train_coaxnn_paper.csv

text
"Contents lists available at ScienceDirect

Journal of Systems Architecture

journal homepage: www.elsevier.com/locate/sysarc

CoAxNN: Optimizing on-device deep learning with conditional approximate
neural networks
Guangli Li a,b,1, Xiu Ma c,d,1, Qiuchu Yu a,b, Lei Liu c,d, Huaxiao Liu c,d, Xueying Wang a,b,∗
a State Key Lab of Processors, Institute of Computing Technology, Chinese Academy of Sciences, Beijing, China
b University of Chinese Academy of Sciences, Beijing, China
c College of Computer Science and Technology, Jilin University, Changchun, China
d MOE Key Laboratory of Symbolic Computation and Knowledge Engineering, Jilin University, Changchun, China

A R T I C L E I N F O

A B S T R A C T

Keywords:
On-device deep learning
Efficient neural networks
Model approximation and optimization

While deep neural networks have achieved superior performance in a variety of intelligent applications, the
increasing computational complexity makes them difficult to be deployed on resource-constrained devices. To"
"Efficient neural networks
Model approximation and optimization

While deep neural networks have achieved superior performance in a variety of intelligent applications, the
increasing computational complexity makes them difficult to be deployed on resource-constrained devices. To
improve the performance of on-device inference, prior studies have explored various approximate strategies,
such as neural network pruning, to optimize models based on different principles. However, when combining
these approximate strategies, a large parameter space needs to be explored. Meanwhile, different configuration
parameters may interfere with each other, damaging the performance optimization effect. In this paper, we
propose a novel model optimization framework, CoAxNN, which effectively combines different approximate
strategies, to facilitate on-device deep learning via model approximation. Based on the principles of different"
"propose a novel model optimization framework, CoAxNN, which effectively combines different approximate
strategies, to facilitate on-device deep learning via model approximation. Based on the principles of different
approximate optimizations, our approach constructs the design space and automatically finds reasonable
configurations through genetic algorithm-based design space exploration. By combining the strengths of
different approximation methods, CoAxNN enables efficient conditional inference for models at runtime. We
evaluate our approach by leveraging state-of-the-art neural networks on a representative intelligent edge
platform, Jetson AGX Orin. The experimental results demonstrate the effectiveness of CoAxNN, which achieves
up to 1.53× speedup while reducing energy by up to 34.61%, with trivial accuracy loss on CIFAR-10/100 and
CINIC-10 datasets.

1. Introduction

Convolutional neural networks (CNNs) have achieved remarkable
success in various intelligent tasks such as image classification [1]."
"up to 1.53× speedup while reducing energy by up to 34.61%, with trivial accuracy loss on CIFAR-10/100 and
CINIC-10 datasets.

1. Introduction

Convolutional neural networks (CNNs) have achieved remarkable
success in various intelligent tasks such as image classification [1].
To pursue superior performance on complex intelligent tasks, CNNs
are becoming wider and deeper, leading to tremendous computational
costs and expensive energy consumption for model execution. Recently,
on-device deep learning has been a mainstay due to its immeasurable
potential for privacy protection and real-time response. However, it is
hard to deploy complicated neural network models on edge devices due
to the limited resources.

Many efforts have been made to enable efficient on-device deep
learning via model approximation. For instance, pruning-based strate-
gies [2] compress a neural network model by reducing redundant
neurons and connections and quantization-based methods [3] improve"
"to the limited resources.

Many efforts have been made to enable efficient on-device deep
learning via model approximation. For instance, pruning-based strate-
gies [2] compress a neural network model by reducing redundant
neurons and connections and quantization-based methods [3] improve

the efficiency of model execution by leveraging low-precision compu-
tations. In addition to these model compression techniques, emerging
staging-based approximate strategies, such as early exiting, improve
model performance by conditional execution at runtime.

While these methods optimize the deep neural network models from
different directions, we found that it is still a challenging problem
to effectively combine them (as described in Section 2.4). To achieve
efficient on-device inference, it is needed to take full advantage of the
superiority of different optimization strategies. Different approximate
strategies, based on distinct principles, have their own configuration"
"to effectively combine them (as described in Section 2.4). To achieve
efficient on-device inference, it is needed to take full advantage of the
superiority of different optimization strategies. Different approximate
strategies, based on distinct principles, have their own configuration
parameters. When combining different strategies, the configuration
parameters of different strategies may affect each other, influencing the
optimization effect of the model, and even leading to poor optimization
results. As such, this paper aims to address the following challenging
problem: How to design an efficient model optimization framework to make

∗ Corresponding author at: State Key Lab of Processors, Institute of Computing Technology, Chinese Academy of Sciences, Beijing, China.

E-mail addresses:

liguangli@ict.ac.cn (G. Li), maxiu18@mails.jlu.edu.cn (X. Ma), yuqiuchu19@mails.ucas.ac.cn (Q. Yu), liulei@jlu.edu.cn (L. Liu),

liuhuaxiao@jlu.edu.cn (H. Liu), wangxueying@ict.ac.cn (X. Wang)."
"E-mail addresses:

liguangli@ict.ac.cn (G. Li), maxiu18@mails.jlu.edu.cn (X. Ma), yuqiuchu19@mails.ucas.ac.cn (Q. Yu), liulei@jlu.edu.cn (L. Liu),

liuhuaxiao@jlu.edu.cn (H. Liu), wangxueying@ict.ac.cn (X. Wang).

1 Guangli Li and Xiu Ma contributed equally to this work.

https://doi.org/10.1016/j.sysarc.2023.102978
Received 24 April 2023; Received in revised form 18 July 2023; Accepted 23 August 2023

JournalofSystemsArchitecture143(2023)102978Availableonline25August20231383-7621/©2023ElsevierB.V.Allrightsreserved.G. Li et al.

full use of various model approximate strategies, so as to optimize on-device
deep learning while meeting accuracy requirements?

In this paper, we present a novel neural network optimization
framework, CoAxNN (Conditional Approximate Neural Networks),
which effectively combines staging-based and pruning-based approx-
imate strategies, for efficient on-device deep learning. The staging-
based approximate strategy optimizes the model structure as multiple"
"framework, CoAxNN (Conditional Approximate Neural Networks),
which effectively combines staging-based and pruning-based approx-
imate strategies, for efficient on-device deep learning. The staging-
based approximate strategy optimizes the model structure as multiple
stages with different complexities by attaching multiple exit branches,
whereas the pruning-based approximate strategy compresses the model
parameters according to the importance of filters. CoAxNN takes ac-
count of both their optimization principles and automatically searches
for reasonable configuration parameters to construct a compressed
multi-stage neural network model, thus taking full advantage of the
superiority of different approximate strategies to achieve efficient
model optimization. The optimization techniques, including pruning
and staging, have been studied individually in the past; the key novelty
of our work is to provide an effective and efficient mechanism to
combine them, so as to optimize the neural network performance with"
"model optimization. The optimization techniques, including pruning
and staging, have been studied individually in the past; the key novelty
of our work is to provide an effective and efficient mechanism to
combine them, so as to optimize the neural network performance with
a reasonable configuration, for a given task and a platform.
The main contributions of this paper are as follows:

• We present a novel neural network optimization framework,
namely CoAxNN, which effectively combines staging-based and
pruning-based approximate strategies, thereby improving actual
performance while meeting accuracy requirements, for efficient
on-device model inference.

• According to the principles of staging-based and pruning-based
approximate strategies, our framework constructs the design
space, and automatically searches for reasonable configuration
parameters, including the number of stages, the position of stages,
the threshold of stages, and the pruning rate, so as to make"
"approximate strategies, our framework constructs the design
space, and automatically searches for reasonable configuration
parameters, including the number of stages, the position of stages,
the threshold of stages, and the pruning rate, so as to make
full use of the advantages of both to achieve efficient model
optimization.

• We validate the effectiveness of CoAxNN by optimizing state-of-
the-art deep neural networks on a commercial edge device, Jetson
AGX Orin, in terms of prediction accuracy, execution latency, and
energy consumption, and experimental results show that CoAxNN
can significantly improve the performance of model inference
with trivial accuracy loss.

The rest of the paper is organized as follows. The background and
motivation are introduced in Section 2. The details of our optimization
framework are described in Section 3. The experimental evaluation
is conducted in Section 4. A discussion is given in Section 5. The
conclusion is presented in Section 6.

2. Background and motivation"
"motivation are introduced in Section 2. The details of our optimization
framework are described in Section 3. The experimental evaluation
is conducted in Section 4. A discussion is given in Section 5. The
conclusion is presented in Section 6.

2. Background and motivation

2.1. Pruning-based approximation

Neural network pruning, one of the most representative model com-
pression techniques, approximates the original neural network model
by reducing redundant neurons or connections making less contribu-
tion to model performance. Most previous works on pruning-based
approximation can be roughly divided into two categories: unstructured
pruning and structured pruning.

Prior works on weight pruning [4,5] achieve high non-structured
sparsity of pruned models by removing single parameters in a fil-
ter. Guo et al. [4] and Hal et al. [5] used magnitude-based pruning
methods, which eliminate weights with the smallest magnitude. Guo
et al. [4] proposed dynamic network surgery to reduce the network"
"sparsity of pruned models by removing single parameters in a fil-
ter. Guo et al. [4] and Hal et al. [5] used magnitude-based pruning
methods, which eliminate weights with the smallest magnitude. Guo
et al. [4] proposed dynamic network surgery to reduce the network
complexity by making on-the-fly connection pruning. Hal et al. [5]
pruned low-weight connections to reduce the storage and computation
demands by an order of magnitude. Some pruning research groups
utilize first-order or second-order derivatives of the loss function with
respect to the weights [6,7]. Hassibi et al. [6] proposed Optimal Brain
Damage (OBD), which uses all second-order derivatives of the loss

function to prune single non-essential weights. Optimal Brain Surgeon
(OBS) [7] have optimized the OBD method, which considers the condi-
tion that the Hessian matrix is non-diagonal. These approaches show
attractive theoretical performance improvement but are difficult to
be supported by existing software and hardware. Unstructured sparse"
"(OBS) [7] have optimized the OBD method, which considers the condi-
tion that the Hessian matrix is non-diagonal. These approaches show
attractive theoretical performance improvement but are difficult to
be supported by existing software and hardware. Unstructured sparse
models require specific matrix multiplication calculations and stor-
age formats, which can hardly leverage existing high-efficiency BLAS
libraries.

Unlike the early efforts on unstructured pruning that may cause
irregular calculation patterns, structured pruning reduces redundant
computations on unimportant filters or channels to produce a struc-
tured sparse model. The corresponding feature maps can be deleted as
the filters are pruned. Therefore, much recent work has focused on filter
pruning methods. SFP [2] and ASFP [8] dynamically pruned the filters
in a soft manner, which zeroizes the unimportant filters and keeps
updating them in the training stage. Li et al. [9] presented a fusion-"
"the filters are pruned. Therefore, much recent work has focused on filter
pruning methods. SFP [2] and ASFP [8] dynamically pruned the filters
in a soft manner, which zeroizes the unimportant filters and keeps
updating them in the training stage. Li et al. [9] presented a fusion-
catalyzed filter pruning approach, which simultaneously optimizes the
parametric and non-parametric operators. Luo et al. [10] pruned filters
based on statistics information computed from its next layer. The filters
of different layers may have different influences on model prediction. Li
et al. [11] proposed a flexible-rate filter pruning approach, FlexPruner,
which automatically selects the number of filters to be pruned for
each layer. Plochaet et al. [12] introduced a hardware-aware pruning
method with the goal of decreasing the inference time for FPGA deep
learning accelerators, adaptively pruning the neural network based on
the size of the systolic array used to calculate the convolutions. To"
"each layer. Plochaet et al. [12] introduced a hardware-aware pruning
method with the goal of decreasing the inference time for FPGA deep
learning accelerators, adaptively pruning the neural network based on
the size of the systolic array used to calculate the convolutions. To
preserve the robustness at a high sparsity ratio in structured pruning,
Zhuang et al. [13] proposed an effective filter importance criterion to
evaluate the importance of filters by estimating their contribution to
the adversarial training loss. Besides, some researchers have found
the value of network pruning in discovering the network architecture
[14,15]. Liu et al. [14] demonstrated that in some cases pruning can
be useful as an architecture search paradigm. Li et al. [15] proposed
a random architecture search to find a good architecture given a
pre-defined model by channel pruning. Li et al. [16] proposed an end-
to-end channel pruning method to search out the desired sub-network"
"be useful as an architecture search paradigm. Li et al. [15] proposed
a random architecture search to find a good architecture given a
pre-defined model by channel pruning. Li et al. [16] proposed an end-
to-end channel pruning method to search out the desired sub-network
automatically and efficiently, which learns per-layer sparsity through
depth-wise binary convolution. Ding et al. [17] presented a neural
architecture search with pruning method, which derives the most po-
tent model by removing trivial and redundant edges from the whole
neural network topology. The structured sparse model can be perfectly
supported by existing libraries to achieve a realistic acceleration. In
this paper, we adopt filter pruning to realize practical performance
improvement for neural network models.

2.2. Staging-based approximation

Prior studies [18,19] found that the difficulty of classifying an image
in real-world scenarios is diverse. The easy samples can be classified"
"this paper, we adopt filter pruning to realize practical performance
improvement for neural network models.

2.2. Staging-based approximation

Prior studies [18,19] found that the difficulty of classifying an image
in real-world scenarios is diverse. The easy samples can be classified
with low effort, and difficult samples consume more computation ef-
forts for prediction. Staging-based approximate strategies, such as early
exiting [18] and layer skipping [20], emerge as a prominent important
technique for separating the classification of easy and hard inputs.
The original neural network uses a fixed computation process for the
prediction of all samples. Staging-based approximate strategies perform
adaptive computing for samples according to conditions at run-time.
Teerapittayanon et al. [18] demonstrated that the deep neural network
with additional side branch classifiers can both improve accuracy and
significantly reduce the inference time of the network. Panda et al. [19]"
"adaptive computing for samples according to conditions at run-time.
Teerapittayanon et al. [18] demonstrated that the deep neural network
with additional side branch classifiers can both improve accuracy and
significantly reduce the inference time of the network. Panda et al. [19]
proposed Conditional Deep Learning cascading a linear network for
each convolutional layer and monitoring the output of the linear net-
work to decide whether classification can be terminated at the current
stage or not. Fang et al. [21] presented an input-adaptive framework
for video analytics, which adopts an architecture search-based scheme
to find the optimal architecture for each early exit branch. Wang

JournalofSystemsArchitecture143(2023)1029782G. Li et al.

et al. [22] designed dynamic layer-skipping mechanisms, which sup-
press unnecessary costs for easy samples and halt inference for all
samples to meet resource constraints for the inference of more compli-"
"JournalofSystemsArchitecture143(2023)1029782G. Li et al.

et al. [22] designed dynamic layer-skipping mechanisms, which sup-
press unnecessary costs for easy samples and halt inference for all
samples to meet resource constraints for the inference of more compli-
cated CNN backbones. Figurnov et al. [23] studied early termination in
each residual unit of ResNets. Farhadi et al. [23] implemented an early-
exiting method on the FPGA platform using partial reconfiguration to
reduce the amount of needed computation. Jayakodi et al. [24] used
Bayesian Optimization to configure the early exit neural networks to
trade off accuracy and energy. To reduce unnecessary intermediate
calculations in the inference process of Branchynet, Liang et al. [25]
directly determined the exit position of the sample in the multi-branch
network according to the difficulty of the sample without intermediate
trial errors. Jo et al. [26] proposed a low-cost early exit network, which"
"calculations in the inference process of Branchynet, Liang et al. [25]
directly determined the exit position of the sample in the multi-branch
network according to the difficulty of the sample without intermediate
trial errors. Jo et al. [26] proposed a low-cost early exit network, which
significantly improves energy efficiencies by reducing the parameters
used in inference with efficient branch structures.
In this paper, we
achieve a multi-stage approximate model by early exiting to accelerate
model inference for input samples in real-world scenarios.

2.3. Design space exploration

Design space exploration (DSE) is a systematic analysis method,
which searches for the optimal solutions in a large design space accord-
ing to the requirements. For example, in the staging-based approximate
strategy, deciding whether or not an exit branch should be inserted at
some position in the middle of the neural network model, and how
the thresholds for each exit point should be set can be seen as a DSE"
"ing to the requirements. For example, in the staging-based approximate
strategy, deciding whether or not an exit branch should be inserted at
some position in the middle of the neural network model, and how
the thresholds for each exit point should be set can be seen as a DSE
problem. Panda et al. [19] and Teerapittayanon et al. [18] empirically
set the location and threshold for each exit in the conditional neural
network model. Jayakodi et al. [24] found the best thresholds via
Bayesian Optimization for the specified trade-off between accuracy
and energy consumption of inference. Park et al. [27] systematically
determined the locations and thresholds of exit branches by genetic
algorithm. Park et al. [28] integrated the once-for-all technique and
BPNet, which consider architectures of base network and exit branches
simultaneously in the same search process. Besides, the fine-grained fil-
ter pruning, that is assign reasonable pruning rates for different layers,"
"algorithm. Park et al. [28] integrated the once-for-all technique and
BPNet, which consider architectures of base network and exit branches
simultaneously in the same search process. Besides, the fine-grained fil-
ter pruning, that is assign reasonable pruning rates for different layers,
can also be considered as a classic DSE problem. Li et al. [11] proposed
a flexible-rate filter pruning method, which selects the filters to be
pruned with a greedy-based strategy. He et al. [29] sampled design
space using reinforcement learning, which performs customizing prun-
ing for each layer, thus improving model compression. Qian et al. [30]
proposed a hierarchical threshold pruning method, which considers
the filter importance within relatively redundant layers instead of all
layers, achieving layerwise pruning for a better network structure. In
this paper, we regard the configuration parameters of staging-based
and pruning-based approximate strategies as the whole design space"
"the filter importance within relatively redundant layers instead of all
layers, achieving layerwise pruning for a better network structure. In
this paper, we regard the configuration parameters of staging-based
and pruning-based approximate strategies as the whole design space
and employ a genetic algorithm(GA)-based DSE to automatically find
the (near-)optimal configuration to effectively combine them, achieving
efficient on-device inference. In the future, we will consider setting
reasonable pruning rates for different layers.

2.4. Motivation

The pruning-based approximate strategy focuses on compressing the
model, which reduces the computation costs by deleting unimportant
parameters in the model, so how to set the pruning rate needs to be
considered. The staging-based approximate strategy concentrates on
improving the execution speed of the model, which allows the inference
of most simple samples to terminate with a good prediction in the"
"parameters in the model, so how to set the pruning rate needs to be
considered. The staging-based approximate strategy concentrates on
improving the execution speed of the model, which allows the inference
of most simple samples to terminate with a good prediction in the
earlier stage by attaching multiple exits in the original model. How
to place the exits and how to set a threshold for each exit should
be considered for the design of a staging-based approximate strategy.
Combining different approximate strategies will involve more configu-
ration parameters and the approximate strategies may affect each other,
which potentially influences the effect of the model optimization.

Fig. 1. The optimization effect for ResNet-56 using different configuration parameters
under the specified accuracy requirement.

Fig. 1 shows the optimization effect of the ResNet-56 using different
configuration parameters under the specified requirements of accuracy"
"Fig. 1. The optimization effect for ResNet-56 using different configuration parameters
under the specified accuracy requirement.

Fig. 1 shows the optimization effect of the ResNet-56 using different
configuration parameters under the specified requirements of accuracy
on the CIFAR-10 dataset, where the triples (𝑥, 𝑦, 𝑧) represent the number
of stages, stage threshold, and pruning rate, respectively. Fig. 1(a), (b),
and (c) show the computational costs (normalized to the computational
cost of the baseline model) of various optimization configurations
when the accuracy is 98.1%, 98.7%, and 98.8% (normalized to the
accuracy of the baseline model). In practice, a certain error can be
allowed in model accuracy (±0.001), for example, 98.09%, and 98.12%
both meet the requirement of 98.1%. The relationship between the
number of stages and the computational cost is not regular, which
will be affected by the stage threshold and pruning rate, for example,"
"allowed in model accuracy (±0.001), for example, 98.09%, and 98.12%
both meet the requirement of 98.1%. The relationship between the
number of stages and the computational cost is not regular, which
will be affected by the stage threshold and pruning rate, for example,
in Fig. 1(c), the computational cost of (3,0.08,0) with more stages is
larger than (2,0.1,0.1), the computational cost of (2,0.1,0.1) with fewer
stages is larger than (3,0.2,0.1). Besides, affected by the staging-based
optimization, the computational costs of the optimized model at a high
pruning rate may be larger than that at low pruning rates, for example,
in Fig. 1(b), the configuration of (2,0.08,0.3) with a pruning rate of 0.3
has more computational costs than (3,0.2,0.2) with a pruning rate of
0.2. In Fig. 1(a), we can observe from the partial experimental results
that at the accuracy requirement of 98.1%, the computation of the
optimized models using three stages is less than that of the model using"
"has more computational costs than (3,0.2,0.2) with a pruning rate of
0.2. In Fig. 1(a), we can observe from the partial experimental results
that at the accuracy requirement of 98.1%, the computation of the
optimized models using three stages is less than that of the model using
two stages. But this law does not apply to the optimization effect of
other accuracy requirements such as 98.7% and 98.8%. It is observed
that the optimization effects of different configuration parameters are
distinct and irregular under the specified accuracy requirement, and
thus it is difficult to find an optimal model. This example shows that

JournalofSystemsArchitecture143(2023)1029783G. Li et al.

Fig. 2. The optimization effect of staging-based strategy, pruning-based strategy, and
CoAxNN for ResNet-56 on the CIFAR-10.

it is challenging to combine different approximate strategies to achieve
efficient optimization for neural network models.

In this paper, for a specified accuracy requirement, we focus on"
"CoAxNN for ResNet-56 on the CIFAR-10.

it is challenging to combine different approximate strategies to achieve
efficient optimization for neural network models.

In this paper, for a specified accuracy requirement, we focus on
combining the principles of different approximate strategies to con-
struct a design space and automatically search for reasonable con-
figuration parameters, giving full play to the advantages of different
approximate strategies to achieve efficient optimization for neural net-
work models. As shown in Fig. 2, at the accuracy requirement of
99.6%, for the staging-based optimization strategy, the two stages are
used and the threshold is set to 0.07 for each stage, the normalized
computational cost is 0.89. For the pruning-based optimization, the
pruning rate is set to 0.1, and the normalized computational cost is
0.89. CoAxNN effectively combines pruning-based and staging-based
strategies, whose computational cost is 0.64, greatly improving the
computational performance."
"computational cost is 0.89. For the pruning-based optimization, the
pruning rate is set to 0.1, and the normalized computational cost is
0.89. CoAxNN effectively combines pruning-based and staging-based
strategies, whose computational cost is 0.64, greatly improving the
computational performance.

3. Methodology

3.1. Overview

In this paper, we propose an efficient optimization framework for
neural network models, CoAxNN, which automatically searches for
reasonable configuration parameters through a GA-based DSE. CoAxNN
effectively combines staging-based with pruning-based approximate
strategies to make full use of the superiority of both, thereby improving
the actual performance while meeting the accuracy requirements for
neural network models.

The overview of the CoAxNN is shown in Fig. 3. First, for the
original deep neural network model, CoAxNN performs staging-based
and pruning-based approximate strategies according to the genes of
the chromosome for each individual, which generates a compressed"
"neural network models.

The overview of the CoAxNN is shown in Fig. 3. First, for the
original deep neural network model, CoAxNN performs staging-based
and pruning-based approximate strategies according to the genes of
the chromosome for each individual, which generates a compressed
multi-stage model. According to the availability of stages in the gene,
CoAxNN attaches exit branches to the original model to build a multi-
stage conditional activation model. According to the threshold of each
stage, CoAxNN predicts input samples, having distinct difficulties, by
multiple stages with different computational complexities, with the
entropy-aware activation manner. The obtained multi-stage model is
compressed by removing unimportant filters, thereby further reducing
computational costs. Next, CoAxNN evaluates the fitness of the cor-
responding individual according to the accuracy and latency of the
compressed multi-stage model and sorts the individuals according to"
"compressed by removing unimportant filters, thereby further reducing
computational costs. Next, CoAxNN evaluates the fitness of the cor-
responding individual according to the accuracy and latency of the
compressed multi-stage model and sorts the individuals according to
their fitness. Then, the chromosome pool is updated, generating the
next generation of individuals. After the evolution of multiple gener-
ations, which repeat the above steps, we can obtain several individuals
that have optimal performance.

3.2. Staging-based approximate optimization

In general, executing a neural network model is a one-staged ap-
proach, which processes all the inputs in the same manner, i.e., starting
from the input operator and performing it operator by operator until
the final exit operator. Prior studies [19] found that classification
difficulty varies widely across inputs in real-world scenarios. Different
computational complexities need to be considered when predicting in-"
"from the input operator and performing it operator by operator until
the final exit operator. Prior studies [19] found that classification
difficulty varies widely across inputs in real-world scenarios. Different
computational complexities need to be considered when predicting in-
puts. Most of the input samples can be correctly classified by employing

a part of a neural network, without the computation effort of the
entire neural network. Early exiting strategy often comes into play,
which allows simple inputs to exit early with a good prediction by the
addition of multiple exit points. By leveraging the early exiting strategy,
CoAxNN achieves a staging-based approximation to give an early exit-
ing opportunity for simple inputs. We denote a neural network model
as  = {𝑓1, 𝑓2, … , 𝑓𝑚} which consists of 𝑚 operators. In CoAxNN, a
multi-stage model,  ∗, can be formalized as follows:

 ∗ =

𝜏
⋃

𝑖=0


𝑖

(1)

where 𝜏 is the number of stages.

The "
"ing opportunity for simple inputs. We denote a neural network model
as  = {𝑓1, 𝑓2, … , 𝑓𝑚} which consists of 𝑚 operators. In CoAxNN, a
multi-stage model,  ∗, can be formalized as follows:

 ∗ =

𝜏
⋃

𝑖=0


𝑖

(1)

where 𝜏 is the number of stages.

The 

𝑖 is an approximate model with the staging-based strategy,

which can be formalized as:



𝑖 =

{

𝑖 + 

𝑖 + 
 , 𝑖 = 𝜏

𝑖, 1 ⩽ 𝑖 < 𝜏

(2)

𝑖 = {𝑓1, 𝑓2, … , 𝑓𝑝𝑖

where 
} represents a part of the original neural
network with 𝑝𝑖 operators, 
, … , 𝑓 ∗
, 𝑓 ∗
} represents an addi-
𝑏𝑖
2
tional exit branch with 𝑏𝑖 operators, and 
𝑖 = {𝑐𝑖, 𝜀𝑖} represents an exit
checker, containing a threshold 𝜀𝑖 and a conditional activation operator
𝜏 =  denotes the original (main)
𝑐𝑖 using threshold 𝜀𝑖. Especially, 
neural network model.

𝑖 = {𝑓 ∗
1

It is non-trivial to design a staging-based approximate strategy
for adaptive conditional inference of a multi-stage model, and the
following factors need to be considered:

• Number of "
"𝜏 =  denotes the original (main)
𝑐𝑖 using threshold 𝜀𝑖. Especially, 
neural network model.

𝑖 = {𝑓 ∗
1

It is non-trivial to design a staging-based approximate strategy
for adaptive conditional inference of a multi-stage model, and the
following factors need to be considered:

• Number of 

𝒊. A multi-stage model with arbitrary exits can be built
by stage availability. However, fewer exits cannot cover the diverse
difficulty of classification of input samples, whereas too many exits
increase the latency of hard samples that do not exit early.

• Selection of Attached Position (𝒑𝒊) for 

𝒊. The exits at a more for-
mer position cannot provide satisfactory accuracy, while redundant
computation may be involved at a more latter exit. Besides, attached
exit branches may also interfere with a variety of computational
graph optimization methods, such as operator fusion and mem-
ory reuse, provided by the deep learning frameworks, increasing
operation counts, data movement, and other system overheads."
"exit branches may also interfere with a variety of computational
graph optimization methods, such as operator fusion and mem-
ory reuse, provided by the deep learning frameworks, increasing
operation counts, data movement, and other system overheads.

• Confidence Threshold (𝜺𝒊) of 

𝒊. The confidence threshold is used
to determine whether the prediction result of stage 
𝑖 has sufficient
confidence. With a higher threshold, complex samples may finish
predictions from the previous exits with lower accuracy, and using
a lower threshold, simple samples may use more complex compu-
tations to complete inference, due to cannot end from the previous
classifiers, incurring additional computational overheads.

• Structure Design for 
𝒊. The structure of each exit branch (
𝑖)
is not identical. Each 
, 𝑓 ∗
𝑖 consists of several operators ({𝑓 ∗
, … ,
2
1
𝑓 ∗
𝑏𝑖−1}) used for feature extraction and a linear classifier 𝑓 ∗
. Feature
𝑏𝑖
extraction operators receive the intermediate feature map from 𝑓𝑝𝑖"
"• Structure Design for 
𝒊. The structure of each exit branch (
𝑖)
is not identical. Each 
, 𝑓 ∗
𝑖 consists of several operators ({𝑓 ∗
, … ,
2
1
𝑓 ∗
𝑏𝑖−1}) used for feature extraction and a linear classifier 𝑓 ∗
. Feature
𝑏𝑖
extraction operators receive the intermediate feature map from 𝑓𝑝𝑖
and extract more high-level features in the form required by a
subsequent linear classifier. The configuration and complexity of the
intermediate feature maps for different depths of the main neural
networks are varying, making the design of 
𝑖 arduous. The 𝑓 ∗
𝑏𝑖
operator is used to produce classification results based on the output
of 𝑓 ∗
is different at
each 
𝑖.

𝑏𝑖−1, and the number of input feature maps for 𝑓 ∗
𝑏𝑖

To effectively utilize the early-exiting method to build an approxi-
mate multi-stage model, our approach carefully designs principles for
each module.

• Setting of Number (𝜏) and Attached Position (𝒑𝒊) of 
𝒊. The
number (𝜏) and the position (𝒑𝒊) of the exit branches (
𝒊) are two"
"𝑏𝑖

To effectively utilize the early-exiting method to build an approxi-
mate multi-stage model, our approach carefully designs principles for
each module.

• Setting of Number (𝜏) and Attached Position (𝒑𝒊) of 
𝒊. The
number (𝜏) and the position (𝒑𝒊) of the exit branches (
𝒊) are two
factors that will affect each other. Some unnecessary exits may
be inserted, having little improvement in accuracy but leading to
non-negligible computational overheads, when the number of exit
branches is large. When the position of the exit branch is not

JournalofSystemsArchitecture143(2023)1029784G. Li et al.

Fig. 3. Overview of CoAxNN.

• Structure Design for 

reasonable, and cannot distinguish the difficulty of the sample, it
is difficult to increase the number of exit branches to reduce the
computational cost while meeting the model accuracy requirement.
To address this problem, CoAxNN puts the availability of each stage
into the design space of the GA, and each available stage corresponds"
"is difficult to increase the number of exit branches to reduce the
computational cost while meeting the model accuracy requirement.
To address this problem, CoAxNN puts the availability of each stage
into the design space of the GA, and each available stage corresponds
to a new exit branch. The availability of the stage can control the
number and position of exit branches at the same time. Besides,
to introduce fewer new model structures and preserve the existing
graph optimizations, CoAxNN chooses to attach exit branches 
𝑖 at
the end of the group of building blocks. It is noted that CoAxNN does
not attach the exit branch after the last group of building blocks, as
there is already an existing original exit for the original backbone.
𝒊. We introduce a feature extractor and
a linear classifier for each exit branch 
𝑖. The structure of the
feature extractor is designed with the building block as granularity.
This design not only retains the original neural network structure"
"𝒊. We introduce a feature extractor and
a linear classifier for each exit branch 
𝑖. The structure of the
feature extractor is designed with the building block as granularity.
This design not only retains the original neural network structure
but also provides more opportunities for system-level optimizations.
Generally, operators for feature extraction also contain non-linear
activation operators such as rectified linear units, and normalization
operators such as batch normalization. Besides, prior studies [24]
revealed that the output feature maps of operators at shallow depths
of a neural network have a relatively large height and width, which
results in a large number of input feature maps being passed to the
linear classifier of former exits, thus leading to a long latency for
easy samples that exit early. As such, in CoAxNN, we add an extra
pooling operator after the last feature extraction operator of shallow

𝑖.

• Confidence Measure in 

𝒊. The 

𝑖. A reliable "
"linear classifier of former exits, thus leading to a long latency for
easy samples that exit early. As such, in CoAxNN, we add an extra
pooling operator after the last feature extraction operator of shallow

𝑖.

• Confidence Measure in 

𝒊. The 

𝑖. A reliable 

𝑖 takes a threshold checking step,
which determines whether an input returns from the current exit
or continues to the next exit according to the prediction result
of 
𝑖 should have the ability to identify whether
the classification results are sufficiently confident. There are var-
ious methods [31], including maximum probability, entropy, and
margin, for the design of 
𝑖. Prior work [24] has demonstrated
the performance of the aforementioned three confidence types is
almost identical. CoAxNN chooses to use the entropy of predicted
probability as the entropy-aware activation operator (𝑐𝑖) to evaluate
the confidence of the prediction result for the input sample (𝑥) of
the 𝑖th stage classifier, as follows:

entropy( ̂𝑦𝑖) =

𝐶
∑

𝑐=1"
"almost identical. CoAxNN chooses to use the entropy of predicted
probability as the entropy-aware activation operator (𝑐𝑖) to evaluate
the confidence of the prediction result for the input sample (𝑥) of
the 𝑖th stage classifier, as follows:

entropy( ̂𝑦𝑖) =

𝐶
∑

𝑐=1

̂𝑦𝑖(𝑐) log ̂𝑦𝑖(𝑐)

(3)

where ̂𝑦𝑖 is the probability distribution of the output of the linear
classifier 𝑓 ∗
on different classification labels, calculated by the soft-
𝑏𝑖
max operator, and 𝐶 is the number of classes. An entropy threshold
𝜀𝑖 is used to decide whether an input returns the prediction of the
current exit or activates the latter operators. A higher confidence
value implies that the input sample that arrived at the current exit is
hard and needs to be processed by a more complex stage to complete
accurate classification.

3.3. Pruning-based approximate optimization

In addition to the staging-based approximate strategy that pro-
vides adaptive computing based on conditional activation at runtime,"
"hard and needs to be processed by a more complex stage to complete
accurate classification.

3.3. Pruning-based approximate optimization

In addition to the staging-based approximate strategy that pro-
vides adaptive computing based on conditional activation at runtime,
CoAxNN also integrates a pruning-based approximate strategy to com-
press model size. The neural network pruning technique has been
widely studied by researchers, which can be broadly categorized as
structured and unstructured pruning. Structured pruning such as filter
pruning has higher computational efficiency than unstructured prun-
ing [32]. Especially, filter pruning is employed, which not only deletes
redundant computations of unimportant filters but also leads to the
removal of corresponding feature maps, providing realistic performance
improvements. In CoAxNN, we utilize the filter pruning method to
compress the multi-stage model and quantify the importance of each
filter in a convolutional operator based on the 𝓁2-norm:



‖
‖

‖2 ="
"removal of corresponding feature maps, providing realistic performance
improvements. In CoAxNN, we utilize the filter pruning method to
compress the multi-stage model and quantify the importance of each
filter in a convolutional operator based on the 𝓁2-norm:



‖
‖

‖2 =
𝑟‖

√
√
√
√
√

𝑘
∑

𝑚
∑

𝑛
∑

𝑡=1

𝑖=1

𝑗=1

𝑤2

𝑡,𝑖,𝑗

(4)

where 
𝑟 indicates the 𝑟th filter in a convolutional operator, 𝑤𝑡,𝑖,𝑗
denotes an element of 
𝑟 that resides in the 𝑖th row and 𝑗th column
in the 𝑡th channel, 𝑘 denotes the input channels, 𝑚 denotes the height
of filters, and 𝑛 denotes the width of filters. The filters with smaller 𝓁2-
norm will be given higher priority to be pruned than those of higher
𝓁2-norm. To keep the model capacity and minimize the loss of accuracy
as much as possible, we utilize a dynamic pruning scheme [2] for
staging-based approximate CNNs, which zeroes the pruned filters and
keeps updating them in the re-training process.

3.4. Training of CoAxNN"
"𝓁2-norm. To keep the model capacity and minimize the loss of accuracy
as much as possible, we utilize a dynamic pruning scheme [2] for
staging-based approximate CNNs, which zeroes the pruned filters and
keeps updating them in the re-training process.

3.4. Training of CoAxNN

Joint training trains all classifiers in a neural network model at
the same time, which is widely used in the training process of neural
network models with exit branches [18,27]. It defines a loss function
for each classifier and minimizes the weighted sum of loss functions
for all classifiers during training. Therefore, each classifier provides
regularization for others to alleviate the overfitting of the model.
CoAxNN utilizes joint training optimization to train the backbone
neural network and exit branches at the same time and minimize the
weighted sum of the cross-entropy loss functions of all stages, denoted
as follows:



joint =

𝜏
∑

𝑖=1

𝜆𝑖

CE(𝑦, ̄𝑦𝑖)


(5)"
"CoAxNN utilizes joint training optimization to train the backbone
neural network and exit branches at the same time and minimize the
weighted sum of the cross-entropy loss functions of all stages, denoted
as follows:



joint =

𝜏
∑

𝑖=1

𝜆𝑖

CE(𝑦, ̄𝑦𝑖)


(5)

where 𝜆𝑖 represents the weight of the loss function of the 𝑖th stage,
𝑦 is the real classification of 𝑥 which is shared by all stages, ̄𝑦𝑖 is the
output of linear classifier 𝑓 ∗
of the 𝑖th stage, and the cross-entropy loss
𝑏𝑖
function 

CE is calculated as follows:

CE(𝑦, ̄𝑦𝑖) = −


𝐶
∑

𝑐=1

𝑦(𝑐) log

e ̄𝑦𝑖(𝑐)
𝑗=1 e ̄𝑦𝑖(𝑗)

∑𝐶

(6)

JournalofSystemsArchitecture143(2023)1029785G. Li et al.

The training process of CoAxNN is summarized in Algorithm 1.
It is given training dataset , training epochs 𝑒𝑝𝑜𝑐ℎ𝑚𝑎𝑥, batch size
𝜌, original deep neural network model  , number of stages 𝜏, the
weights for loss functions of all stages 𝜆, and the chromosome pool
𝑃 . First, based on the genes on the chromosome, CoAxNN performs a"
"It is given training dataset , training epochs 𝑒𝑝𝑜𝑐ℎ𝑚𝑎𝑥, batch size
𝜌, original deep neural network model  , number of stages 𝜏, the
weights for loss functions of all stages 𝜆, and the chromosome pool
𝑃 . First, based on the genes on the chromosome, CoAxNN performs a
staging-based optimization strategy, which approximates the original
neural network model as a multi-stage conditional activation model
by attaching exit branches (Lines 1–11). Then, the generated multi-
stage model is initialized randomly (Lines 12–13). Next, the model
is compressed and tuned according to the training data  and the
gene of the pruning rate (𝑃 [𝑝].𝑝𝑟𝑢𝑛𝑖𝑛𝑔_𝑟𝑎𝑡𝑒) on the chromosome in
𝑒𝑝𝑜𝑐ℎ𝑚𝑎𝑥 epochs (Lines 14–34). In each epoch, CoAxNN calculates the
loss function according to Eq. (5) and updates the weights by the
traditional backpropagation algorithm (Lines 15–26). Besides, for each
convolutional operator in the approximate multi-stage model, CoAxNN
obtains the number of filters (𝑡) and calculates the 𝓁2-norm of each"
"loss function according to Eq. (5) and updates the weights by the
traditional backpropagation algorithm (Lines 15–26). Besides, for each
convolutional operator in the approximate multi-stage model, CoAxNN
obtains the number of filters (𝑡) and calculates the 𝓁2-norm of each
filter according to Eq. (4), then the dynamic pruning scheme is used
to prune ⌊𝑡 × 𝑃 [𝑝].𝑝𝑟𝑢𝑛𝑖𝑛𝑔_𝑟𝑎𝑡𝑒⌋ filters with low 𝓁2-norm (Lines 27–33).
The pruned filters can be updated once it is found to be important at
any time, thus maintaining the learning ability of the model. In the
pruning process of each epoch, CoAxNN will reorder the importance
of filters for each convolutional operator, and select the filters to be
pruned. Finally, the trained model  ′ is obtained (Line 36).

Algorithm 1: CoAxNN training

Input: training data: , training epoch: 𝑒𝑝𝑜𝑐ℎ𝑚𝑎𝑥, batch size: 𝜌,

original model backbone:  , the number of stages: 𝜏, the
weights for loss functions: 𝜆, chromosomes: 𝑃

Output: trained models:  ′

1 for 𝑝 = 1 → 𝑃 .𝑠𝑖𝑧𝑒() do"
"Algorithm 1: CoAxNN training

Input: training data: , training epoch: 𝑒𝑝𝑜𝑐ℎ𝑚𝑎𝑥, batch size: 𝜌,

original model backbone:  , the number of stages: 𝜏, the
weights for loss functions: 𝜆, chromosomes: 𝑃

Output: trained models:  ′

1 for 𝑝 = 1 → 𝑃 .𝑠𝑖𝑧𝑒() do

// Generate model structure
 ′[𝑝] =  ;
for 𝑖 = 1 → 𝜏 − 1 do

if P[p][i] is available then
𝑖 from  ;
Construct 
𝑖 according to 
Construct 
Construct 
𝑖 according to 𝑃 [𝑝][𝑖].𝑡ℎ𝑟𝑒𝑠ℎ𝑜𝑙𝑑;

𝑖 + 
𝑖 + 
𝑖 = 
𝑖;
 ′[𝑝] =  ′[𝑝] ∪ 

𝑖;

𝑖;

end

end
// Tune and prune model parameters
 ′[𝑝] = LoadModel( ′[𝑝], 𝑖𝑛𝑖𝑡𝑎𝑙_𝑤𝑒𝑖𝑔ℎ𝑡𝑠);
𝑡𝑟𝑎𝑖𝑛_𝑏𝑎𝑡𝑐ℎ𝑒𝑠 = make_batch(, 𝜌);
for 𝑒𝑝𝑜𝑐ℎ = 1 → 𝑒𝑝𝑜𝑐ℎ𝑚𝑎𝑥 do

foreach (𝑖𝑛𝑝𝑢𝑡, 𝑡𝑎𝑟𝑔𝑒𝑡) ∈ 𝑡𝑟𝑎𝑖𝑛_𝑏𝑎𝑡𝑐ℎ𝑒𝑠 do
𝑜𝑢𝑡𝑝𝑢𝑡 =  ′[𝑝].forward(𝑖𝑛𝑝𝑢𝑡);
𝑤𝑒𝑖𝑔ℎ𝑡𝑒𝑑_𝑙𝑜𝑠𝑠 ← 0;
for 𝑖 = 1 → 𝜏 do

if P[p][i] is available then

𝑙𝑜𝑠𝑠 = CrossEntropy(𝑜𝑢𝑡𝑝𝑢𝑡[𝑖], 𝑡𝑎𝑟𝑔𝑒𝑡);
𝑤𝑒𝑖𝑔ℎ𝑡𝑒𝑑_𝑙𝑜𝑠𝑠 += 𝜆[𝑖] × 𝑙𝑜𝑠𝑠;

end

end
𝑤𝑒𝑖𝑔ℎ𝑡𝑒𝑑_𝑙𝑜𝑠𝑠 = 𝑤𝑒𝑖𝑔ℎ𝑡𝑒𝑑_𝑙𝑜𝑠𝑠∕𝑠𝑢𝑚(𝜆);
 ′[𝑝].backward(𝑤𝑒𝑖𝑔ℎ𝑡𝑒𝑑_𝑙𝑜𝑠𝑠);

end
foreach 𝑓 ∈  ′[𝑝] do

if 𝑓 .𝑡𝑦𝑝𝑒 == 𝐶𝑂𝑁𝑉 then"
"𝑤𝑒𝑖𝑔ℎ𝑡𝑒𝑑_𝑙𝑜𝑠𝑠 ← 0;
for 𝑖 = 1 → 𝜏 do

if P[p][i] is available then

𝑙𝑜𝑠𝑠 = CrossEntropy(𝑜𝑢𝑡𝑝𝑢𝑡[𝑖], 𝑡𝑎𝑟𝑔𝑒𝑡);
𝑤𝑒𝑖𝑔ℎ𝑡𝑒𝑑_𝑙𝑜𝑠𝑠 += 𝜆[𝑖] × 𝑙𝑜𝑠𝑠;

end

end
𝑤𝑒𝑖𝑔ℎ𝑡𝑒𝑑_𝑙𝑜𝑠𝑠 = 𝑤𝑒𝑖𝑔ℎ𝑡𝑒𝑑_𝑙𝑜𝑠𝑠∕𝑠𝑢𝑚(𝜆);
 ′[𝑝].backward(𝑤𝑒𝑖𝑔ℎ𝑡𝑒𝑑_𝑙𝑜𝑠𝑠);

end
foreach 𝑓 ∈  ′[𝑝] do

if 𝑓 .𝑡𝑦𝑝𝑒 == 𝐶𝑂𝑁𝑉 then

𝑡 ← the filters number of 𝑓 ;
Calculate the 𝓁2-norm for the filters;
Zeroize the lowest filters ⌊𝑡 × 𝑃 [𝑝].𝑝𝑟𝑢𝑛𝑖𝑛𝑔_𝑟𝑎𝑡𝑒⌋ filters;

2

3

4

5

6

7

8

9

10

11

12

13

14

15

16

17

18

19

20

21

22

23

24

25

26

27

28

29

30

31

32

33

34

3.5. GA-based design space exploration

To effectively combine the staging-based with the pruning-based ap-
proximate strategies, the design space of CoAxNN includes the number
of stages, the position of the stage, the threshold of the stage, and the
pruning rate, which is a very large search space. When the number of
stages is 𝜏, the search space for determining which stage is available
is 2𝜏 , the search space for thresholds is 𝑄𝜏 where 𝑄 is the number"
"of stages, the position of the stage, the threshold of the stage, and the
pruning rate, which is a very large search space. When the number of
stages is 𝜏, the search space for determining which stage is available
is 2𝜏 , the search space for thresholds is 𝑄𝜏 where 𝑄 is the number
of candidate thresholds, and the search space for pruning rate is 𝑅,
which indicates the number of candidate pruning rates. The parameter
configurations are searched independently, making the search space as
large as 2𝜏 × 𝑄𝜏 × 𝑅. It is laborious to explore the large parameter
space by brute force search. CoAxNN adopts the genetic algorithm
for the design space exploration. Genetic algorithm [33] is inspired by
biological evolution based on Charles Darwin’s theory of natural selec-
tion, which is often used to find the (near-)optimal solution in a large
search space. In CoAxNN, the number of genes on each chromosome is
2 × (𝜏 − 1) + 1. For the first 𝜏 − 1 stages, CoAxNN uses two genes, one for"
"biological evolution based on Charles Darwin’s theory of natural selec-
tion, which is often used to find the (near-)optimal solution in a large
search space. In CoAxNN, the number of genes on each chromosome is
2 × (𝜏 − 1) + 1. For the first 𝜏 − 1 stages, CoAxNN uses two genes, one for
whether the stage is available, and the other for the threshold of the
stage. In addition, CoAxNN also uses a gene to represent the pruning
ratio. The fitness of the single individual is represented by a 2-tuple
(𝑎𝑐𝑐𝑢𝑟𝑎𝑐𝑦, 𝑙𝑎𝑡𝑒𝑛𝑐𝑦). GA-based DSE aims to increase accuracy and reduce
latency, finding the (near-)optimal solutions for model performance.

Algorithm 2 shows that how accuracy and latency are evaluated
for individuals. It is given the test dataset , the number of stages 𝜏,
and the chromosome set 𝑃 . For each individual, CoAxNN obtains the
model 𝑛𝑒𝑡 configured with the corresponding gene (Line 4). Then, the
test dataset  is predicted by the model, and the result of prediction"
"for individuals. It is given the test dataset , the number of stages 𝜏,
and the chromosome set 𝑃 . For each individual, CoAxNN obtains the
model 𝑛𝑒𝑡 configured with the corresponding gene (Line 4). Then, the
test dataset  is predicted by the model, and the result of prediction
𝑜𝑢𝑡𝑝𝑢𝑡 is got (Line 6). For each input sample, CoAxNN traverses all
available stages and calculates the confidence 𝑒 of corresponding output
at this stage according to Eq. (3) (Lines 7–12). If the confidence (𝑒)
is less than the confidence threshold (𝜀𝑖) of this stage, the prediction
is ended, and the accuracy of the sample at this stage is added to
the accuracy score (𝛿[𝑝]) of the current individual (𝑝) (Lines 13–16).
The accuracy function returns 1 if the prediction is correct, and 0
otherwise. When the sample does not exit from the first 𝜏 − 1 stages,
it must be exited from the 𝜏th stage. Therefore, in the 𝜏th stage, the
accuracy is directly added to the accuracy score (𝛿[𝑝]) (Lines 17–19)."
"The accuracy function returns 1 if the prediction is correct, and 0
otherwise. When the sample does not exit from the first 𝜏 − 1 stages,
it must be exited from the 𝜏th stage. Therefore, in the 𝜏th stage, the
accuracy is directly added to the accuracy score (𝛿[𝑝]) (Lines 17–19).
The evaluation of latency is similar to accuracy. CoAxNN evaluates the
latency score (𝜇[𝑝]) using a similar manner as the accuracy score (𝛿[𝑝]),
which accumulates the latency of the backbone neural network and exit
branches until the end of the prediction (Line 8–10). For the latency,
we test the original network with all possible exit branches attached on
the target edge devices. The execution time of all operators is recorded.
Finally, the average accuracy score (𝛿) and average latency score (𝜇) for
all individuals are obtained (Lines 23–26).

GA-based DSE gets the (near-)optimal solutions about the goal of
accuracy and latency. Users choose the (near-)optimal solution among"
"Finally, the average accuracy score (𝛿) and average latency score (𝜇) for
all individuals are obtained (Lines 23–26).

GA-based DSE gets the (near-)optimal solutions about the goal of
accuracy and latency. Users choose the (near-)optimal solution among
them according to their requirements. If the accuracy requirement is
high, the model with the least computation cost is selected under a triv-
ial accuracy loss. If a certain accuracy loss can be tolerated, the model
with greatly less computation cost is selected. Finally, unavailable
branches and unimportant filters are removed to obtain an optimized
neural network model.

4. Evaluation

4.1. Experimental setting

end

end

end

Evaluation Platforms. We conduct optimization with PaddlePaddle,2
an open-sourced deep learning framework, for neural network models
on a server with Intel Xeon CPUs and an Nvidia A100 GPU. We evaluate

35 end
36 return  ′;

2 https://www.paddlepaddle.org.cn/en.

JournalofSystemsArchitecture143(2023)1029786G. Li et al."
"an open-sourced deep learning framework, for neural network models
on a server with Intel Xeon CPUs and an Nvidia A100 GPU. We evaluate

35 end
36 return  ′;

2 https://www.paddlepaddle.org.cn/en.

JournalofSystemsArchitecture143(2023)1029786G. Li et al.

Algorithm 2: Performance Collection

Input: test data: , the number of stages: 𝜏, chromosomes: 𝑃
Output: accuracy for each configuration of neural network
models: 𝛿, latency for each configuration of neural
network models: 𝜇

1 for 𝑝 = 1 → 𝑃 .𝑠𝑖𝑧𝑒() do

2

3

4

5

6

7

8

9

10

11

12

13

14

15

16

17

18

19

20

21

22

23

24

𝛿[𝑝] ← 0;
𝜇[𝑝] ← 0;
𝑛𝑒𝑡 = getModel(𝑃 [𝑝]);
foreach (𝑖𝑛𝑝𝑢𝑡, 𝑡𝑎𝑟𝑔𝑒𝑡) ∈  do
𝑜𝑢𝑡𝑝𝑢𝑡 = 𝑛𝑒𝑡.forward(𝑖𝑛𝑝𝑢𝑡);
for 𝑖 = 1 → 𝜏 do

𝜇[𝑝] += computeLatency(
if P[p][i] is available then

𝑖);

𝜇[𝑝] += computeLatency(⋃𝑖
if 𝑖 ≠ 𝜏 then



𝑗 );

𝑗=1

𝑒 ← Compute entropy of 𝑜𝑢𝑡𝑝𝑢𝑡[𝑖];
if 𝑒 < 𝜀𝑖 then

𝛿[𝑝] += accuracy(𝑜𝑢𝑡𝑝𝑢𝑡[𝑖], 𝑡𝑎𝑟𝑔𝑒𝑡);
break;

end

else

𝛿[𝑝] += accuracy(𝑜𝑢𝑡𝑝𝑢𝑡[𝑖], 𝑡𝑎𝑟𝑔𝑒𝑡);

end

end

end

end
𝛿[𝑝] = 𝛿[𝑝]∕.𝑠𝑖𝑧𝑒();"
"𝜇[𝑝] += computeLatency(
if P[p][i] is available then

𝑖);

𝜇[𝑝] += computeLatency(⋃𝑖
if 𝑖 ≠ 𝜏 then



𝑗 );

𝑗=1

𝑒 ← Compute entropy of 𝑜𝑢𝑡𝑝𝑢𝑡[𝑖];
if 𝑒 < 𝜀𝑖 then

𝛿[𝑝] += accuracy(𝑜𝑢𝑡𝑝𝑢𝑡[𝑖], 𝑡𝑎𝑟𝑔𝑒𝑡);
break;

end

else

𝛿[𝑝] += accuracy(𝑜𝑢𝑡𝑝𝑢𝑡[𝑖], 𝑡𝑎𝑟𝑔𝑒𝑡);

end

end

end

end
𝛿[𝑝] = 𝛿[𝑝]∕.𝑠𝑖𝑧𝑒();
𝜇[𝑝] = 𝜇[𝑝]∕.𝑠𝑖𝑧𝑒();

25 end
26 return (𝛿, 𝜇);

the realistic speedup and energy consumption of optimized models on
a representative intelligent edge platform, Jetson AGX Orin, integrated
with Ampere GPUs and Arm Cortex CPUs. For the genetic algorithm,
we adopted the OpenGA [34] and the NSGA-III [35].

Benchmark Datasets and Models. We demonstrate the effectiveness
of our proposed method on the CIFAR [36] dataset and the CINIC-
10 [37] dataset. The CIFAR dataset, which consists of 50,000 images for
training and 10,000 images for testing, contains two datasets: CIFAR-10
and CIFAR-100. The CIFAR-10 and CIFAR-100 datasets are categorized
into 10 and 100 classes, respectively. CINIC-10 consisting of 27 000"
"10 [37] dataset. The CIFAR dataset, which consists of 50,000 images for
training and 10,000 images for testing, contains two datasets: CIFAR-10
and CIFAR-100. The CIFAR-10 and CIFAR-100 datasets are categorized
into 10 and 100 classes, respectively. CINIC-10 consisting of 27 000
images is split into three equal-sized train, validation, and test subsets
and is categorized into 10 classes. We adopt the state-of-the-art residual
neural network (ResNet) [1], which has less redundancy and is more
challenging to be compressed and accelerated than conventional model
structures, as model architectures. ResNet-20/32/56/110 models are
evaluated for the CIFAR-10 dataset, ResNet-56/110 models are evalu-
ated for the CIFAR-100 dataset and ResNet-18/50 models are evaluated
for the CINIC-10 dataset.

Hyper-parameters Setting. For staging-based approximation, we at-
tach exit branches after each residual block by default for building a
multi-stage model, and the weight of the loss function of each stage is"
"for the CINIC-10 dataset.

Hyper-parameters Setting. For staging-based approximation, we at-
tach exit branches after each residual block by default for building a
multi-stage model, and the weight of the loss function of each stage is
set to 1.0 by default. For pruning-based approximation, we follow the
same data argumentation strategies and scheduling settings as [1].

4.2. GA-based design space exploration

The GA-based DSE, taking increasing accuracy and reducing latency
as the goal, evaluates and sorts the solutions in the design space, and
obtains the (near-)optimal solutions about accuracy and latency after
multiple generations of individuals. After the process of survival of

the fittest for multiple generations of individuals, the (near-)optimal
solutions about accuracy and latency are obtained.

Fig. 4 shows solutions, obtained by GA-based DSE, for ResNet-20,
ResNet-32, ResNet-56, and ResNet-110 on the CIFAR-10 dataset. The
𝑥-axis and 𝑦-axis represent the normalized top-1 accuracy and latency,"
"solutions about accuracy and latency are obtained.

Fig. 4 shows solutions, obtained by GA-based DSE, for ResNet-20,
ResNet-32, ResNet-56, and ResNet-110 on the CIFAR-10 dataset. The
𝑥-axis and 𝑦-axis represent the normalized top-1 accuracy and latency,
normalized to the top-1 accuracy and latency of the corresponding
baseline model, respectively. The data, marked by the green dot, are
the design points of the brute-force algorithm, and the data, marked
by the red triangle, are the (near-)optimal results found by CoAxNN.
The optimal solutions found by brute force are plotted by the boundary
of the green and red regions. It can be observed that the (near-
)optimal solutions searched by CoAxNN are close to this boundary,
which demonstrates the effectiveness of CoAxNN. Therefore, CoAxNN
can search for the model having the least computational cost in most
cases and meeting the accuracy requirements by GA-based DSE.

4.3. Performance of optimized models"
"which demonstrates the effectiveness of CoAxNN. Therefore, CoAxNN
can search for the model having the least computational cost in most
cases and meeting the accuracy requirements by GA-based DSE.

4.3. Performance of optimized models

We compare CoAxNN with state-of-the-art optimization methods
such as ASRFP [38]. For the sake of fairness, the accuracy numbers
are directly cited from their original papers. Different hyperparameters,
such as learning rate, are used by distinct optimization methods, so the
accuracy of the baseline model may be slightly different. Therefore,
both the accuracy of the baseline model and the optimized model
are shown in our experimental results, and ‘‘ACC. Drop’’ is used to
represent the accuracy dropping of the model after optimization. A
smaller number of ‘‘ACC. Drop’’ is better, and a negative number
indicates the optimized model has higher accuracy than the baseline
model. This is because model optimization has a regularization effect,"
"represent the accuracy dropping of the model after optimization. A
smaller number of ‘‘ACC. Drop’’ is better, and a negative number
indicates the optimized model has higher accuracy than the baseline
model. This is because model optimization has a regularization effect,
which can reduce the overfitting of neural network models [2,18]. To
avoid interference, we run each experiment three times and report the
mean and standard deviation (mean ±std) of accuracy. Besides, we
employ FLOPs to quantify the computational costs of neural network
models.

4.3.1. ResNets on CIFAR-10

Table 1 shows the accuracy and FLOPs of ResNet-20/32/56/110 on
the CIFAR-10 dataset. CoAxNN reduces the computational complexity
of the original neural network model while meeting the accuracy
requirements. The optimized ResNet-20, ResNet-32, ResNet-56, and
ResNet-110 by CoAxNN achieves the FLOPs reduction from 4.06E7,
6.89E7, 1.25E8, 2.53E8 (refer to Table 2) to 3.00E7, 4.89E7, 8.06E7,"
"of the original neural network model while meeting the accuracy
requirements. The optimized ResNet-20, ResNet-32, ResNet-56, and
ResNet-110 by CoAxNN achieves the FLOPs reduction from 4.06E7,
6.89E7, 1.25E8, 2.53E8 (refer to Table 2) to 3.00E7, 4.89E7, 8.06E7,
1.63E8, reduced by 25.94%, 28.93%, 35.76%, 35.57% in computa-
tional complexity, with the accuracy loss of 0.67%, 0.84%, 0.74%, and
0.63%, respectively. Moreover, CoAxNN can exploit less computation
to achieve top-1 accuracy that is comparable to other state-of-the-art
model optimization methods. For example, ResNet-20 optimized by
SFP demands the computational complexity of 2.43E7 FLOPs while
reducing the top-1 accuracy by 1.37%. The optimized ResNet-20 by
CoAxNN consumes less computation cost, i.e., 2.27E7 FLOPs, drops by
1.39% in top-1 accuracy. CoAxNN reduces the computational cost of
ResNet-32 to 3.44E7 FLOPs with a 1.58% accuracy drop. MIL spends
more computations (4.70E7 FLOPs), reducing the top-1 accuracy by"
"CoAxNN consumes less computation cost, i.e., 2.27E7 FLOPs, drops by
1.39% in top-1 accuracy. CoAxNN reduces the computational cost of
ResNet-32 to 3.44E7 FLOPs with a 1.58% accuracy drop. MIL spends
more computations (4.70E7 FLOPs), reducing the top-1 accuracy by
1.59%. The compressed ResNet-56 by SFP achieves the FLOPs reduction
of 52.60% and the accuracy loss of 1.33%. CoAxNN decreases the
computational cost of ResNet-56 by 54.88% with a 1.22% accuracy
drop. The optimized ResNet-110 by GAL reduces FLOPs by 48.50% with
a 0.81% drop in top-1 accuracy. CoAxNN achieves a similar accuracy
loss (0.88%) while reducing the computational complexity by 62.09%.
For original neural network models, CoAxNN automatically searches
for a reasonable configuration to effectively optimize the computational
complexity while meeting the accuracy requirements. For the same ac-
curacy requirement, CoAxNN reduces more computations than existing
methods, achieving less resource consumption."
"for a reasonable configuration to effectively optimize the computational
complexity while meeting the accuracy requirements. For the same ac-
curacy requirement, CoAxNN reduces more computations than existing
methods, achieving less resource consumption.

We also analyze the FLOPs and the percentage of predicted images
for different stages of optimized ResNet-20, ResNet-32, ResNet-56,

JournalofSystemsArchitecture143(2023)1029787G. Li et al.

Fig. 4. The solutions with GA-based DSE on the CIFAR-10 dataset. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of
this article.)

Table 1
Performance of optimized neural network models on CIFAR-10 (see [39–43]).

Model

Method

Top-1 Acc.
Baseline (%)

Top-1 Acc.
Accelerated (%)

Top-1 Acc.
Drop (%)

ResNet-20

ResNet-32

ResNet-56

ResNet-110

MIL [39]
SFP [2]
FPGM [40]
TAS [41]
CoAxNN (0.67%)
CoAxNN (1.39%)

MIL [39]
SFP [2]
TAS [41]
CoAxNN (0.84%)
CoAxNN (1.58%)

SFP [2]
ASFP [8]
CP [42]
AMC [29]"
"Model

Method

Top-1 Acc.
Baseline (%)

Top-1 Acc.
Accelerated (%)

Top-1 Acc.
Drop (%)

ResNet-20

ResNet-32

ResNet-56

ResNet-110

MIL [39]
SFP [2]
FPGM [40]
TAS [41]
CoAxNN (0.67%)
CoAxNN (1.39%)

MIL [39]
SFP [2]
TAS [41]
CoAxNN (0.84%)
CoAxNN (1.58%)

SFP [2]
ASFP [8]
CP [42]
AMC [29]
CoAxNN (0.74%)
CoAxNN (1.22%)

SFP [2]
ASRFP [38]
TAS [41]
GAL [43]
CoAxNN (0.63%)
CoAxNN (0.88%)

92.49
92.20
92.20
92.88
92.68
92.68

92.33
92.63
93.89
93.56
93.56

93.59
93.59
92.8
92.8
94.15
94.15

93.68
94.33
94.97
93.26
94.42
94.42

91.43
90.83
91.09
90.97
92.01 (±0.43)
91.29 (±0.26)

90.74
90.08
91.48
92.72 (±0.13)
91.98 (±0.41)

92.26
92.44
90.9
91.9
93.41 (±0.05)
92.93 (±0.25)

92.90
93.69
94.33
92.74
93.79 (±0.36)
93.54 (±0.17)

1.06
1.37
1.11
1.91
0.67
1.39

1.59
2.55
2.41
0.84
1.58

1.33
1.15
1.90
0.90
0.74
1.22

0.78
0.67
0.64
0.81
0.63
0.88

#FLOPs

2.61E7
2.43E7
2.43E7
2.19E7
3.00E7
2.27E7

4.70E7
4.03E7
4.08E7
4.89E7
3.44E7

5.94E7
5.94E7
–
6.29E7
8.06E7
5.66E7

1.21E8
1.21E8
1.19E8
–
1.63E8
9.59E7"
"93.54 (±0.17)

1.06
1.37
1.11
1.91
0.67
1.39

1.59
2.55
2.41
0.84
1.58

1.33
1.15
1.90
0.90
0.74
1.22

0.78
0.67
0.64
0.81
0.63
0.88

#FLOPs

2.61E7
2.43E7
2.43E7
2.19E7
3.00E7
2.27E7

4.70E7
4.03E7
4.08E7
4.89E7
3.44E7

5.94E7
5.94E7
–
6.29E7
8.06E7
5.66E7

1.21E8
1.21E8
1.19E8
–
1.63E8
9.59E7

FLOPs ↓
(%)

36.00
42.20
42.20
46.20
25.94
44.02

31.20
41.50
41.00
28.93
49.98

52.60
52.60
50.00
50.00
35.76
54.88

52.30
52.30
53.00
48.50
35.57
62.09

and ResNet-110, with an accuracy loss of 0.67%, 0.84%, 0.74%, and
0.63%, respectively, as shown in the Table 2. Weighted average FLOPs
(‘‘Avg. #FLOPs’’) are computed by exit percentage and exit FLOPs for
each stage (e.g., 3.00E7 = 58.71% × 1.93E7 + 41.29% × 4.53E7),
which indicates the average model performance on the entire dataset.
CoAxNN employs distinct stages for different neural network models.
The two stages are used for ResNet-20, and three stages are used
for more complex ResNet-32, ResNet-56, and ResNet-110. The neural"
"which indicates the average model performance on the entire dataset.
CoAxNN employs distinct stages for different neural network models.
The two stages are used for ResNet-20, and three stages are used
for more complex ResNet-32, ResNet-56, and ResNet-110. The neural
network prediction finished at earlier stages costs less computational
effort. Simple images, making up most of the dataset, are predicted by

the first few stages, which reduces the computational complexity while
ensuring accuracy.

Besides, we show the configurations of the optimized models
searched by the GA-based DSE, as shown in Table 3. The pruning rate,
the number of stages, and the position and the threshold for each stage
are reported. For the optimized ResNet-20, the pruning rate is 0, i.e., no
pruning is performed, the number of stages is two, the position of the
first stage is the end of the fourth residual block, the corresponding
threshold is 0.3, and the second stage refers to the backbone neural"
"are reported. For the optimized ResNet-20, the pruning rate is 0, i.e., no
pruning is performed, the number of stages is two, the position of the
first stage is the end of the fourth residual block, the corresponding
threshold is 0.3, and the second stage refers to the backbone neural
network with no confidence threshold since images must be exited from

JournalofSystemsArchitecture143(2023)1029788G. Li et al.

Table 2
Analysis of optimized models on CIFAR-10.

Model (Acc.Drop)

Stages

CoAxNN

Percentage

#FLOPs

Avg. #FLOPs

Baseline

#FLOPs

FLOPs ↓ (%)

ResNet-20
(0.67%)

ResNet-32
(0.84%)

ResNet-56
(0.74%)

ResNet-110
(0.63%)
















1

2

1

2

3

1

2

3

1

2

3

Table 3
Configurations optimized by GA-based DSE for CIFAR-10.

Model (Acc.Drop)

Configurations

ResNet-20
(0.67%)

ResNet-32
(0.84%)

ResNet-56
(0.74%)

ResNet-110
(0.63%)

Rate
Stage
Position
Threshold

Rate
Stage
Position
Threshold

Rate
Stage
Position
Threshold

Rate
Stage
Position
Threshold

0
1
4
0.3

0
1
6
0.09"
"Model (Acc.Drop)

Configurations

ResNet-20
(0.67%)

ResNet-32
(0.84%)

ResNet-56
(0.74%)

ResNet-110
(0.63%)

Rate
Stage
Position
Threshold

Rate
Stage
Position
Threshold

Rate
Stage
Position
Threshold

Rate
Stage
Position
Threshold

0
1
4
0.3

0
1
6
0.09

0
1
10
0.07

0
1
19
0.07

58.71%
41.29%

41.72%
38.78%
19.50%

44.81%
36.79%
18.40%

42.66%
30.93%
26.41%

2
–
–

2
11
0.1

2
19
0.08

2
37
0.015

3
–
–

3
–
–

3
–
–

3
–
–

the last stage. Although ResNet-32, ResNet-56, and ResNet-110 are all
optimized into three stages with a pruning rate of 0, the position and
threshold of each stage are different. For the optimized ResNet-32, the
threshold is 0.09, 0.1 for each stage where the position is the end of
the 6, 11th residual block of the backbone network, respectively. For
the optimized ResNet-56, the position of three stages is the end of the
10, 19th residual block with the threshold of 0.07, 0.08. The optimized
ResNet-110 uses three-stage with the threshold of 0.07, 0.017, where"
"the 6, 11th residual block of the backbone network, respectively. For
the optimized ResNet-56, the position of three stages is the end of the
10, 19th residual block with the threshold of 0.07, 0.08. The optimized
ResNet-110 uses three-stage with the threshold of 0.07, 0.017, where
the position is the end of the 19, 37th residual block.

4.3.2. ResNets on CIFAR-100

We evaluate CoAxNN on the CIFAR-100 dataset by ResNet-56 and
ResNet-110, as shown in Table 4. Similarly, CoAxNN outperforms other
state-of-the-art methods. For example, the computational complexity of
optimized ResNet-110 by ASFP is 1.82E8 FLOPs, reduced by 28.20%
compared to the original neural network model, leading to a 1.48%
drop in top-1 accuracy. CoAxNN consumes 1.69E8 FLOPs, achieving
a higher computation reduction of 33.34% and a lower accuracy loss
of 1.30%. Although GHFP achieves a lower accuracy drop of 1.10%, it
uses a higher computational complexity of 1.82E8 FLOPs. These results
demonstrate the effectiveness of CoAxNN."
"a higher computation reduction of 33.34% and a lower accuracy loss
of 1.30%. Although GHFP achieves a lower accuracy drop of 1.10%, it
uses a higher computational complexity of 1.82E8 FLOPs. These results
demonstrate the effectiveness of CoAxNN.

In addition, Table 6 shows the configurations of the optimized mod-
els with the accuracy loss of 0.98% and 1.30%, searched by CoAxNN,
on the CIFAR-100 dataset. Despite the optimized ResNet-56 employing
a three-stage and deactivating pruning-based strategy, which is as same
as CIFAR-10, the thresholds are distinct. The optimized ResNet-56 uses
three-stage with the threshold of 0.7 and 0.065. Besides, The optimized
ResNet-110 adopts three-stage with a pruning rate of 0.1.

We also study the FLOPs and the percentage of predicted images of
the optimized model at each stage on CIFAR-100, as shown in Table 5.
CoAxNN uses three stages for the ResNet-56 and the ResNet-110 as

1.93E7
4.53E7

2.88E7
5.59E7
7.83E7

4.76E7
9.36E7
1.35E8

9.01E7
1.79E8
2.62E8

3.00E7

4.06E7"
"We also study the FLOPs and the percentage of predicted images of
the optimized model at each stage on CIFAR-100, as shown in Table 5.
CoAxNN uses three stages for the ResNet-56 and the ResNet-110 as

1.93E7
4.53E7

2.88E7
5.59E7
7.83E7

4.76E7
9.36E7
1.35E8

9.01E7
1.79E8
2.62E8

3.00E7

4.06E7

25.94

4.89E7

6.89E7

28.93

8.06E7

1.25E8

35.76

1.63E8

2.53E8

35.57

1 and 

same as CIFAR-10. But, since the CIFAR-100 is more complex, more
complex models are required, leading to a smaller percentage of images
predicted at 
2 than CIFAR-10. For example, for ResNet-56 on
CIFAR-10, the percentages of predicted images by 
3 are
44.81%, 36.79%, and 18.40%, respectively. For ResNet-56 on CIFAR-
100, the percentages of predicted images by 
3 are 29.67%,
32.85%, and 37.48%, respectively. For both CIFAR-10 and CIFAR-100,
most of the images on the whole dataset are predicted by the first few
stages with less computation. On the CIFAR-100, CoAxNN reduces the"
"100, the percentages of predicted images by 
3 are 29.67%,
32.85%, and 37.48%, respectively. For both CIFAR-10 and CIFAR-100,
most of the images on the whole dataset are predicted by the first few
stages with less computation. On the CIFAR-100, CoAxNN reduces the
FLOPs by 23.93% and 33.34%, with an accuracy drop of 0.98% and
1.30%, for ResNet-56 and ResNet-110, respectively.

2, and 

2, and 

1, 

1, 

4.3.3. ResNets on CINIC-10

We utilize the CINIC-10 dataset, which consists of images from both
CIFAR and ImageNet [46], avoiding the time-consuming process of
model training on the entire ImageNet dataset, to facilitate experiments
for complicated image classification scenarios. We evaluate CoAxNN on
the CINIC-10 dataset by ResNet-18 and ResNet-50 models that are in
line with the model structures on the ImageNet dataset.

Table 7 shows the accuracy and computational cost of optimized
models. For ResNet-18, when the FLOPs are reduced from 5.49E8"
"the CINIC-10 dataset by ResNet-18 and ResNet-50 models that are in
line with the model structures on the ImageNet dataset.

Table 7 shows the accuracy and computational cost of optimized
models. For ResNet-18, when the FLOPs are reduced from 5.49E8
(i.e., the computational cost of the original ResNet-18, refer to Table 8)
to 2.21E8, reduced by 59.80%, the top-1 accuracy is dropped by 1.01%.
If the accuracy requirement is higher, CoAxNN can achieve 0.50%
accuracy loss while reducing the computational complexity by 43.71%
for the ResNet-18. ResNet-50 with a large number of computations is
improved by 0.10% in top-1 accuracy, and the corresponding FLOPs
is reduced from 1.18E9 (i.e., the computational cost of the original
ResNet-50, refer to Table 8) to 4.63E8, reduced by 60.75% in compu-
tational complexity. We compare CoAxNN with state-of-the-art model
optimization methods, FPC [47] and CCPrune [48]. FPC reduces the
computational complexity by 40.48% (7.76E8 FLOPs) while increas-"
"ResNet-50, refer to Table 8) to 4.63E8, reduced by 60.75% in compu-
tational complexity. We compare CoAxNN with state-of-the-art model
optimization methods, FPC [47] and CCPrune [48]. FPC reduces the
computational complexity by 40.48% (7.76E8 FLOPs) while increas-
ing the top-1 accuracy by 1.14% for the ResNet-50 model. CCPrune
increases the top-1 accuracy of the ResNet-50 model by 0.23% with
a computational complexity of 7.44E8 FLOPs. CoAxNN reduces the
computational complexity by 49.73% (5.93E8 FLOPs) with a 0.38%
improvement in top-1 accuracy. By effectively combining stage-based
with pruning-based approximate strategies, CoAxNN achieves better
performance than existing methods.

Moreover, we analyze the FLOPs and predicted images at each stage
for the optimized ResNet-18 and ResNet-50 with a 1.01% and −0.10%
accuracy drop respectively, as shown in Table 8. For the CINIC-10
dataset, both the ResNet-18 and the ResNet-50 use four stages. More"
"Moreover, we analyze the FLOPs and predicted images at each stage
for the optimized ResNet-18 and ResNet-50 with a 1.01% and −0.10%
accuracy drop respectively, as shown in Table 8. For the CINIC-10
dataset, both the ResNet-18 and the ResNet-50 use four stages. More
than 80% of the images are finished in the previous two stages, and
less than 10% of images are predicted in the last stage.

Table 9 shows the configurations of the ResNet-18 and the ResNet-
50. ResNet-18 uses four-stage with thresholds of 0.23, 0.2, and 0.4,
whose position is the end of the 3, 5, and 7th residual block, and the
pruning rate is 0.3. When the sample does not exit from the first few
stages, it must be exited from the last stage. Therefore, the last stage

JournalofSystemsArchitecture143(2023)1029789G. Li et al.

Table 4
Performance of optimized neural network models on CIFAR-100 (see [44,45]).

Model

Method

Top-1 Acc.
Baseline (%)

Top-1 Acc.
Accelerated (%)

Top-1 Acc.
Drop (%)

ResNet-56

ResNet-110

MIL [39]
CoAxNN (0.98%)"
"JournalofSystemsArchitecture143(2023)1029789G. Li et al.

Table 4
Performance of optimized neural network models on CIFAR-100 (see [44,45]).

Model

Method

Top-1 Acc.
Baseline (%)

Top-1 Acc.
Accelerated (%)

Top-1 Acc.
Drop (%)

ResNet-56

ResNet-110

MIL [39]
CoAxNN (0.98%)
CoAxNN (2.36%)

MIL [39]
SFP [2]
ASFP [8]
ASRFP [38]
GHFP [44]
AHSG-HT [45]
CoAxNN (1.30%)
CoAxNN (3.42%)

71.33
72.75
72.75

72.79
74.14
74.39
74.39
74.39
74.46
74.17
74.17

68.37
71.77 (±0.28)
70.39 (±0.11)

70.78
71.28
72.91
73.02
73.29
72.74
72.87 (±0.19)
70.75 (±0.38)

2.96
0.98
2.36

2.01
2.86
1.48
1.37
1.10
1.72
1.30
3.42

#FLOPs

7.63E7
9.55E7
7.46E7

1.73E8
1.21E8
1.82E8
1.82E8
1.82E8
–
1.69E8
1.15E8

FLOPs ↓
(%)

39.30
23.93
40.53

31.30
52.30
28.20
28.20
28.20
29.30
33.34
54.47

Table 5
Analysis of optimized models on CIFAR-100.

Model (Acc.Drop)

Stages

CoAxNN

Percentage

#FLOPs

Avg. #FLOPs

Baseline

#FLOPs

FLOPs ↓ (%)

ResNet-56
(0.98%)

ResNet-110
(1.30%)


1

2

3

1

2

3

Table 6"
"(%)

39.30
23.93
40.53

31.30
52.30
28.20
28.20
28.20
29.30
33.34
54.47

Table 5
Analysis of optimized models on CIFAR-100.

Model (Acc.Drop)

Stages

CoAxNN

Percentage

#FLOPs

Avg. #FLOPs

Baseline

#FLOPs

FLOPs ↓ (%)

ResNet-56
(0.98%)

ResNet-110
(1.30%)


1

2

3

1

2

3

Table 6
Configurations optimized by GA-based DSE for CIFAR-100.

Model (Acc.Drop)

Configurations

ResNet-56
(0.98%)

ResNet-110
(1.30%)

Rate
Stage
Position
Threshold

Rate
Stage
Position
Threshold

0
1
10
0.7

0.1
1
19
0.73

29.67%
32.85%
37.48%

27.60%
30.18%
42.22%

2
19
0.65

2
37
0.62

3
–
–

3
–
–

has no threshold value. The ResNet-34 uses four-stage with thresholds
of 0.08, 0.09, and 0.09, whose position is the end of the 4, 8, and 14th
residual block, and the pruning rate is 0.2.

Summary. As shown in Tables 1, 4, and 7, CoAxNN, which auto-
matically finds (near)-optimal configurations for effectively combining
staging-based and pruning-based approximate strategies, is comparable"
"residual block, and the pruning rate is 0.2.

Summary. As shown in Tables 1, 4, and 7, CoAxNN, which auto-
matically finds (near)-optimal configurations for effectively combining
staging-based and pruning-based approximate strategies, is comparable
to the state-of-the-art methods. The staging-based approximate strate-
gies perform adaptive inference for inputs according to conditions at
run-time. The inference of simple input can be terminated with a good
prediction confidence in the earlier stage, thereby avoiding remaining
layerwise computations, so that the overall computation cost can be
significantly reduced. However, the number of model parameters is
still too large to be deployed on mobile devices. The pruning-based
approximate strategies remove the unimportant weights or filters to
gain a thinner model. However, the pruning method lacks the ability
to configure the neural network dynamically, which will miss the
opportunities to optimize the model inference. Based on these previ-"
"approximate strategies remove the unimportant weights or filters to
gain a thinner model. However, the pruning method lacks the ability
to configure the neural network dynamically, which will miss the
opportunities to optimize the model inference. Based on these previ-
ous mentioned optimization principles, CoAxNN automatically finds
(near-)optimal configurations by GA-based DSE, making full use of the
advantages of both, thus achieving efficient model optimization.

4.4. Realistic performance of on-device inference

To demonstrate the realistic speedup and energy savings of our
approximate compressed multi-stage models, we evaluate the perfor-
mance of models on a representative intelligent edge device, Jetson
AGX Orin.

4.76E7
9.36E7
1.35E8

8.23E7
1.59E8
2.32E8

9.55E7

1.25E8

23.93

1.69E8

2.53E8

33.34

For the measurement of inference latency, on the one hand, we pre-
execute each neural network model 10 times to warm up the machine,
and then repeat the single-batch inference 100 times to record the"
"4.76E7
9.36E7
1.35E8

8.23E7
1.59E8
2.32E8

9.55E7

1.25E8

23.93

1.69E8

2.53E8

33.34

For the measurement of inference latency, on the one hand, we pre-
execute each neural network model 10 times to warm up the machine,
and then repeat the single-batch inference 100 times to record the
average execution time to reduce the interference, such as system
initialization. On the other hand, after executing all the operators on
the device we insert synchronous instructions to obtain timestamps,
thus avoiding inaccurate measurements for inference time. Table 10
depicts the inference latency for the optimized ResNet-20, ResNet-
32, ResNet-56, and ResNet-110 by CoAxNN, respectively dropped by
0.67%, 0.84%, 0.74%, and 0.63% in top-1 accuracy on the CIFAR-
10 dataset. The results show that CoAxNN can accelerate ResNet-20,
ResNet-32, ResNet-56, and ResNet-110 models by 1.33×, 1.34×, 1.53×,
and 1.51×, respectively. In general, the larger models can obtain a more
significant speedup."
"0.67%, 0.84%, 0.74%, and 0.63% in top-1 accuracy on the CIFAR-
10 dataset. The results show that CoAxNN can accelerate ResNet-20,
ResNet-32, ResNet-56, and ResNet-110 models by 1.33×, 1.34×, 1.53×,
and 1.51×, respectively. In general, the larger models can obtain a more
significant speedup.

To analyze the energy consumption of optimized models, we use
the jetson-stats3 to monitor the power of the system. We per-
form 10 000 times single-batch inference for ResNet-20, ResNet-32,
ResNet-56, and ResNet-110 on Jetson AGX Orin, and the instantaneous
powers are obtained to multiply the average inference time per image
to compute the energy consumption of models. Table 11 shows the
energy consumption for ResNet-20, ResNet-32, ResNet-56, and ResNet-
110 with the accuracy loss of 0.67%, 0.84%, 0.74%, and 0.63% on
the CIFAR-10 dataset. CoAxNN reduces the energy consumption of
ResNet-20, ResNet-32, ResNet-56, and ResNet-110 by 25.17%, 25.68%,
34.61%, and 33.81%, respectively. The experimental results show that"
"110 with the accuracy loss of 0.67%, 0.84%, 0.74%, and 0.63% on
the CIFAR-10 dataset. CoAxNN reduces the energy consumption of
ResNet-20, ResNet-32, ResNet-56, and ResNet-110 by 25.17%, 25.68%,
34.61%, and 33.81%, respectively. The experimental results show that
the optimized models by CoAxNN can be improved in terms of energy
consumption, and the more complex neural network models can save
more energy.

We also evaluate the realistic speedup and energy reduction of mod-
els optimized by existing filter pruning approaches [2,8,11]. Tables 12
and 13 show the execute latency and energy consumption of single-
batch inference of optimized ResNet-20, ResNet-32, ResNet-56, and
ResNet-110 by filter pruning with the accuracy loss of 2.32%, 1.12%,
0.23%, and 0.10%, on the CIFAR-10 dataset, respectively. Compared
with the baseline models, the optimized models have higher execution
latency and more energy consumption. Although filter pruning can
reduce theoretical computation costs and memory footprint, the opti-"
"0.23%, and 0.10%, on the CIFAR-10 dataset, respectively. Compared
with the baseline models, the optimized models have higher execution
latency and more energy consumption. Although filter pruning can
reduce theoretical computation costs and memory footprint, the opti-
mized models cannot obtain actual acceleration and energy reduction

3 https://pypi.org/project/jetson-stats/.

JournalofSystemsArchitecture143(2023)10297810G. Li et al.

Table 7
Performance of optimized neural network models on CINIC-10.

Model

Method

Top-1 Acc.
Baseline (%)

Top-1 Acc.
Accelerated (%)

Top-1 Acc.
Drop (%)

ResNet-18

ResNet-50

CoAxNN (0.50%)
CoAxNN (1.01%)

FPC [47]
CCPrune [48]
CoAxNN (−0.38%)
CoAxNN (−0.10%)

87.57
87.57

86.63
88.30
88.52
88.52

87.07 (±0.29)
86.56 (±0.43)

87.77
88.53
88.14 (±0.15)
88.62 (±0.34)

0.50
1.01

−1.14
−0.23
−0.38
−0.10

#FLOPs

3.09E8
2.21E8

7.76E8
7.44E8
5.93E8
4.63E8

FLOPs ↓
(%)

43.71
59.80

40.48
–
49.73
60.75

Table 8
Analysis of optimized models on CINIC-10.

Model (Acc.Drop)"
"86.63
88.30
88.52
88.52

87.07 (±0.29)
86.56 (±0.43)

87.77
88.53
88.14 (±0.15)
88.62 (±0.34)

0.50
1.01

−1.14
−0.23
−0.38
−0.10

#FLOPs

3.09E8
2.21E8

7.76E8
7.44E8
5.93E8
4.63E8

FLOPs ↓
(%)

43.71
59.80

40.48
–
49.73
60.75

Table 8
Analysis of optimized models on CINIC-10.

Model (Acc.Drop)

Stages

CoAxNN

Percentage

#FLOPs

Avg. #FLOPs

Baseline

#FLOPs

FLOPs ↓ (%)

ResNet-18
(1.01%)

ResNet-50
(−0.10%)


1

2

3

4

1

2

3

4

50.86%
30.96%
13.13%
5.05%

39.90%
41.58%
9.27%
9.26%

1.35E8
2.57E8
3.79E8
4.56E8

2.11E8
4.91E8
8.66E8
1.02E9

2.21E8

5.49E8

59.80

4.63E8

1.18E9

60.75

Table 9
Configurations optimized by GA-based DSE for CINIC-10.

Model (Acc.Drop)

Configurations

ResNet-18
(1.01%)

ResNet-50
(−0.10%)

Rate
Stage
Position
Threshold

Rate
Stage
Position
Threshold

0.3
1
3
0.23

0.2
1
4
0.08

2
5
0.2

2
8
0.09

3
7
0.4

3
14
0.09

4
–
–

4
–
–

Table 12
Speedups of optimized models by existing pruning approaches [2,8,11] on Jetson AGX
Orin.

Model (Acc.Drop)

Latency (ms)"
"ResNet-50
(−0.10%)

Rate
Stage
Position
Threshold

Rate
Stage
Position
Threshold

0.3
1
3
0.23

0.2
1
4
0.08

2
5
0.2

2
8
0.09

3
7
0.4

3
14
0.09

4
–
–

4
–
–

Table 12
Speedups of optimized models by existing pruning approaches [2,8,11] on Jetson AGX
Orin.

Model (Acc.Drop)

Latency (ms)

Speedup

Baseline

Filter pruning

ResNet-20
(2.32%)

ResNet-32
(1.12%)

ResNet-56
(0.23%)

ResNet-110
(0.10%)

6.26

9.55

16.89

32.33

8.70

13.73

22.51

42.59

0.72

0.70

0.75

0.76

Table 10
Speedups of optimized models by CoAxNN on Jetson AGX Orin.

Model (Acc.Drop)

Latency (ms)

Speedup

Baseline

CoAxNN

Table 13
Energy reductions of optimized models by existing pruning approaches [2,8,11] on
Jetson AGX Orin.

Model (Acc.Drop)

Energy (mJ)

Reduction

ResNet-20
(0.67%)

ResNet-32
(0.84%)

ResNet-56
(0.74%)

ResNet-110
(0.63%)

6.26

9.55

16.89

32.33

4.69

7.11

11.05

21.4

1.33

1.34

1.53

1.51

ResNet-20
(2.32%)

ResNet-32
(1.12%)

ResNet-56
(0.23%)

ResNet-110
(0.10%)

Baseline

27.89

42.79

76.57"
"Energy (mJ)

Reduction

ResNet-20
(0.67%)

ResNet-32
(0.84%)

ResNet-56
(0.74%)

ResNet-110
(0.63%)

6.26

9.55

16.89

32.33

4.69

7.11

11.05

21.4

1.33

1.34

1.53

1.51

ResNet-20
(2.32%)

ResNet-32
(1.12%)

ResNet-56
(0.23%)

ResNet-110
(0.10%)

Baseline

27.89

42.79

76.57

Filter pruning

45.74

72.47

−63.99%

−69.36%

119.24

−55.73%

146.69

225.34

−53.61%

Table 11
Energy reductions of optimized models by CoAxNN on Jetson AGX Orin.

Model (Acc.Drop)

Energy (mJ)

Reduction

ResNet-20
(0.67%)

ResNet-32
(0.84%)

ResNet-56
(0.74%)

ResNet-110
(0.63%)

Baseline

27.89

42.79

76.57

146.69

CoAxNN

20.87

31.8

50.07

97.10

25.17%

25.68%

34.61%

33.81%

Fig. 5. Accuracy of the optimization model at different stages. ‘‘CoAxNN-ALL’’ and
‘‘CoAxNN-ACT’’ denote the accuracy of the model at each stage on the whole dataset
and on the images that satisfy the activation condition of the corresponding stage,
respectively.

JournalofSystemsArchitecture143(2023)10297811G. Li et al."
"‘‘CoAxNN-ACT’’ denote the accuracy of the model at each stage on the whole dataset
and on the images that satisfy the activation condition of the corresponding stage,
respectively.

JournalofSystemsArchitecture143(2023)10297811G. Li et al.

Fig. 6. Example images predicated correctly at different stages.

Table 14
Overheads of GA-based DSE.

Model

ResNet-20
ResNet-32
ResNet-56
ResNet-110

GA time (s)

Training time (s)

1.15
1.60
1.69
1.46

5472
1813
2720
7712

on Jetson AGX Orin. Therefore, the critical motivation of CoAxNN is to
find a satisfying optimization configuration for practical scenarios.

4.5. Ablation study

Accuracy of CoAxNN models at different stages. We study the accu-
racy of ResNet-56 optimized by CoAxNN at different stages, as shown
in Fig. 5. In ‘‘CoAxNN-ALL’’, the accuracy of the model in the first few
stages is lower than that of the baseline model. As the computational
complexity of the model increases, the accuracy in the later stages"
"racy of ResNet-56 optimized by CoAxNN at different stages, as shown
in Fig. 5. In ‘‘CoAxNN-ALL’’, the accuracy of the model in the first few
stages is lower than that of the baseline model. As the computational
complexity of the model increases, the accuracy in the later stages
gradually converges to that of the baseline model. CoAxNN separates
the prediction of simple and complex images by conditional activation,
allowing simple images to exit from the first few stages and complex
images to exit from the latter stages. In ‘‘CoAxNN-ACT’’, the accuracy
of the first few stages becomes higher and even exceeds that of the
baseline model, which indicates that the first few stages have sufficient
ability to classify simple images. Besides, since complex images are
predicted by the later stages, the accuracy of the last stage of the
optimization model is lower than that of the baseline model.

Visualization results at different stages. Fig. 6 depicts the predicted"
"ability to classify simple images. Besides, since complex images are
predicted by the later stages, the accuracy of the last stage of the
optimization model is lower than that of the baseline model.

Visualization results at different stages. Fig. 6 depicts the predicted
sample images for each stage of optimized ResNet-56 on CIFAR-10.
The samples predicated at stage 
1 are relatively ‘‘easy’’, which have
a small number of objects and clear background, whereas the samples
predicated at stage 
3 are relatively ‘‘hard’’, which have various
objects and complex background. CoAxNN can separate ‘‘easy’’ images
consuming less effort from ‘‘hard’’ ones consuming more computation,
significantly reducing computation costs for neural network models.

2 and 

Overheads of GA-based DSE. We collect the latency of each operator
of the neural network model on the edge device in the profiling phase
beforehand to be used in GA-based search. We perform the model"
"significantly reducing computation costs for neural network models.

2 and 

Overheads of GA-based DSE. We collect the latency of each operator
of the neural network model on the edge device in the profiling phase
beforehand to be used in GA-based search. We perform the model
optimization processes, including model training and GA-based search,
on a server with Intel Xeon CPUs and an Nvidia A100 GPU. The
inference of optimized models is performed on edge devices such as
Jetson AGX Orin. Table 14 shows the times for GA-based search and
the time to train the model once during the model optimization. The
GA-based DSE takes 1–2 s on the CPU platform, which is greatly less
than model training (e.g., ResNet-20 takes 5472 s for training once).
Therefore, the runtime overhead of the GA is negligible.

5. Discussion

Generality. CoAxNN is a generic framework for optimizing on-device
deep learning via model approximation, which can be generalized
to other intelligent tasks such as object detection [49]. In addition,"
"Therefore, the runtime overhead of the GA is negligible.

5. Discussion

Generality. CoAxNN is a generic framework for optimizing on-device
deep learning via model approximation, which can be generalized
to other intelligent tasks such as object detection [49]. In addition,
more approximate strategies such as knowledge distillation [50] can
be integrated into CoAxNN to further optimize neural network models.

Applicability. CoAxNN is system-independent, which not requires spe-
cific software implementations and hardware design support. The op-
timized models by CoAxNN can be directly deployed on the target
platform, especially intelligent edge accelerators. Users can choose
the (near)-optimal model according to the accuracy and performance
requirements of intelligent tasks. Moreover, the time-consuming opti-
mization process can be performed offline on high-performance servers,
achieving efficient fine-tuning.

Limitations. Although CoAxNN shows the advantages of combining"
"requirements of intelligent tasks. Moreover, the time-consuming opti-
mization process can be performed offline on high-performance servers,
achieving efficient fine-tuning.

Limitations. Although CoAxNN shows the advantages of combining
staging-based with pruning-based approximate strategies for model
optimization, there is still room for further improvement. On one hand,
the NSGA-III used in GA-based DSE cannot always find the optimal
solutions for the goals of increasing accuracy and decreasing latency.
We will explore other genetic algorithms such as NPGA [51] for multi-
objective optimization. On the other hand, the fixed-rate filter pruning
strategy is used in CoAxNN. Prior works [11] demonstrated that differ-
ent layers have different sensitives for model accuracy. Setting different
pruning ratios for different layers can potentially further improve the
performance, which will be explored in future studies.

6. Conclusion

In this paper, we proposed an efficient optimization framework,"
"ent layers have different sensitives for model accuracy. Setting different
pruning ratios for different layers can potentially further improve the
performance, which will be explored in future studies.

6. Conclusion

In this paper, we proposed an efficient optimization framework,
CoAxNN, which effectively combines staging-based with pruning-based
approximate strategies for efficient model
inference on resource-
constrained edge devices. Evaluation with state-of-the-art CNN models
demonstrates the effectiveness of CoAxNN, which can significantly im-
prove the performance with trivial accuracy loss. We plan to integrate
more model approximate strategies into CoAxNN in future work.

Declaration of competing interest

The authors declare that they have no known competing finan-
cial interests or personal relationships that could have appeared to
influence the work reported in this paper.

Data availability

Data will be made available on request.

JournalofSystemsArchitecture143(2023)10297812G. Li et al."
"The authors declare that they have no known competing finan-
cial interests or personal relationships that could have appeared to
influence the work reported in this paper.

Data availability

Data will be made available on request.

JournalofSystemsArchitecture143(2023)10297812G. Li et al.

Acknowledgments

This work is supported by the National Key R&D Program of China
(2021ZD0110101), the National Natural Science Foundation of China
(62232015, 62302479), the China Postdoctoral Science Foundation
(2023M733566), and the CCF-Baidu Open Fund, China."
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