Scan internals

Context

This document is meant to act as reference material for developers working on Aesara’s loop mechanism. This mechanism is called Scan and its internals are highly complex, hence the need for a centralized repository of knowledge regarding its inner workings.

The aesara.scan function is the public-facing interface for looping in Aesara. Under the hood, this function will perform some processing on its inputs and instantiate the Scan Op class which implements the looping mechanism. It achieves this by compiling its own Aesara function representing the computation to be done at every iteration of the loop and calling it as many times as necessary.

The correspondence between the parameters and behaviors of the function and the Op is not always simple since the former is meant for usability and the second for performance. Since this document is intended to be used by developers working inside Scan itself, it will mostly discuss things from the point of view of the Scan Op class. Nonetheless, it will attempt to link those elements to their corresponding concepts in the Scan function as often as is reasonably practical.

Pre-requisites

The following sections assumes the reader is familiar with the following :

1. Aesara’s graph structure (Apply nodes, Variable nodes and Ops)

2. The interface and usage of Aesara’s scan function

Additionally, the Rewrites section below assumes knowledge of:

3. Aesara’s graph rewriting

Relevant code files

The implementation of Scan is spread over several files in aesara/scan. The different files, and sections of the code they deal with, are :

• basic.py implements the scan function. The scan function arranges the arguments of scan correctly, constructs the Scan Op and afterwards calls the constructed Scan Op on the arguments. This function takes care of figuring out missing inputs and shared variables.

• op.py implements the Scan Op class. The Scan respects the Op interface, and contains most of the logic of the Scan operator.

• utils.py contains several helpful functions used throughout out the other files that are specific of the Scan operator.

• views.py contains different views of the Scan Op that have simpler and easier signatures to be used in specific cases.

• opt.py contains the list of all Aesara graph rewrites for the Scan operator.

Notation

Scan being a sizeable and complex module, it has its own naming convention for functions and variables which this section will attempt to introduce.

A Scan Op contains an Aesara function representing the computation that is done in a single iteration of the loop represented by the Scan Op (in other words, the computation given by the function provided as value to aesara.scan’s fn argument ). Whenever we discuss a Scan Op, the outer function refers to the Aesara function that contains the Scan Op whereas the inner function refers to the Aesara function that is contained inside the Scan Op.

In the same spirit, the inputs and outputs of the Apply node wrapping the `Scan` `Op` (or `Scan` node for short) are referred to as outer inputs and outer outputs, respectively, because these inputs and outputs are variables in the outer function graph. The inputs and outputs of Scan’s inner function are designated inner inputs and inner outputs, respectively.

Scan variables

The following are the different types of variables that Scan has the capacity to handle, along with their various caracteristics.

Sequence : A sequence is an Aesara variable which Scan will iterate over and give sub-elements to its inner function as input. A sequence has no associated output. For a sequence variable X, at timestep t, the inner function will receive as input the sequence element X[t]. These variables are used through the argument sequences of the aesara.scan function.

Non-sequences : A non-sequence is an Aesara variable which Scan will provide as-is to its inner function. Like a sequence, a non-sequence has no associated output. For a non-sequence variable X, at timestep t, the inner function will receive as input the variable X. These variables are used through the argument non_sequences of the aesara.scan function.

NITSOT (no input tap, single output tap) : A NITSOT is an output variable of the inner function that is not fed back as an input to the next iteration of the inner function. NITSOTs are typically encountered in situations where Scan is used to perform a ‘map’ operation (every element in a tensor is independently altered using a given operation to produce a new tensor) such as squaring every number in a vector.

SITSOT (single input tap, single output tap) : A SITSOT is an output variable of the inner function that is fed back as an input to the next iteration of the inner function. A typical setting where a SITSOT might be encountered is the case where Scan is used to compute the cumulative sum over the elements of a vector and a SITSOT output is employed to act as an accumulator.

MITSOT (multiple input taps, single output tap) : A MITSOT is an output variable of the inner function that is fed back as an input to future iterations of the inner function (either multiple future iterations or a single one that isn’t the immediate next one). For example, a MITSOT might be used in the case where Scan is used to compute the Fibonacci sequence, one term of the sequence at every timestep, since every computed term needs to be reused to compute the two next terms of the sequence.

MITMOT (multiple input taps, multiple output taps) : These outputs exist but they cannot be directly created by the user. They can appear in an Aesara graph as a result of taking the gradient of the output of a Scan with respect to its inputs: This will result in the creation of a new Scan node used to compute the gradients of the first Scan node. If the original Scan had SITSOTs or MITSOTs variables, the new Scan will use MITMOTs to compute the gradients through time for these variables.

To synthesize :

Type of Scan variables	Corresponding outer input	Corresponding inner input at timestep t (indexed from 0)	Corresponding inner output at timestep t (indexed from 0)	Corresponding outer output t	Corresponding argument of the aesara.scan function
Sequence	Sequence of elements X	Individual sequence element X[t]	No corresponding inner output	No corresponding outer output	sequences
Non-Sequence	Any variable X	Variable identical to X	No corresponding inner output	No corresponding outer output	non_sequences
Non-recurring output (NITSOT)	No corresponding outer input	No corresponding inner input	Output value at timestep t	Concatenation of the values of the output at all timestep	outputs_info
Singly-recurrent output (SITSOT)	Initial value (value at timestep -1)	Output value at previous timestep (t-1)	Output value at timestep t	Concatenation of the values of the output at all timestep	outputs_info
Multiply-recurrent output (MITSOT)	Initial values for the required timesteps where t<0	Output value at previous required timesteps	Output value at timestep t	Concatenation of the values of the output at all timestep	outputs_info
Multiply-recurrent multiple outputs (MITMOT)	Initial values for the required timesteps where t<0	Output value at previous required timesteps	Output values for current and multiple future timesteps	Concatenation of the values of the output at all timestep	No corresponding argument

Rewrites

remove_constants_and_unused_inputs_scan

This rewrite serves two purposes, The first is to remove a ScanOp’s unused inputs. The second is to take a Scan Op’s constant inputs and remove them, instead injecting the constants directly into the graph or the Scan Op’s inner function. This will allow constant folding to happen inside the inner function.

PushOutNonSeqScan

This rewrite pushes sub-graphs that depends only on non-sequence inputs out of Scan’s inner function and into the outer function. Such computation ends up being done every iteration on the same values so moving it to the outer function to be executed only once, before the ScanOp, reduces the amount of computation that needs to be performed.

PushOutSeqScan

This rewrite resembles PushOutNonSeqScan but it tries to push, out of the inner function, the computation that only relies on sequence and non-sequence inputs. The idea behind this rewrite is that, when it is possible to do so, it is generally more computationally efficient to perform a single operation on a large tensor rather then perform that same operation many times on many smaller tensors. In many cases, this rewrite can increase memory usage but, in some specific cases, it can also decrease it.

PushOutScanOutput

This rewrite attempts to push out some of the computation at the end of the inner function to the outer function, to be executed after the Scan node. Like PushOutSeqScan, this rewrite aims to replace many operations on small tensors by few operations on large tensors. It can also lead to increased memory usage.

PushOutDot1

This is another rewrite that attempts to detect certain patterns of computation in a ScanOp’s inner function and move this computation to the outer graph.

ScanInplaceOptimizer

This rewrite attempts to make Scan compute its recurrent outputs inplace on the input tensors that contain their initial states. This rewrite can improve runtime performance as well as reduce memory usage.

ScanSaveMem

This rewrite attempts to determine if a Scan node, during its execution, for any of its outputs, can get away with allocating a memory buffer that is large enough to contain some of the computed timesteps of that output but not all of them.

By default, during the execution of a Scan node, memory buffers will be allocated to store the values computed for every output at every iteration. However, in some cases, there are outputs for which there is only really a need to store the most recent N values, not all of them.

For instance, if a Scan node has a SITSOT output (last computed value is fed back as an input at the next iteration) and only the last timestep of that output is ever used in the outer function, the ScanSaveMem rewrite could determine that there is no need to store all computed timesteps for that SITSOT output. Only the most recently computed timestep ever needs to be kept in memory.

ScanMerge

This rewrite attempts to fuse distinct Scan nodes into a single Scan node that performs all the computation. The main advantage of merging Scan nodes together comes from the possibility of both original ScanOps having some computation in common. In such a setting, this computation ends up being done twice. The fused Scans, however, would only need to do it once and could therefore be more computationally efficient. Also, since every Scan node involves a certain overhead, at runtime, reducing the number of Scan nodes in the graph can improve performance.

scan_merge_inouts

This rewrite attempts to merge a Scans identical outer inputs as well as merge its identical outer outputs (outputs that perform the same computation on the same inputs). This can reduce the amount of computation as well as result in a simpler graph for both the inner function and the outer function.

Helper classes and functions

Because of the complexity involved in dealing with Scan, a large number of helper classes and functions have been developed over time to implement operations commonly needed when dealing with the ScanOp. The ScanOp itself defines a large number of them and others can be found in the file utils.py. This sections aims to point out the most useful ones sorted by usage.

Accessing/manipulating Scan’s inputs and outputs by type

Declared in utils.py, the class ScanArgs handles the parsing of the inputs and outputs (both inner and outer) to a format that is easier to analyze and manipulate. Without this class, analyzing Scan’s inputs and outputs can require convoluted logic which make for code that is hard to read and to maintain. Because of this, you should favor using ScanArgs when it is practical and appropriate to do so.

The Scan Op extends ScanPropertiesMixin, which defines a few helper methods for this purpose, such as inner_nitsot_outs or mitmot_out_taps, but they are often poorly documented and easy to misuse. These should be used with great care.

Navigating between outer inputs/outputs and inner inputs/outputs

Navigation between these four sets of variables can be done in two ways, depending on the type of navigation that is required.

If the goal is to navigate between variables that are associated with the same states (e.g. going from an outer sequence input to the corresponding inner sequence input, going from an inner output associated with a recurrent state to the inner input(s) associated with that same recurrent state, etc.), then the get_oinp_iinp_iout_oout_mappings_mappings method of the Scan Op can be used.

This method returns a dictionary with 12 key/value pairs. The keys are listed below :

• “outer_inp_from_outer_out”

• “inner_inp_from_outer_out”

• “inner_out_from_outer_out”

• “inner_inp_from_outer_inp”

• “inner_out_from_outer_inp”

• “outer_out_from_outer_inp”

• “outer_inp_from_inner_inp”

• “inner_out_from_inner_inp”

• “outer_out_from_inner_inp”

• “outer_inp_from_inner_out”

• “inner_inp_from_inner_out”

• “outer_out_from_inner_out”

Every corresponding value is a dictionary detailing a mapping from one set of variables to another. For each of those dictionaries the keys are indices of variables in one set and the values are the indices of the corresponding variables in another set. For mappings to outer variables, the values are individual indices or -1 if there is not corresponding outer variable. For mappings to inner variables, the values are list of indices because multiple inner variables may be associated with the same state.

If the goal is to navigate between variables that are connected (meaning that one of them is used to compute the other), the method Scan.connection_pattern can be used. The method Scan.connection_pattern returns a list of lists detailing, for every pair of outer input and outer output whether they are connected or not.