
’s application. TheVariable
s found in*inputs
must be operated on using Aesara’s symbolic language to compute the symbolic outputVariable
s. This method should put these outputs into anApply
instance, and return theApply
instance.This method creates an
Apply
node representing the application of theOp
on the inputs provided. If theOp
cannot be applied to these inputs, it must raise an appropriate exception.The inputs of the
Apply
instance returned by this call must be ordered correctly: a subsequentself.make_node(*apply.inputs)
must produce something equivalent to the firstapply
.
- perform(node, inputs, output_storage)#
This method computes the function associated to this
Op
.node
is anApply
node created by theOp
’sOp.make_node()
method.inputs
is a list of references to data to operate on using non-symbolic statements, (i.e., statements in Python, NumPy).output_storage
is a list of storage cells where the variables of the computation must be put.More specifically:
node
: This is a reference to anApply
node which was previously obtained via theOp.make_node()
method. It is typically not used in simpleOp
s, but it contains symbolic information that could be required for complexOp
s.inputs
: This is a list of data from which the values stored inoutput_storage
are to be computed using non-symbolic language.output_storage
: This is a list of storage cells where the output is to be stored. A storage cell is a one-element list. It is forbidden to change the length of the list(s) contained inoutput_storage
. There is one storage cell for each output of theOp
.The data put in
output_storage
must match the type of the symbolic output. This is a situation where thenode
argument can come in handy.A function
Mode
may allowoutput_storage
elements to persist between evaluations, or it may resetoutput_storage
cells to hold a value ofNone
. It can also pre-allocate some memory for theOp
to use. This feature can allowOp.perform()
to reuse memory between calls, for example. If there is something preallocated in theoutput_storage
, it will be of the good dtype, but can have the wrong shape and have any stride pattern.
This method must be determined by the inputs. That is to say, if it is evaluated once on inputs A and returned B, then if ever inputs C, equal to A, are presented again, then outputs equal to B must be returned again.
You must be careful about aliasing outputs to inputs, and making modifications to any of the inputs. See Views and inplace operations before writing a
Op.perform()
implementation that does either of these things.
- __eq__(other)#
other
is also anOp
.Returning
True
here is a promise to the rewrite system that the otherOp
will produce exactly the same graph effects (e.g. from itsOp.perform()
) as this one, given identical inputs. This means it will produce the same output values, it will destroy the same inputs (sameOp.destroy_map
), and will alias outputs to the same inputs (sameOp.view_map
). For more details, see Views and inplace operations.Note
If you set
__props__
, this will be automatically generated.
- __hash__()#
If two
Op
instances compare equal, then they must return the same hash value.Equally important, this hash value must not change during the lifetime of self.
Op
instances should be immutable in this sense.
Note
If you set Op.__props__
, this will be automatically generated.
Optional methods or attributes#
- __props__#
Default: Undefined
Must be a tuple. Lists the name of the attributes which influence the computation performed. This will also enable the automatic generation of appropriate
__eq__
,__hash__
and__str__
methods. Should be set to()
if you have no attributes that are relevant to the computation to generate the methods.New in version 0.7.
- default_output#
Default: None
If this member variable is an integer, then the default implementation of
__call__
will returnnode.outputs[self.default_output]
, wherenode
was returned byOp.make_node()
. Otherwise, the entire list of outputs will be returned, unless it is of length 1, where the single element will be returned by itself.
- make_thunk(node, storage_map, compute_map, no_recycling, impl=None)#
This function must return a thunk, that is a zero-arguments function that encapsulates the computation to be performed by this
Op
on the arguments of the node.- Parameters:
node –
Apply
instance The node for which a thunk is requested.storage_map – dict of lists This maps variables to a one-element lists holding the variable’s current value. The one-element list acts as pointer to the value and allows sharing that “pointer” with other nodes and instances.
compute_map – dict of lists This maps variables to one-element lists holding booleans. If the value is 0 then the variable has not been computed and the value should not be considered valid. If the value is 1 the variable has been computed and the value is valid. If the value is 2 the variable has been garbage-collected and is no longer valid, but shouldn’t be required anymore for this call.
no_recycling – WRITEME WRITEME
impl – None, ‘c’ or ‘py’ Which implementation to use.
The returned function must ensure that is sets the computed variables as computed in the
compute_map
.Defining this function removes the requirement for
perform()
or C code, as you will define the thunk for the computation yourself.
- __call__(*inputs, **kwargs)#
By default this is a convenience function which calls
make_node()
with the supplied arguments and returns the result indexed bydefault_output
. This can be overridden by subclasses to do anything else, but must return either an AesaraVariable
or a list ofVariable
s.If you feel the need to override
__call__
to change the graph based on the arguments, you should instead create a function that will use yourOp
and build the graphs that you want and call that instead of theOp
instance directly.
- infer_shape(fgraph, node, shapes)#
This function is needed for shape rewrites.
shapes
is a list with one tuple for each input of theApply
node (which corresponds to the inputs of theOp
). Each tuple contains as many elements as the number of dimensions of the corresponding input. The value of each element is the shape (number of items) along the corresponding dimension of that specific input.While this might sound complicated, it is nothing more than the shape of each input as symbolic variables (one per dimension).
The function should return a list with one tuple for each output. Each tuple should contain the corresponding output’s computed shape.
Implementing this method will allow Aesara to compute the output’s shape without computing the output itself, potentially sparing you a costly recomputation.
- flops(inputs, outputs)#
It is only used to have more information printed by the memory profiler. It makes it print the mega flops and giga flops per second for each apply node. It takes as inputs two lists: one for the inputs and one for the outputs. They contain tuples that are the shapes of the corresponding inputs/outputs.
- __str__()#
This allows you to specify a more informative string representation of your
Op
. If anOp
has parameters, it is highly recommended to have the__str__
method include the name of theOp
and theOp
’s parameters’ values.Note
If you set
__props__
, this will be automatically generated. You can still override it for custom output.
- do_constant_folding(fgraph, node)#
Default: Return
True
By default when rewrites are enabled, we remove during function compilation
Apply
nodes whose inputs are all constants. We replace theApply
node with an Aesara constant variable. This way, theApply
node is not executed at each function call. If you want to force the execution of anOp
during the function call, make do_constant_folding return False.As done in the Alloc
Op
, you can return False only in some cases by analyzing the graph from the node parameter.
- debug_perform(node, inputs, output_storage)#
Undefined by default.
If you define this function then it will be used instead of C code or
Op.perform()
to do the computation while debugging (currently DebugMode, but others may also use it in the future). It has the same signature and contract asOp.perform()
.This enables
Op
s that cause trouble with DebugMode with their normal behaviour to adopt a different one when run under that mode. If yourOp
doesn’t have any problems, don’t implement this.
If you want your Op
to work with aesara.gradient.grad()
you also
need to implement the functions described below.
Gradient#
These are the function required to work with aesara.gradient.grad()
.
- grad(inputs, output_gradients)#
If the
Op
being defined is differentiable, its gradient may be specified symbolically in this method. Bothinputs
andoutput_gradients
are lists of symbolic AesaraVariable
s and those must be operated on using Aesara’s symbolic language. TheOp.grad()
method must return a list containing oneVariable
for each input. Each returnedVariable
represents the gradient with respect to that input computed based on the symbolic gradients with respect to each output.If the output is not differentiable with respect to an input then this method should be defined to return a variable of type
NullType
for that input. Likewise, if you have not implemented the gradient computation for some input, you may return a variable of typeNullType
for that input.aesara.gradient
contains convenience methods that can construct the variable for you:aesara.gradient.grad_undefined()
andaesara.gradient.grad_not_implemented()
, respectively.If an element of
output_gradient
is of typeaesara.gradient.DisconnectedType
, it means that the cost is not a function of this output. If any of theOp
’s inputs participate in the computation of only disconnected outputs, thenOp.grad()
should returnDisconnectedType
variables for those inputs.If the
Op.grad()
method is not defined, then Aesara assumes it has been forgotten. Symbolic differentiation will fail on a graph that includes thisOp
.It must be understood that the
Op.grad()
method is not meant to return the gradient of theOp
’s output.aesara.grad()
computes gradients;Op.grad()
is a helper function that computes terms that appear in gradients.If an
Op
has a single vector-valued outputy
and a single vector-valued inputx
, then theOp.grad()
method will be passedx
and a second vectorz
. DefineJ
to be the Jacobian ofy
with respect tox
. TheOp.grad()
method should returndot(J.T,z)
. Whenaesara.grad()
calls theOp.grad()
method, it will setz
to be the gradient of the costC
with respect toy
. If thisOp
is the onlyOp
that acts onx
, thendot(J.T,z)
is the gradient of C with respect tox
. If there are otherOp
s that act onx
,aesara.grad()
will have to add up the terms ofx
’s gradient contributed by the otherOp.grad()
method.In practice, an
Op
’s input and output are rarely implemented as single vectors. Even if anOp
’s output consists of a list containing a scalar, a sparse matrix, and a 4D tensor, you can think of these objects as being formed by rearranging a vector. Likewise for the input. In this view, the values computed by theOp.grad()
method still represent a Jacobian-vector product.In practice, it is probably not a good idea to explicitly construct the Jacobian, which might be very large and very sparse. However, the returned value should be equal to the Jacobian-vector product.
So long as you implement this product correctly, you need not understand what
aesara.gradient.grad()
is doing, but for the curious the mathematical justification is as follows:In essence, the
Op.grad()
method must simply implement through symbolicVariable
s and operations the chain rule of differential calculus. The chain rule is the mathematical procedure that allows one to calculate the total derivative of the final scalar symbolicVariable
C
with respect to a primitive symbolicVariable
x found in the listinputs
. TheOp.grad()
method does this usingoutput_gradients
which provides the total derivative ofC
with respect to a symbolicVariable
that is returned by theOp
(this is provided inoutput_gradients
), as well as the knowledge of the total derivative of the latter with respect to the primitiveVariable
(this has to be computed).In mathematics, the total derivative of a scalar variable with respect to a vector of scalar variables , i.e. the gradient, is customarily represented as the row vector of the partial derivatives, whereas the total derivative of a vector of scalar variables with respect to another , is customarily represented by the matrix of the partial derivatives, i.e. the Jacobian matrix. In this convenient setting, the chain rule says that the gradient of the final scalar variable with respect to the primitive scalar variables in through those in is simply given by the matrix product: .
Here, the chain rule must be implemented in a similar but slightly more complex setting: Aesara provides in the list
output_gradients
one gradient for each of theVariable
s returned by theOp
. Where is one such particularVariable
, the corresponding gradient found inoutput_gradients
and representing is provided with a shape similar to and thus not necessarily as a row vector of scalars. Furthermore, for eachVariable
of theOp
’s list of input variablesinputs
, the returned gradient representing must have a shape similar to that ofVariable
x.If the output list of the
Op
is , then the listoutput_gradients
is . Ifinputs
consists of the list , thenOp.grad
should return the list , where (and can stand for multiple dimensions).In other words,
Op.grad()
does not return , but instead the appropriate dot product specified by the chain rule: . Both the partial differentiation and the multiplication have to be performed byOp.grad()
.Aesara currently imposes the following constraints on the values returned by the
Op.grad()
method:They must be
Variable
instances.When they are types that have dtypes, they must never have an integer dtype.
The output gradients passed to
Op.grad()
will also obey these constraints.Integers are a tricky subject. Integers are the main reason for having
DisconnectedType
,NullType
or zero gradient. When you have an integer as an argument to yourOp.grad()
method, recall the definition of a derivative to help you decide what value to return:.
Suppose your function f has an integer-valued output. For most functions you’re likely to implement in Aesara, this means your gradient should be zero, because for almost all . (The only other option is that the gradient could be undefined, if your function is discontinuous everywhere, like the rational indicator function)
Suppose your function has an integer-valued input. This is a little trickier, because you need to think about what you mean mathematically when you make a variable integer-valued in Aesara. Most of the time in machine learning we mean “ is a function of a real-valued , but we are only going to pass in integer-values of ”. In this case, exists, so the gradient through should be the same whether is an integer or a floating point variable. Sometimes what we mean is “ is a function of an integer-valued , and is only defined where is an integer.” Since doesn’t exist, the gradient is undefined. Finally, many times in Aesara, integer valued inputs don’t actually affect the elements of the output, only its shape.
If your function has both an integer-valued input and an integer-valued output, then both rules have to be combined:
If is defined at , then the input gradient is defined. Since would be equal to almost everywhere, the gradient should be zero (first rule).
If is only defined where is an integer, then the gradient is undefined, regardless of what the gradient with respect to the output is.
Examples:
is a dot product between and . and are integers. Since the output is also an integer, is a step function. Its gradient is zero almost everywhere, so
Op.grad()
should return zeros in the shape of and .is a dot product between and . is floating point and is an integer. In this case the output is floating point. It doesn’t matter that is an integer. We consider to still be defined at . The gradient is exactly the same as if were floating point.
is the argmax of along axis . The gradient with respect to is undefined, because is not defined for floating point . How could you take an argmax along a fractional axis? The gradient with respect to is 0, because almost everywhere.
is a vector with elements, each of which taking on the value The
Op.grad()
method should returnDisconnectedType
for , because the elements of don’t depend on . Only the shape of depends on . You probably also want to implement a connection_pattern method to encode this.converts float into an integer. converts an integer into a float. If the final cost , then the gradient with respect to will be 0.5, even if is an integer. However, the gradient with respect to will be 0, because the output of is integer-valued.
- connection_pattern(node):
Sometimes needed for proper operation of
aesara.gradient.grad()
.Returns a list of list of booleans.
Op.connection_pattern[input_idx][output_idx]
is true if the elements ofinputs[input_idx]
have an effect on the elements ofoutputs[output_idx]
.The
node
parameter is needed to determine the number of inputs. SomeOp
s such asSubtensor
take a variable number of inputs.If no connection_pattern is specified,
aesara.gradient.grad()
will assume that all inputs have some elements connected to some elements of all outputs.This method conveys two pieces of information that are otherwise not part of the Aesara graph:
Which of the
Op
’s inputs are truly ancestors of each of theOp
’s outputs. Suppose anOp
has two inputs, and , and outputs and . is not really an ancestor of , but it appears to be so in the Aesara graph.Whether the actual elements of each input/output are relevant to a computation. For example, the shape
Op
does not read its input’s elements, only its shape metadata. should thus raise a disconnected input exception (if these exceptions are enabled). As another example, the elements of theAlloc
Op
’s outputs are not affected by the shape arguments to theAlloc
Op
.
Failing to implement this function for an
Op
that needs it can result in two types of incorrect behavior:aesara.gradient.grad()
erroneously raising aTypeError
reporting that a gradient is undefined.aesara.gradient.grad()
failing to raise aValueError
reporting that an input is disconnected.
Even if connection_pattern is not implemented correctly, if
aesara.gradient.grad()
returns an expression, that expression will be numerically correct.
- R_op(inputs, eval_points)#
Optional, to work with
aesara.gradient.R_op()
.This function implements the application of the R-operator on the function represented by your
Op
. Let assume that function is , with input , applying the R-operator means computing the Jacobian of and right-multiplying it by , the evaluation point, namely: .inputs
are the symbolic variables corresponding to the value of the input where you want to evaluate the Jacobian, andeval_points
are the symbolic variables corresponding to the value you want to right multiply the Jacobian with.Same conventions as for the
Op.grad()
method hold. If yourOp
is not differentiable, you can return None. Note that in contrast to the methodOp.grad()
, forOp.R_op()
you need to return the same number of outputs as there are outputs of theOp
. You can think of it in the following terms. You have all your inputs concatenated into a single vector . You do the same with the evaluation points (which are as many as inputs and of the shame shape) and obtain another vector . For each output, you reshape it into a vector, compute the Jacobian of that vector with respect to and multiply it by . As a last step you reshape each of these vectors you obtained for each outputs (that have the same shape as the outputs) back to their corresponding shapes and return them as the output of theOp.R_op()
method.