Tensor Program Optimization with Probabilistic Programs
Anonymous Author(s)
AﬀiliationAddress email
Abstract
Automatic optimization for tensor programs becomes increasingly important as we deploy deep learning in various environments, and eﬀicient optimization relies on a rich search space and effective search.
Most existingefforts adopt a search space which lacks the ability to eﬀiciently enabledomain experts to grow the search space.
This paper introduces SpaceCraft, a domain-specific probabilistic programming language abstractionto construct a rich search space of tensor programs.
Our abstraction allows domain experts to analyze the program, and easily propose stochasticchoices in a modular way to compose program transformation accordingly.
We also build an end-to-end learning-driven framework to find an optimizedprogram for a given search space.
Experimental results show that SpaceCraft can cover the search space used in the state-of-the-art tensor programoptimization frameworks in a modular way.
Additionally, it empowers domain experts to conveniently grow the search space and modularly enhancethe system, which brings 48% speedup on end-to-end deep learning workloads.
Introduction
Deep learning has become pervasive in daily life.
From video understanding [28], naturallanguage understanding [15], and recommendation system [29] to autonomous driving [21],different deep learning models are deployed on different hardware platforms and devices.
Deep learning frameworks usually rely on manually optimized libraries [13, 22] to accelerate deployment.
Engineers need to choose from many tensor programs that are logicallyequivalent but differ significantly in performance due to memory access, threading, and theuse of specialized hardware primitives.
The engineering effort required for tensor programoptimization has become a significant bottleneck for machine learning deployment with thegrowing number of models and hardware backends.
Automatic program optimization [10, 43, 1] is a recent sequence of efforts that aims touse machine learning to solve this problem.
There are two vital elements of automatictensor program optimizations.
First, a search space is defined to provide a possible set ofequivalent tensor programs.
Then existing systems use learning-based search algorithms tofind an optimized tensor program in the search space with feedback from the deploymentenvironment.
Most of the current approaches [10, 43, 1, 25, 4] use pre-defined search spacesthat effectively encode the domain knowledge of the authors once and focus on developing eﬀicient search algorithms
Initial Tensor Program e
Search Space S(e 0 )
Optimal Programe * Search Algorithm for i in range(1024): for j in range(1024): for k in range(1024): C[i, j] += A[i, k] * B[j, k] for i 0 , j 0 in grid(16, 8): for i 1 , j 1 in grid(8, 16): for k 0 in range(1024): for i 2 , j 2 in grid(8, 8): C[...] += ...
for i 0 , j 0 in grid(64, 8): for i 1 , j 1 in grid(4, 32): for k 0 in range(64): for i 2 , j 2 in grid(4, 4): for k 1 in range(16): C[...]
+= ...
[Figure]
Examples and the search algorithm that finds the optimal tensor program 𝑒 ⋆ .
The search space usually incorporates choices over loop transformation, vectorization, threading patterns, and hardware acceleration.
While eﬀicient search is essential, the search space itself fundamentally limits the best possible performance search algorithms can get.
To construct a good search space, domainexperts have to make numerous choices over loop transformation, vectorization, threadingpatterns, and hardware acceleration.
Additionally, the best search space itself evolves asnew tensor program optimization techniques [24] and hardware primitives [30] grow.
As aresult, there is a strong need to enable easy customization and construction of the searchspace at scale by taking inputs from system engineers and domain experts.
Unfortunately,any change to search space construction currently requires surgical modifications to theautomatic program optimization frameworks.
This research asks the following question: can we decouple the search space constructionfrom search and provide an adequate abstraction for domain experts and the learning systemto collaborate on search space construction?
We give an aﬀirmative answer to the questionwith two key observations.
First, we can parameterize an optimization search space by theinitial program followed by a sequence of transformations on the program.
Next, usingthis parameterization, domain experts can then provide probabilistic choices that representpossible transformations after examining the program state.
These two observations lead toa simple yet powerful abstraction for search space construction through a domain-specificprobabilistic language.
Finally, our framework composes multiple possible probabilistictransformations to form a rich search space.
We make the following contributions:
• We introduce a simple yet powerful probabilistic language abstraction for tensor
program search space construction.
• We build a learning-driven framework to find optimized tensor programs specified
by the search space constructed using our abstraction.
• We build an end-to-end system that can take prior knowledge from domain experts
to construct optimization search space to optimize deep learning deployment on
multiple platforms.
Our end-to-end system can easily expand search space that matches previous approacheswithout surgical changes and achieve comparable performance on multiple hardware backends.
Experimental results show that our abstraction is expressive enough to cover the optimization space of a diverse set of tensor programs, delivering a competitive performance ofpopular deep learning models, and convenient to incorporate hardware-specific knowledgeinto the search space to outperform the state-of-the-art frameworks.
Background and Problem Overview
Figure 1 shows a typical workflow for tensor program optimization.
For a given program 𝑒,a typical tensor program optimization framework will generate candidates from a pre-definedsearch space 𝑆(𝑒 0 ) containing semantically-equivalent programs.
Then the framework findsoptimized tensor program 𝑒 ∗ ∈ 𝑆(𝑒 0 ) with the minimum latency on the target hardware.
for i in range(1024): B[i] =
ReLU(A[i])
Initial tensor program: eparallel for i 0 in range(32):
for i 1 in range(8): i = i
* 32 + i 1 * 4 B[i : i + 4] =
ReLU(A[i : i + 4])
Equivalent optimizedprogram: e* ① for i 0 in range(32): for i 1 in range(8): for i 2 in range(4): i = i
* 32 + i 1 * 4 + i
B[i] =
ReLU(A[i]) ② ③ ③ Vectorize ② Parallelize ① Split
i, 32, 8, 4 ii
Transformation
Parameterization parameterized by: e 0 + ① parameterized by: e 0 + ①②③
Equivalent intermediate program: e
Parameterization
Equivalent Programs Induced by Parameterized Transformation
[Figure]
𝑒is parameterized by the initial program 𝑒, plus Step 1 ! , which is further parameterized by loop 𝑖 and resulting loop extents 32, 8, 4, respectively.
A typical search space 𝑆(𝑒 0 ) contains choices over threading, loop ordering, memory access, and hardware primitives.
Defining the search space 𝑆(𝑒 0 ) for a wide range of tensorprograms brings several challenges.
First, 𝑆(𝑒 0 ) is highly dependent on 𝑒. For example,𝑆(𝑒 0 ) of a compute-intensive program (e.g., Dense ) needs to consider many more possibleconfigurations than a communication-intensive program such as ReLU .
The space also differssignificantly in different hardware domains.
For example, 𝑆(𝑒 0 ) on CPU involves multi-coreparallelism and vectorization, while 𝑆(𝑒 0 ) on GPU involves thread binding and tensorization.
Finally, as the hardware and model settings change, we need to bring in fresh domainexperts to update the 𝑆(𝑒 0 ) to leverage the latest improvements.
This paper aims to provide a programmable abstraction to construct 𝑆(⋅) in a composableand modular way.
Our key goals are listed as follows:
Expressiveness.
We need to be ableto build a rich search space that covers the optimization programs that domain experts willwrite.
Modularity.
Tensor program optimization likely will involve inputs from multipledomain experts over different periods.
Therefore, we need to be able to combine priorknowledge in a composable and modular way.
Designed for learning.
We need to build ageneric learning-based framework to enable diverse variations of the cost model and searchfor search space specified in our abstraction.
We will address the above goals in the followingtwo sections.
Composable Search Space Parameterization
This section presents SpaceCraft, a probabilistic approach to search space parameterization.
3.1 Stochastic Search Space Construction
SpaceCraft constructs a search space 𝑆(⋅) with stochastic program transformations as theprimitive building blocks.
Traditional program optimization can usually be represented bya sequence of transformations 𝜏 , where at step 𝑖 , the program 𝑒 𝑖− is transformed into asemantically-equivalent program 𝑒 𝑖 , which finally leads to the optimized program
𝑒 𝑛 .
SpaceCraft generalizes this idea by allowing further parameterization of each transformation stepin 𝜏 .
Taking Figure 2 as an example: 𝑒is the initial program for the program 𝐵 =
ReLU (𝐴) 1 .
In SpaceCraft, transformation 𝑡 1 =
Split is parameterized by a loop 𝑖 and a sequence ofintegers indicating the loop extents after splitting; Similarly, transforms 𝑡 2 =
Parallelizeand 𝑡 3 =
Vectorize are parameterized by loops respectively.
As a result, an optimizedprogram 𝑒 𝑛 is obtained by applying a sequence of parameterized transformations 𝜏 to theinitial program 𝑒.
Accordingly, the search space 𝑆(𝑒 0 ) is composed of 𝑒and all possiblesequences of parameterized transformations.
[Figure]
' in grid( θ 4 , θ 5 ): D[...]
= ...
θ ReLU = θ ReLU =
Probabilistic Program
Stochastic Transformation
An example execution instance: Step 1 ! :
Draw tile sizes of and then organize the loops into a two-level tiling structure.
Step 2 !
: Decide where to fuse the ReLU operator.
On the other hand, it could be less practical for practitioners to determine the best combination of the parameter values in transformations.
For instance, in Figure 2, it is usuallyeﬀicient to use 512-bit vectorization over the inner loop when AVX-512 2 vector instructionsare available on Intel CPUs, or other vectorization lengths may lead to better performance,otherwise.
Therefore, deep knowledge of the target hardware is mandatory to enumerateplausible parameter combinations to control the search space size while covering the optimalprogram.
To let practitioners eﬀiciently define parameterized transformations without worrying aboutcandidate values, SpaceCraft introduces random variables drawn from analysis, sampling .
Parameterized by random variables, a transformation naturally becomes stochastic, and theunderlying probabilistic space reflects the space of possible transformations.
As illustrated in Figure 3, when creating Split transformations to tile loop 𝑖 and 𝑗 in the Dense operator, the tile sizes are drawn by random variables 𝜃 0− defined from
Sample-Tile .
In this way, the Split transformation becomes stochastic.
Similarly, we use
Sample-Compute-Location to enumerate valid loops in Dense after splitting for ReLU tofuse its computation.
In summary, 7 lines of SpaceCraft program covers a family of possibleoptimized tensor programs with stochastic transformations in its search space 𝑆(𝑒 0 ) , where𝑒is Dense-ReLU .
Notably, unlike orthogonal grid space in hyperparameter tuning, SpaceCraft captures longterm structural and arithmetic dependency between random variables and the tensor program 𝑒 𝑖 being transformed.
As demonstrated on Step 2 !
in Figure 3, sampling distributionfrom Sample-Compute-Location depends on the latest tensor program 𝑒, whose structuredepends on all previous random variables.
3.2 Modular Search Space Composition
Although the search space constructed by stochastic transformations proposed in the previous subsection is eﬀicient and is capable of covering the optimal tensor program, it is hardfor other developers to learn how the search space is constructed by reading a long sequenceof transformations.
It makes transformations designed for a workload hard to be reused byother workloads.
Meanwhile, we observe that it is straightforward to group a sub-sequenceof transformations for a particular fine-grained optimization.
For example, some transformations implement multi-level tiling for better memory locality in compute-intensive operatorslike Conv2d and Dense ; some other transformations are used to fold/inline elementwise op-
[Figure]
Stochastic
[Figure]
Compose procedure Compose (modules, program) while locations are not exhausted do location ← next-location (program) 𝑚 ∼
Sample ( modules ) program ← 𝑚 (location, program) end while end procedure
[Figure]
erations such as activation functions into their predecessors or successors for better memorybandwidth eﬀiciency.
To improve the usability and make SpaceCraft more practical, we introduce transformationmodule .
Just like the convolutional module with Conv2D , BiasAdd and ReLU in ResNet, atransformation module in SpaceCraft is defined as either atomic stochastic transformation,or composition of program analysis, sampling as well as smaller transformations.
Eachtransformation module can have a meaningful name so that it can be easily adopted bymany workloads to hierarchically construct a search space.
Figure 4 shows hierarchical composition of transformation modules.
Specifically,
Multi-Level-Tiling interleaves program analysis on loops and the stochastic tiling ofthe loop structure and organizes the original tensor program into a 5-level tiling structure.
Notably, the transformation module is generic to tensor programs and thus could be appliedto a variety of operators, including conv1d , conv3d , matmul , etc.
Figure 5 depicts an example of composing a search space with transformation modules.
Inthis simple example, we select a set of transformation modules, which are implemented inadvance by practitioners with prior domain knowledge, and apply them to every availablelocation in the tensor program to form a search space.
Consequently, the formed searchspace covers common optimizations on diverse hardware.
3.3 Relation to Existing Tensor Program Optimization Methods
In this subsection, we discuss prior approaches for automatic tensor program optimizationand illustrate that many of them can be covered by the SpaceCraft framework.
Domain specific languages for program transformations used by prior frameworks [32, 5, 9, 37] allow developers to easily optimize a program manually.
When there is
The Probabilistic
[Figure]
no random variable sampling in the program, SpaceCraft reduces to a DSL for deterministicprogram transformations and achieves the functionality.
Template-guided auto-tuning [10, 2, 25, 27] fully relies on developers to define a searchspace.
In SpaceCraft, it means all random variables in a search space are defined aheadof the transformations, so there is no interaction between program analysis and follow-uprandom sampling choices conditioned on the program state.
Auto-scheduling [43, 45, 1, 19] requires developers to implement workload agnostic transformation rules.
SpaceCraft achieves the same programmability and functionality throughspecific probabilistic transformation modules that correspond to the search space generationrules.
Notably, all approaches mentioned above have important use-cases in tensor program optimizations, depending on how much domain knowledge we want to incorporate for a particular scenario.
By decoupling the search space construction from the search, we effectivelybuild a single framework for all the use cases and enable further customization withoutsurgical changes to the system.
Learning-driven Search
The last section provides a modular abstraction for search space.
We still need to do aneffective search to find an optimized program within the search space.
This section providesa generic learning-driven framework to find an optimized program.
Objective formalization.
For a given probabilistic program 𝑒, let us use 𝜏 to denotethe transformations performed on 𝑒. 𝜏 can be sampled from a prior distribution specifiedby the probabilistic program.
We define 𝑔(𝑒 0 , 𝜏) to be the tensor program after applyingtransformation 𝜏 to 𝑒. Let 𝑓(𝑒) be the latency of the particular program 𝑒 on the hardwareenvironment.
We define a posterior probability of an optimized program as:
[Equation]
Intuitively, we want to assign a higher probability to the programs that perform well.
Ourfinal goal is to find
[Equation]
that maximizes the posterior through maximuma posteriori estimation (MAP) estimation.
Execution tracing.
To enable domain experts to express their knowledge via transformations modules productively, we embed SpaceCraft in Python.
We introduced executiontracing to reduce the cost of repetitive re-execution of the Python program.
Figure 6demonstrates an example tracing process.
During program execution, the system recordsall samplings and transformations while ignoring control flow and other constructs of thehost language.
The resulting trace is a sequence of SpaceCraft primitives with only sam-
[Figure]
pling and transformation instructions, which could be re-executed as a normal SpaceCraftprogram.
We can then continue to explore different sampling choices for a given collectionof initial traces.
Conceptually, this is equivalent to dividing up our support set and thensampling the program condition on the execution sequence of the program.
End-to-end search.
Figure 7 shows the overall workflow of our learning-driven framework.
The search algorithm first samples the SpaceCraft program to obtain a collection of traces.
Then it continues to explore the space condition on the traces.
Notably, there is a significantly higher cost measuring 𝑓(𝑒) directly on the hardware, so we also incorporated a proxycost model ̂ 𝑓(𝑒) , which is updated throughout the process, similar to previous efforts ontensor program optimization [10, 43].
At each iteration, we adopt an evolutionary searchalgorithm that proposes a new variant of the trace by mutating the random variables, thenaccept or reject the proposal based on the cost model.
While evolutionary search could beviewed as parallel chain MCMC, we also made our system modular enough to incorporateother ways to select the probabilistic choices, such as those through Bayesian optimizationand reinforcement learning.
Cost model.
Our approach allows extensive cost models, enabling us to supply those pretrained from existing datasets [44].
We pick a tree-boosting-based cost model in ̂ 𝑓(⋅) bydefault and leverage a common set of features that are used in previous works [43].
Trace validation.
Importantly, invalid traces may show up as we propose updates.
Sucha scenario can happen when some of the random variable choices go beyond the physicalhardware limit or a variable that induces changes to the execution sequence.
Instead ofenforcing a conservative proposal, we introduce a validator that validates the correctness ofthe trace.
The trace validation allows us to move around the space more freely while stillensuring the correctness of the sample outcome to be on the right support set.
Related Work
Tensor Program Transformations are proposed by many prior works, such as Halide [32],
TVM [9], Tiramisu [5] and TACO [23, 37].
Note that all the previous transformation languages are deterministic and cannot be directly used for search space construction, meaningthat they have to introduce a separate programming model to express a search space.
Thispaper makes a simple but powerful generalization to domain-specific probabilistic language.
The resulting abstraction enables a unified approach to deterministic transformation andsearch space construction.
Black-box optimization has been adopted in high-performance computing libraries [16, 3].
Recent advances in automatic tensor program optimization brought a rich set of techniquesto accelerate search through better cost modeling [10, 4, 34] and learning-based search [2,25, 1, 19, 44], which could be incorporated into SpaceCraft search.
Different variations ofpre-defined search spaces have also been proposed that couple with the automatic tensor program optimization frameworks [10, 43, 1].
Polyhedral model [40, 6, 39] is one useful wayto construct a rich pre-defined search space.
This paper focuses on modular search spaceconstruction and provides orthogonal contributions to these prior works.
Probabilistic programming language is a powerful abstraction for incorporating domainknowledge and probabilistic inference.
There are many general-purpose probabilistic languages, such as Church [17], Stan [8], Pyro [7], NumPyro [31], PyMC3 [35] and Edward [38].
This paper proposes a domain-specific probabilistic language for tensor program optimization with specializations such as tensor program analysis that would otherwise be opaqueto the previous systems.
Our learning-driven search can be viewed as an application ofprevious works [42, 33, 41, 46] that use tracing to divide programs into subprograms withfixed support.
We focus on the MAP inference problem where the posterior depends on anunknown cost function.
We solve the problem through a learned cost-model-driven evolutionary search over traces and with validation.
Automatic neural program synthesis [12, 18, 11] has seen large progress recently.
Alphacode [26] builds a system that can output creative and sound solutions to problems thatrequire deeper-level human understanding.
These approaches generate abstract syntax trees(ASTs) that can be incorrect and use input-output pairs to filter out those erroring programs.
Our compiler approach requires us to ensure the correctness of all transformations.
However,some ideas like validation after creation might be reusable.
Experiments
6.1 Expressiveness to Cover Common Optimization Techniques
This section aims to answer the following question: Is SpaceCraft expressive enough tocapture the search space of the state-of-the-art optimization techniques?
To answer thisquestion, we evaluate our work on a diverse set of operators and subgraphs extracted frompopular deep learning models, including variants of convolution, dense, and normalization.
As baselines, PyTorch (v1.11.0) results are provided to compare performance with vendorlibraries; TVM (commit: 8d4f4dd73f ), which incorporates AutoTVM [10] and Ansor [43],is used as the state-of-the-art tensor program optimization system, and we pick the bestamong the two in each respective setups.
Full operators and hardware configurations aredocumented in Appendix A.2.
[Figure] Figure 8 shows that, in all cases on CPU and GPU, SpaceCraft delivers performance comparable with or even better than TVM, from which we could infer that SpaceCraft couldexpress optimization techniques comparable to TVM on diverse workloads.
Additionally, inmost of the cases, SpaceCraft outperforms PyTorch by a significant margin except for SFM,which is highly optimized manually in PyTorch.
6.2 Optimizing End-to-End Deep Learning Models
Operator performance does not always translate directly to full model optimization.
Therefore, this section is dedicated to answering the following question: Can SpaceCraft delivercompetitive performance with state-of-the-art works for end-to-end models?
Therefore, a series of experiments are conducted to compare SpaceCraft and TVM, including
BERT-Base [14], ResNet-50 [20], and MobileNet-v2 [36] on both CPU and GPU.
As shownin Figure 9, SpaceCraft performance is on parity with TVM, while surpassing PyTorchin all cases, which indicates that SpaceCraft framework delivers end-to-end performance.
Additionally, tuning time is provided in Appendix A.5.
6.3 Search Space Composition and Hardware-Specific Modules
Besides performance parity with existing work, in this section, we demonstrate the extravalue of modular search space composition by answering the following question: How convenient is it to compose transformation modules, and how does it translate to performance?
We design an ablation study for transformation modules composition.
As indicated in Figure 10a, by progressively enriching the search space, the performance of optimized tensor programs consistently increases.
Composed with a hardware-specific module Use-Tensor-Core ,
SpaceCraft delivers significantly better performance compared with generic search space.
The performance gain, brought by search space composition with customized rules, doestranslate to end-to-end model performance, as shown in Figure 10b.
Specifically, on BERTlarge workloads, SpaceCraft with Use-Tensor-Core delivers 48% speedup over TVM.
Notably, it took a graduate student only 2 days to craft the 82-line Use-Tensor-Core modulein Python (see supplementary materials), which provides strong evidence of the convenienceof customization and composition. More details are in Appendix A.4.
Conclusion
This paper presents SpaceCraft, a programming model to describe search space construction in tensor program optimization.
Our method abstracts search space as a probabilisticlanguage and enables flexible incorporation of domain knowledge by allowing practitioners to implement customized probabilistic programs.
A learning-driven search algorithmis developed on top of the probabilistic language abstraction, which delivers competitiveperformance with state-of-the-art frameworks.
In the future, we will explore and modularize declarative API for various hardware environments.
Therefore, we will open-source our framework and hope it could enable broader collaboration between the machine learningdeployment engineers and intelligent machine learning algorithms for tensor programs.
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