NSGA-N ET : A M ULTI -O BJECTIVE G ENETIC A LGORITHM FOR N EURAL A RCHITECTURE S EARCH
Anonymous authors
Paper under double-blind review
A BSTRACT
This paper introduces NSGA-Net, an evolutionary approach for neural architecture search (NAS).
NSGA-Net is designed with three goals in mind: (1) a NAS procedure for multiple, possibly conﬂicting, objectives, (2) efﬁcient exploration and exploitation of the space of potential neural network architectures, and (3) output of a diverse set of network architectures spanning a trade-off frontier of the objectives in a single run.
NSGA-Net is a population-based search algorithm that explores a space of potential neural network architectures in three steps, namely, a population initialization step that is based on prior-knowledge from hand-crafted architectures, an exploration step comprising crossover and mutation of architectures and ﬁnally an exploitation step that applies the entire history of evaluated neural architectures in the form of a Bayesian Network prior.
Experimental results suggest that combining the objectives of minimizing both an error metric and computational complexity, as measured by FLOPS, allows NSGA-Net to ﬁnd competitive neural architectures near the Pareto front of both objectives on two different tasks, object classiﬁcation and object alignment.
NSGA-Net obtains networks that achieve 3.72% (at 4.5 million FLOP) error on CIFAR-10 classiﬁcation and 8.64% (at 26.
million FLOP) error on the CMU-Car alignment task.
I NTRODUCTION
Deep convolutional neural networks have been overwhelmingly successful at a variety of image analysis tasks.
One of the key driving forces behind this success is the introduction of many CNN architectures such as, AlexNet (Krizhevsky et al., 2012), VGG (Simonyan & Zisserman, 2015),
GoogLeNet (Szegedy et al., 2015), ResNet (He et al., 2016a), DenseNet (Huang et al., 2017) etc.
in the context of object classiﬁcation and Hourglass (Newell et al., 2016), and Convolutional Pose
Machines (Wei et al., 2016a) in the context of object alignment.
Concurrently, network designs such as MobileNet (Howard et al., 2017), XNOR-Net (Rastegari et al., 2016),
BinaryNets
(Courbariaux et al., 2016), LBCNN (Juefei-Xu et al., 2017) etc. have been developed with the goal of enabling realworld deployment of high performance models on resource constrained devices.
These developments are the fruits of years of painstaking efforts and human ingenuity.
Neural architecture search (NAS) methods, on the other hand, seek to automate the process of designing network architectures.
State-of-the-art reinforcement learning (RL) (Baker et al., 2017;
Zhong et al., 2017; Zoph & Le, 2016; Zoph et al., 2017; Hsu et al., 2018a; Cai et al., 2018; Pham et al., 2018) and evolutionary algorithm (EA) (Miikkulainen et al., 2017; Real et al., 2017; 2018;
Xie & Yuille, 2017; Kim et al., 2017; Liu et al., 2018; Elsken et al., 2018; Liu et al., 2017; Dong et al., 2018) approaches for NAS focus on the single objective of minimizing an error metric on a task and cannot be easily adapted to handle minimizing multiple, possibly conﬂicting, objectives.
Methods like (Real et al., 2017) and (Zoph et al., 2017) are inefﬁcient in their use of their search space and require 3,150 and 2,000 GPU days, respectively.
Furthermore, most state-of-the-art approaches search over a single computation block, similar to an Inception block (Szegedy et al., 2015), and repeat it as many times as necessary to form a complete network.
In this paper, we present NSGA-Net, a genetic algorithm for NAS to address the aforementioned limitations of current approaches.
The salient features of NSGA-Net are, (1) Multiobjective Optimization : Real-world deployment of NAS models is seldom guided by a single objective and often has to balance between multiple, possibly competing, objectives.
For instance, we seek to maximize
[Figure]
performance on compute devices that are often constrained by hardware resources in terms of power consumption, available memory, available FLOPS, and latency constraints, to name a few.
NSGA-Net is explicitly designed to optimize such competing objectives.
(2) Complete Architecture Search
Space :
The search space for most existing methods is restricted to a block that is repeated as many times as desired.
In contrast, NSGA-Net searches over the entire structure of the network.
This scheme overcomes the limitations inherent to repeating the same computation block throughout an entire network, namely, that a single block may not be optimal for every application and it is desirable to allow NAS to discover architectures with different blocks in different parts of the network.
(3) Non-Dominated Sorting :
The core component of NSGA-Net is the Non-Dominated Sorting
Genetic Algorithm II (NSGA-II) (Deb et al., 2000), a multi-objective optimization algorithm that has been successfully employed for solving a variety of multiobjective problems (Tapia & Coello, 2007; Pedersen & Yang, 2006).
Here, we leverage it’s ability to maintain a diverse trade-off frontier between multiple, possibly conﬂicting, objectives, thereby resulting in a more effective and efﬁcient exploration of the search space, (4) Crossover : To fully utilize the diverse frontier of solutions provided by non-dominated sorting, we employ crossover (in addition to mutation) to combine networks with desirable qualities across multiple objectives, and ﬁnally (5) BOA : We construct and employ a Bayesian Network inspired by the Bayesian Optimization Algorithm (BOA) (Pelikan et al., 1999) to fully utilize the promising solutions present in our search history archive and the inherent correlations between the layers of the network architecture.
We demonstrate the efﬁcacy of NSGA-Net on two tasks: image classiﬁcation (CIFAR-10 (Krizhevsky & Hinton, 2009)) and object alignment or key-point prediction (CMU-Car (Boddeti et al., 2013)).
For both tasks we minimize two objectives: an error metric and computational complexity.
Here, computational complexity is deﬁned by the number of ﬂoating-point operations or multiply-adds (FLOPS).
Experimentally, we observe NSGA-Net can ﬁnd a set of network architectures containing solutions that are signiﬁcantly better than hand-crafted methods in both objectives while being competitive in single objectives to state-of-the-art NAS approaches.
Furthermore, by fully utilizing a population of networks through crossover and utilization of the search history, NSGA-Net explores the search space efﬁciently and requires less computation time for searching than competing methods.
R ELATED W ORK
Recent research efforts in NAS have produced a plethora of methods to automate the design of networks.
Broadly speaking, these methods can be divided into EA approaches and RL approaches—with a few methods not falling into either category.
Due to space constraints, we provide a comprehensive overview of relevant methods and a table summarizing their contributions in Appendix D. Here, we focus on multiobjective methods, and relate NSGA-Net to recent contributions.
Kim et al.
(2017) presented NEMO, one of the earliest multiobjective approaches involving neural networks.
NEMO used NSGA-II (Deb et al., 2000) to minimize error and inference time of a network and searched over the space of the number of output channels from each layer within a restricted space of 7 different architectures.
In contrast, NSGA-Net seeks to use NSGA-II to minimize error and computational complexity (as measured by FLOPS), while searching over the vast space of possible network architectures but with ﬁxed hyperparameters.
Dong et al.
(2018) proposed PPP-Net as a multiobjective extension to the progressive NAS method in (Liu et al., 2017) that uses a predictive model to choose promising networks to train in an effort to reduce computational strain.
PPP-Net differs from NSGA-Net by not using crossover and the use of this predictive model.
Elsken et al.
(2018) present the LEMONADE method which is formulated to develop networks with low error and number of parameters through custom designed approximate network morphisms (Wei et al., 2016b).
This allows newly generated networks to share parameters with their forerunners, obviating network training from scratch.
NSGA-Net differs from LEMONADE in terms of the encoding scheme, the network morphisms as well in the selection scheme.
NSGA-Net relies on, (1) genetic operations like mutation and crossover, encouraging population diversity, instead of custom network morphism operations, and (2) NSGA-II, rather than novelty-based sampling as the selection scheme in LEMONADE, with the former affording a more efﬁcient exploration of the search space.
Finally, we highlight MONAS, presented by Hsu et al.
(2018a).
This approach is one of the ﬁrst—if not the only—RL NAS method to use a multiobjective scheme.
MONAS searches the same space that is presented in (Zoph et al., 2017) while using a reward signal to optimize a recurrent neural network to generate convolutional networks.
The reward signal however is a linear combination of accuracy and energy consumption of a network.
However, it is well known in multiobjective optimization literature that a simple linear combination of objectives suffers from a number of drawbacks, including sub-optimality (Koski, 1985) of the search (see Section 3 for a more detailed discussion).
P ROPOSED A PPROACH
Compute devices are often constrained by hardware resources in terms of power consumption, available memory, available FLOPS, and latency constraints.
Hence, real-world design of DNNs are required to balance multiple, possibly competing, objectives (e.g., predictive performance and computational complexity).
Often, when multiple design criteria are considered simultaneously, there may not exist a single solution that performs optimally in all desired criteria, especially with competing objectives.
Under such circumstances, a set of solutions that provides the entire trade-off information between the objectives is more desirable.
This enables a practitioner to analyze the importance of each criterion, depending on the application, and to choose an appropriate solution on the trade-off frontier.
We propose NSGA-Net, a genetic algorithm based architecture search method to automatically generate a set of DNN architectures that approximate the Pareto-front between performance and complexity on classiﬁcation and regression tasks.
The rest of this section describes the design principles, encoding scheme, and main components of NSGA-Net in detail.
3. D ESIGN P RINCIPLES
Population Based Methods: The two main approaches to obtaining an efﬁcient set of trade-off solutions are: i) classical point-based methods like a weighted sum of the objective functions (e.g., αf 1 ( x ) + (1 − α )
f 2 ( x ) ); and ii) population-based methods like genetic algorithms.
Despite the well characterized convergence properties of pointbased methods, they often present the following challenges as illustrated in Fig.2: 1) They require prior knowledge of the range of values each objective function can take for appropriate normalization before weighting, else, the solutions are often biased towards the objective with large values; 2) weighted combinations of objectives can only ﬁnd solutions on convex regions of the Pareto-front and are incapable of discovering solutions on concave parts of the Pareto-front (Koski, 1985); ﬁnally 3) obtaining each solution on the Pareto-front requires the repetition of the entire search procedure for each combination weight α .
On the other hand, population-based methods, like genetic algorithms, offer a ﬂexible, viable alternative to ﬁnd multiple efﬁcient trade-off solutions in one execution.
A population-based method is capable of introducing an implicit parallel search by processing multiple solution candidates jointly at each iteration.
However, optimizing with N population members processes O ( N 3 ) sub-regions of the search space in parallel (Goldberg, 1989; Holland, 1975).
The efﬁciency afforded by such parallelism cannot be matched by point-based methods like weighted combinations.
Recent studies
[Figure]
population millions et al
Multi-Criterion Based Selection:
The general form of a multiobjective optimization problem is
[Equation]
where f i ( · ) are the criterion that we wish to optimize and x is the representation of a neural network architecture (described in Section 3.2).
For the aforementioned problem, given solutions x 1 and x 2 , x 1 is said to dominate x 2 (i.e., x 1 (cid:22) x 2 ) if both of the following conditions are satisﬁed,
x 1 is no worse than x 2 for all objectives ( f i ( x 1 ) ≤
f i ( x 2 ) ∀ i ∈ { 0 , . . .
, M } ) 2.
x 1 is strictly better than x 2 in at least one objective ( ∃ i ∈ { 0 , . . .
, M } | f i ( x 1 ) <
f i ( x 2 ))
Therefore, a solution x i is non-dominated if these conditions hold for all x j and j (cid:54)
=
i .
The core of NSGA-Net is a selection criterion that leverages non-dominated solutions.
Specifically, given a population of network architectures { x 1 , . . .
, x n } and their ﬁtness functions { f 1 x i , . . .
, f M ( x i ) } , the ranking and selection procedure consists of two stages: (1) non-dominated solutions are selected over dominated solutions; (2) explicitly ranking of solutions that are diverse w.r.t.
to the trade-off between the objectives higher than solutions that are “crowded" on the trade-off front (see Fig.
3c) i.e., how close a given solution is to its neighbors in the objective space.
We adopt the non-domination ranking and crowded-ness measurements proposed by Deb et al.
The non-domination ranking indicates the front number that a solution belongs to; these fronts are composed by the set of non-dominated solutions at the current search iteration.
An example of the non-domination fronts and non-domination ranking is shown in Fig.
3a and Fig.
3b, respectively.
It’s worth noting that both non-domination ranking and crowded-ness are relative measurements which need to be re-assessed when new solutions are created.
Elitist-preserving : This term refers to the fact that the best solution (in terms of objective values and crowded-ness) in the parent population will always be carried on to the next population.
This allows the previous best solution to always have a chance to share its genetic information with the next generation, without the risk of losing the information to the newly generated child population.
As a consequence of this scheme, the best solution encountered during the entirety of the search will always be present in the ﬁnal population.
3. E NCODING
Genetic algorithms, like any other biologically inspired search methods, do not directly operate on phenotypes .
From the biological perspective, we may view the DNN architecture as a phenotype, and the representation it is mapped from as its genotype .
As in the natural world, genetic operations like
I npu t
P oo li n g P oo li n g
A v g P oo li n g
L i n e a r x (1) o =0-01-000-0010-00101x (2) o =0-00-000-0101-10101x (3) o
=0-00-111-0111-00000- crossover and mutation are only carried out in the genotype space; such is the case in NSGA-Net as well.
We refer to the interface between the genotype and the phenotype as encoding in this paper.
Most existing CNN architectures can be viewed as a composition of computational blocks (e.g.
ResNet blocks (He et al., 2016a), DenseNet block (Huang et al., 2017), and Inception block (Szegedy et al., 2015), etc.) and resolution of information
ﬂow path.
For example, down-sampling is often used after a computational block to reduce the information resolution going into the next computational block in networks designed for classiﬁcation tasks.
In NSGA-Net, each computational block, referred to as a phase , is encoded using the method presented by Xie & Yuille (2017), with the small change of adding a bit to represent a skip connection.
And we name it as the operation encoding x o .
To handle the resolutions of the information ﬂowing paths, we present a novel encoding, named the path encoding x p , that takes inspiration from the Hourglass (Newell et al., 2016) architecture.
Hence, each
DNN architecture in NSGA-Net is presented as a tuple x = ( x p , x o ) .
Though NSGA-Net operates in the genotype space, the performance of each solution is assessed based on its phenotype.
3.2. O PERATION E NCODING x o
Unlike most of the hand-crafted and NAS generated architectures, we do not repeat the same phase (computational block) to construct a network.
Instead, the operations of a network are encoded by x o = (cid:16) x ( 1 ) o , x ( 2 ) o , . . .
, x ( n p ) o (cid:17) where n p is the number of phases.
Each x ( i ) o encodes a directed acyclic graph consisting of n o number of nodes that describes the operation within a phase using a binary string.
Here, a node is a basic computational unit, which can be a single operation like convolution, pooling, batch-normalization or a sequence of operations.
This encoding scheme offers a compact representation of the network architectures in genotype space, yet is ﬂexible enough that many of the computational blocks in hand-crafted networks can be encoded, e.g VGG (Simonyan &
Zisserman, 2015), ResNet (He et al., 2016a) and DenseNet (Huang et al., 2017).
Figure 4 shows an example of the operation encoding and more details are provided in Appendix E.1.
3.2. P ATH E NCODING x p
In NSGA-Net, the path encoding, x p = (cid:16) x ( 1 ) p , x ( 2 ) p , . . .
, x ( n p ) p (cid:17) , is a n p -dimensional integer vector whose entries are in the range [ − r, r ] , where r is the original input information resolution (e.g. for
CIFAR-10, r = 32) and n p is the number of phases (computational blocks) in the network.
Each x p ( i ) indicates the stride value for the pooling operation after phase i , where positive and negative values denote up-sampling and down-sampling respectively, and zero encodes no change in resolution.
3.2. S EARCH S PACE
The search domain, Ω x o , deﬁned by our operation encoding x o consists of n p × 2 n o ( n o − 1) / 2+1 strings.
And the search domain, Ω x p , deﬁned by our path encoding x p consists of (2 r ) n p combinations.
Hence, the total search domain in the genotype space is:
[Equation]
Parent    ⊗
Parent    =
Child    (Parent 2) DenseNet: 1-11-111- (Parent 1) VGG: 1-01-001- ——————————————(Child)ResNet: 1-01-101- :
common : VGG : DenseNet where n p is the number of phases (computational blocks), n o is the number of nodes (computational units) in each phase, and r is the original input information resolution.
However, for computationally tractability, we constrain the search space in two ways: (1) each node (computational unit) in a phase (computational block) carriers the same sequence of operations, i.e. a 3 × 3 convolution followed by batch-normalization and ReLU; (2) we restrict the search space to only the operation encoding i.e, x o and ﬁx the path encoding to 2 × 2 max-pooling with stride 2, i.e., x ( i ) p = − 2 and use a global averaging pool before classiﬁcation layer.
It is worth noting that, as a result of nodes in each phase having identical operations, the encoding between genotype and phenotype is a many-to-one mapping.
Given the prohibitive computational expense required to train each network architecture before its performance can be assessed, it is essential to avoid evaluating genomes that decode to the same architecture.
We develop an algorithm to quickly and approximately identify these duplicate genomes (see Appendix E.3 for details).
[Equation]
ROCEDURE
NSGA-Net is an iterative process in which initial solutions are made gradually better as a group, called population .
In every iteration, the same number of offspring (new network architectures) are generated from parents selected from the population.
Each population member (including both parents and offspring) compete for both survival and reproduction (becoming a parent) in the next iteration.
The initial population may be generated randomly or guided by prior-knowledge (e.g. seeding the hand-crafted network architectures into the initial population).
The overall NSGA-Net search proceeds in two sequential stages, an exploration and exploitation .
3.3. E XPLORATION
The goal of this stage is to discover diverse ways of connecting nodes to form a phase (computational block).
Genetic operations, crossover 1 and mutation, offer an effective mean to realize this goal.
Crossover: The implicit parallelism of population-based search approaches can be unlocked when the population members can effectively share (through crossover) building-blocks (Holland, 1975).
In the context of NAS, a phase or the sub-structure of a phase can be viewed as a building-block.
We design a homogeneous crossover operator, which takes two selected population members as parents, to create offspring (new network architectures) by inheriting and recombining the building-blocks from parents.
The main idea of this crossover operator is to 1) preserve the common building-blocks shared between both parents by inheriting the common bits from both parents’ binary bit-strings; 2) maintain, relatively, the same complexity between the parents and their offspring by restricting the p (cid:16) x (1) o (cid:17) p (cid:16) x (3) o
(cid:12)(cid:12)(cid:12) x (2) o (cid:17) p (cid:16) x (2) o
(cid:12)(cid:12)(cid:12) x (1) o (cid:17)
[Figure]
number of “1" bits in the offspring’s bit-string to lie between the number of “1" bits in both parents.
An example of the crossover operator is provided in Figure 5.
Mutation: To enhance the diversity (having different network architectures) of the population and the ability to escape from local optima, we use a bit-ﬂipping mutation operator, which is commonly used in binary-coded genetic algorithms.
Due to the nature of our encoding, a one bit ﬂip in the genotype space could potentially create a completely different architecture in the phenotype space.
Hence, we restrict the number of bits that can be ﬂipped to be at most one for each mutation operation.
3.3. E XPLOITATION
The exploitation stage follows exploration in NSGA-Net.
The goal of this stage is to exploit the archive of solutions explored in the previous stage.
The exploitation step in NSGA-Net is heavily inspired by the Bayesian Optimization Algorithm (BOA) (Pelikan et al., 1999) which is explicitly designed for problems with inherent correlations between the optimization variables.
In the context of our NAS encoding, this translates to correlations in the blocks and paths across the different phases.
Exploitation uses past information across all networks evaluated to guide the ﬁnal part of the search.
More speciﬁcally, say we have a network with three phases, namely x ( 1 ) o , x ( 2 ) o , and x ( 3 ) o .
We would like to know the relationship of the three phases.
For this purpose, we construct a Bayesian Network (BN) relating these variables, modeling the probability of networks beginning with a particular phase x ( 1 ) o , the probability that x ( 2 ) o follows x ( 1 ) o , and the probability that x ( 3 ) o follows x ( 2 ) o .
In other words we estimate the distributions p (cid:16) x ( 1 ) o (cid:17) , p (cid:16) x ( 2 ) o | x ( 1 ) o (cid:17) , and p (cid:16) x ( 3 ) o | x ( 2 ) o (cid:17) by using the population history, and update these estimates during the exploitation process.
New offspring solutions are created by sampling from this BN.
Figure 6 shows a pictorial depiction of this process.
E XPERIMENTS
In this section, we demonstrate the efﬁcacy of NSGA-Net to automate the NAS process for two tasks: image classiﬁcation and object alignment (regression).
Due to space constraints, we only present classiﬁcation results here and present all the regression results in Appendix G.
4. P ERFORMANCE M ETRICS
We consider two objectives to guide NSGA-Net based NAS, namely, classiﬁcation error and computational complexity.
A number of metrics can serve as proxies for computational complexity: number of active nodes, number of active connections between the nodes, number of parameters, inference time and number of multiply-add operations (FLOPs) needed to execute the forward pass of a given network.
Our initial experiments considered each of these different metrics.
We concluded from extensive experimentation that inference time cannot be estimated reliably due differences and inconsistencies in computing environment, GPU manufacturer and temperature etc.
Similarly, the number of parameters, active connections or active nodes only relate to one aspect of computational complexity.
In contrast, we found an estimate of FLOPs to be a more accurate and reliable proxy for network complexity.
See Appendix F for more details.
Therefore, classiﬁcation error and FLOPs serve as the twin objectives for selecting networks.
For the purpose of quantitatively comparing different multi-objective search methods or different conﬁguration setups of NSGA-Net, we use the hypervolume (HV) performance metric, which calculates the dominated area (hypervolume in the general case) from the a set of solutions (network architectures) to a reference point which is usually an estimate of the nadir point—a vector concatenating worst objective values of the Pareto-frontier.
It has been proved that the maximum HV can only be achieved when all solutions are on the
Pareto-frontier (Fleischer, 2003).
Hence, the higher the HV measures, the better solutions that are being found in terms of both objectives.
See Figure 7 for a hypothetical example.
4. I MPLEMENTATION D ETAILS Dataset
We consider the
CIFAR-
(Krizhevsky et al.) dataset for our classiﬁcation task.
We split the original training set (80%-20%) to create our training and validation sets.
The original CIFAR-10 testing set is only utilized at the conclusion of the search to obtain the test accuracy for the models on the ﬁnal trade-off front.
NSGA-Net hyper-parameters:
We set the number of phases n p to three and the number of nodes in each phase n o to six.
We also ﬁx the path encoding x p =
[ − 2 , − 2 , − 8] , which decodes to having a max-pooling with stride 2 after the ﬁrst and the second phase, and a global average pooling layer after the last phase.
The initial population is generated by uniform random sampling.
The population size is 40 and the number of iterations is 20 for the exploration stage.
During exploitation, we reduce the number of iterations by half.
Hence, a total of 1,200 network architectures are searched by
NSGA-Net.
Network training: For training each generated network architecture, we use standard stochastic gradient descent (SGD) and a cosine annealing learning rate schedule (Loshchilov &
Hutter, 2016).
Our initial learning rate is 0.025 and we train for 25 epochs, which takes aboutminutes on a NVIDIA 1080Ti GPU implementation in PyTorch (Paszke et al., 2017).
4. R ESULTS
Figure 8b shows the bi-objective frontiers obtained by NSGA-Net through the various stages of the search, clearly showcasing a gradual improvement of the whole population.
Figure 8c shows two metrics: normalized HV and offspring survival rate, through the different generations of the population.
The monotonic increase in the former suggests that a better set of trade-off network architectures have been found over the generations.
The monotonic decrease in the latter metric suggests that, not surprisingly, it is increasingly difﬁcult to create better off-springs (than their parents).
We can use a threshold on the offspring survival rate as a potential criterion to terminate the current stage of the search process and switch between the exploration and exploitation.
To compare the trade-off front of network architecture obtained from NSGA-Net to other hand-crafted and search-generated architectures, we pick the network architectures with the lowest classiﬁcation error from the ﬁnal frontier (the dot in the lower right corner on the green curve in Figure 8b) and extrapolate the network by increasing the number of ﬁlters of each node in the phases, and train with the entire ofﬁcial CIFAR-10 training set.
The chosen network architecture results in 3.85% classiﬁcation error on the CIFAR-10 testing set with 3.3 Millions of parameters and
[Figure]
I
[Figure]
Method
Error
Other Objective
Compute
PPP-Net
(Dong et al., 2018) 4.
FLOPs or
Params or Inference Time
Nvidia Titan X
MONAS (Hsu et al., 2018a) 4.
Power
Nvidia 1080Ti
NSGA-Net (Ours) 3.
FLOPS
Nvidia 1080Ti
[Table]
MFLOPs (green star in Figure 8a).
Table 4 provides a summary that compares NSGA-Net with other multi-objective NAS methods.
We include the comprehensive comparison table in Appendix G.
4. A BLATION S TUDIES
Here, we ﬁrst present results comparing NSGA-Net with uniform random sampling (RSearch) from our encoding as a sanity check.
It’s clear from Figure 9a that much better set of network architectures are obtained using NSGA-Net.
Then we present additional results to showcase the beneﬁts of the two main components of our approach: crossover and Bayesian network based offspring creation.
Crossover Operator: Current state-of-the-art NAS search results (Liu et al., 2018; Real et al., 2018) using evolutionary algorithms use mutation alone with enormous computation resources.
We quantify the importance of crossover operation in an EA by conducting the following small-scale experiments on different datasets, including MNIST, SVHN, and CIFAR-10.
From ﬁgs.
9b and 9c, we observe that crossover helps achieve a better trade-off frontier and performance w.r.t.
to both criteria across these datasets.
Bayesian Network (BN) based Offspring Creation:
Here we quantify the beneﬁts of the exploitation stage i.e., off-spring creation by sampling from BN.
We uniformly samplednetwork architectures each from our encoding and from the BN constructed on the population archive generated by NSGA-Net at the end of exploration.
The architectures sampled from the BN dominate (see Fig.9d) all network architectures created through uniform sampling.
4. D ISCUSSION
We analyze the intermediate solutions of our search and the trade-off frontiers and make some observations.
Upon visualizing networks, like the one in Figure 4, we observe that as network
[Figure]
complexity decreases along the front, the search process gravitates towards reducing the complexity by minimizing the amount of processing at higher image resolutions
i
[Figure]
e., remove nodes from the phases that are closest to the input to the network.
As such, NSGA-Net outputs a set of network architectures that are optimized for wide range of complexity constraints.
On the other hand, approaches that search over a single repeated computational block can only control the complexity of the network by manually tuning the number of repeated blocks used.
Therefore, NSGA-Net provides a more ﬁne-grained control over the two objectives as opposed to the control afforded by arbitrary repetition of blocks.
Moreover, some objectives, for instance susceptibility to adversarial attacks, may not be easily controllable by simple repetition of blocks.
Figure 20 in the Appendix shows a subset of networks discovered on the trade-off frontier for CIFAR-10.
C ONCLUSION
This paper presented NSGA-Net, a multi-objective evolutionary approach for neural architecture search.
NSGA-Net affords a number of practical beneﬁts: (1) the design of neural network architectures that can effectively optimize and trade-off multiple, possibly, competing objectives, (2) advantages afforded by population based methods being more effective than optimizing weighted linear combination of objectives, (3) more efﬁcient exploration and exploitation of the search space through a novel crossover scheme and leveraging the entire search history through BOA, and ﬁnally (4) output a set of solutions spanning a trade-off front in a single run.
Experimentally, by optimizing both prediction performance and computational complexity NSGA-Net ﬁnds networks that are signiﬁcantly better than hand-crafted networks on both objectives and is compares favorably to other state-of-the-art single objective NAS methods for classiﬁcation on CIFAR-10 and object alignment (regression) on CMUCars.
Appendix A U TILITY OF NSGA-N ET : R OBUSTNESS AGAINST A DVERSARIAL A TTACK
Hand designing a neural network architecture for a given task is an intensive and time consuming process.
When the task at hand, target datasets or the goal of the network architecture changes, the whole complex process needs to be repeated again.
In reality, one simply modiﬁes existing network architectures developed on other tasks and adopts them for the task at hand.
This procedure while successful, has limited scalability.
In this section, we provide another proof-of-concept example of the utility of NSGA-Net.
We consider the task of ﬁnding neural architectures that not only demonstrate high accuracy for image classiﬁcation but simultaneously be robust against adversarial attacks.
Our choice of this problem is motivated by the following factors: (i) To demonstrate the utility of NSGA-Net on a task that is not easily controlled by hand.
Single objective based NAS methods are able to control a second objective (computational complexity) by manually controlling the number of repeated blocks or number of ﬁlters etc.
Robustness to adversarial attacks on the other hand cannot be achieved in the same way.
Furthermore, we conjecture that these two objectives are competing and hence are a good test of the utility of NSGA-Net, (ii) Explore the effect of network architecture on their robustness to adversarial attacks.
[Figure]
=
(clean images).
Designing a measure/objective for robustness against adversarial robustness is an area of active research (e.g., (Carlini & Wagner, 2016)).
For our purposes here, we present a possible measure here, illustrated in Figure 10.
Using the fast gradient sign method (FSGM) presented by Goodfellow et al.
(2014), this robustness objective progressively increases noise produced by FSGM.
The (cid:15) axis in
Figure 10 refers to the hyper-parameter in the FSGM equation,
[Equation]
where x is the original image, x (cid:48) adversarial image, y true is true class label, and J cross-entropy loss.
Therefore, for this experiment, we seek to maximize two objectives, namely, classiﬁcation accuracy and the robustness objective deﬁned above.
The setup for the robustness experiment is as follows.
For training we use 40,000 CIFAR-10 images from the ofﬁcial CIFAR-10 training data, 10,000 of which are reserved for validation.
Each network is encoded with three phases (computational blocks).
However, for this constrained experiment each phase has the same architecture i.e., we repeat the computational block.
In each phase a maximum of size nodes may be active—where the computation at each node is as in the main paper: 3x convolution followed by ReLU and batch normalization.
Each network is trained for 20 epochs with SGD on a cosine annealed learning rate schedule.
The epsilon values used in the FSGM robustness calculation are [0.0, 0.01, 0.03, 0.05, 0.07, 0.1, 0.15].
As before, NSGA-Net initiates the search with 40 randomly created network architecture, and 40 new network architectures are created at each generation (iteration) via genetic operations (see main paper for details).
The search is terminated at 30 generations.
A.0. E XPERIMENT R ESULTS
A clear trade-off between accuracy and robustness is shown in Figures 11 and 12c.
NSGA-Net is capable of identifying a set of network architectures that provide an efﬁcient trade-off between accuracy and robustness that would not have easily been obtained by hand.
“Wide” networks (like
ResNeXt (Xie et al., 2016) or Inception blocks (Szegedy et al., 2015)) appear to provide good accuracy on standard benchmark images, but are fragile to the FSGM attack.
On the other hand, “deep” networks (akin to ResNet (He et al., 2016a) or VGG (Simonyan & Zisserman, 2015)) are more robust to FSGM attack, while having less accuracy.
This phenomenon is illustrated with examples in
Figures 12a and 12b, respectively.
Furthermore, the skip connection provided by the modiﬁcation we made to the original encoding proposed by Xie & Yuille (2017) appears to be critical in obtaining a network that is robust to adversarial attacks; see Figure ?? . [Figure]
[Figure]
B A DDITIONAL A BLATION E XPERIMENTS
To provide further analysis of the efﬁcacy of the main components in NSGA-Net, we perform some additional ablation experiments.
For these experiments, unlike the experiments in the main paper, we search for a single computational block that is then repeated as desired. We compare NSGA-Net with ﬁve other alternative approaches; 1) random search i.e., uniform sampling from our encoding, 2) mutation only, without crossover, but with a higher mutation probability (maximum 5 bits change in each mutation operation, as opposed to 1 bit change in NSGA-Net; 3) No crossover but allow up to two mutations; 4) mutation only, without crossover, but with a small population size of four individuals, and 5) a one-point crossover followed by mutation with low mutation probability (maximum 1 bit change). For all of the above variations we use the same experimental setup.
We encode every network architecture with three phases and a maximum of six nodes in each of phase.
We also restrict the networks to have the same architectures in all three phases, i.e., we repeat a block three times.
Again, we split 40,000 images from the ofﬁcial CIFAR-10 training set for training and the remaining 10, images for validation.
Each network architecture is then trained for 20 epochs with SGD method on a cosine annealing learning rate schedule.
All ablation experiments share the same initial population to begin the search and all hyper-parameters associated to NSGA-Net, like population size, number of generations, etc., are the same unless speciﬁcally mentioned.
The results of our experiment are shown in Fig.
We make the following observations: it is clear empirically that i) crossover is beneﬁcial as the two strategies that use crossover out-perform the other three strategies; ii) mutation with high probability is harmful to the overall search.
C C ROWDING D ISTANCE VS . H YPERVOLUME - CONTRIBUTION
The goal of multi-objective optimization is to obtain a set of converged, diverse, and evenly distributed solutions along the pareto-front.
Hypervolume (HV) is a popular metric to quantify these aspects (convergence and diversity) simultaneously.
However, choosing points during a multi-objective search that maximize the HV metric may not necessarily result in a well-distributed trade-off frontier.
Ishibuchi et al.
(2018) have shown that algorithms driven purely by HV metric can lead to inferior outcomes (speciﬁcally, Figures 1 and 2 in (Ishibuchi et al., 2018) demonstrate the complexities of using a HV based approach).
We have provided an example in Figure 15a to show the potential discrepancy between the HV metric value and the desired outcome i.e., obtaining a set of converged, diverse, and evenly distributed solutions along the pareto-front. (a) A hypothetical two-objective-minimization problem with a concave front Both
axes being (b) A hypothetical two-objective-minimization problem with a convex pareto front. Both objectives along the axes are being minimized.
50 points were selected from 1,500 randomly sampled points from the
front using crowding distance (CD) and HV-contribution.
It’s clear that HV
contribution need not necessarily ensure a better HV value as compared to selection with In addition, HV-contribution is not necessarily consistent with the HV metric depending on the number of objectives and concavity of the pareto-front. Finding a subset that maximizes the HV metric is an NP-hard process that involves solving a N choose k problem and in general a greedy solution to this search is a sub-optimal solution.
A large body of the literature exists under the category of hypervolume subset selection problem, (Guerreiro et al., 2016; Ishibuchi et al., 2017).
In short, using HV-contribution as opposed to CD will not ensure a better HV metric in the end nor a monotonic increase in the HV metric over the generations.
We have provided an example in Figure 15b to show that HV-contribution can result in a worse HV metric value as compared to CD.
D R ELATED W ORK C ONTINUED
As mentioned in the main paper, NAS has seen a methodological explosion in the past 2 years.
Table 2 attempts to summarize the published and relevant methods in this area.
A more complete listing, including unpublished methods, would be much longer.
It is natural to group methods into EA and
RL approaches, with some more exotic methods that ﬁt into neither.
The main motivation of EA methods is to treat structuring a network as a combinatorial optimization problem.
EAs operate with a population that makes small changes (mutation) and mixes parts (crossover) of solutions to guide a search toward optimal solutions.
RL, on the other hand, views constructing a network as a decision process.
Usually, an agent is trained to optimally choose the pieces of a network in a particular order.
We brieﬂy review a few methods here.
Reinforcement Learning: Q -learning
Watkins, 1989) is a widely popular value iteration method used for RL.
The MetaQNN method (Baker et al., 2017) employs an (cid:15) -greedy Q -learning strategy with experience replay to search connections between convolution, pooling, and fully connected layers, and the operations carried out inside the layers.
Zhong et al.
(Zhong et al., 2017) extended this idea with the BlockQNN method.
BlockQNN searches the design of a computational block with the same Q -learning approach.
The block that is then repeated to construct a network.
This method allows for a much more general network and achieves better results than its predecessor on
CIFAR-10.
A policy gradient method seeks to approximate some non-differentiable reward function to train a model that requires parameter gradients, like a neural network.
Zoph & Le (2016)
ﬁrst applied this method in architecture search to train a recurrent neural network controller that constructs networks.
The original method in (Zoph & Le, 2016) uses the controller to generate the entire network at once.
This contrasts from its successor, NASNet (Zoph et al., 2017), which designs a convolutional and pooling block that is repeated to construct a network.
NASNet outperforms its predecessor and produces a network achieving state-of-the-art error rate on CIFAR-10.
NSGA-Net differs from
RL methods by using precise selection criteria—in fact, any EA method shares this characteristic.
More speciﬁcally, networks are selected for their accuracy on a task, rather than an approximation of accuracy, along with computational complexity.
Furthermore, the most successful RL methods search only a computational block that is repeated to create a network, NSGA-Net allows for search across computational blocks and combinations of blocks.
Hsu et al.
(2018a) extend the NASNet approach to a multiobjective domain to optimize a linear combination of accuracy and energy consumption.
However, a linear combination of objectives has been characterized as suboptimal (Deb et al., 2000).
Evolutionary Algorithms: Designing neural networks through evolution, or neuroevolution , has been a topic of interest for some time, ﬁrst showing popular success in 2002 with the advent of the neuroevolution of augmenting topologies (NEAT) algorithm (Stanley & Miikkulainen, 2002).
In its original iteration, NEAT only performs well on comparatively small networks.
Miikkulainen et al.
(2017) attempt to extend NEAT to deep networks with CoDeepNEAT using a co-evolutionary approach that achieved limited results on the CIFAR-10 dataset.
CoDeepNEAT does, however, produce state-of-the-art results in the Omniglot multi-task learning domain (Liang et al., 2018).
Real et al.
(2017) introduced perhaps the ﬁrst truly large scale application of a simple evolutionary algorithm.
The extension of this method presented in (Real et al., 2018), called AmoebaNet, provided the ﬁrst large scale comparison of EC and RL methods.
Their simple EA searches over the same space as NASNet (Zoph et al., 2017) and has shown faster convergence to an accurate network when compared to RL and random search.
Furthermore, AmoebaNet obtains the state-of-the-art results on
CIFAR-10.
Conceptually, NSGA-Net is closest to the Genetic CNN (Xie & Yuille, 2017) algorithm.
Genetic
CNN uses a binary encoding that corresponds to connections in convolutional blocks.
In NSGA-Net we augment the original encoding and genetic operations by (1) adding an extra bit for a residual connection, (2) introducing an encoding scheme for multi-resolution processing, and (3) introducing within-phase crossover.
We also introduce a multi-objective based selection scheme.
Moreover, we also diverge from Genetic CNN by incorporating a Bayesian network in our search to fully utilize past population history.
Evolutionary multiobjective approaches have been limited in past work.
Kim et al.
(2017) present an algorithm utilizing NSGA-II (Deb et al., 2000), however their method only searches over hyperparam- eters and a small ﬁxed set of architectures.
The evolutionary method shown in (Elsken et al., 2018) uses weight sharing through network morphisms (Wei et al., 2016b) and approximate morphisms as mutations and uses a biased sampling to select for novelty from the objective space rather than a principled selection scheme like NSGA-II (Deb et al., 2000).
Network morphisms are also an important method in many approaches.
They allow for a network to be “widened" or “deepened" in a manner that maintains functional equivalence.
For architecture search, this allows for easy parameter sharing after a perturbation in a network’s architecture.
Other Methods: Methods that do not subscribe to either an EA or RL paradigm have also shown success in architecture search.
Liu et al.
(2017) present a method that progressively expands networks from simple cells and only trains networks the best K networks that are predicted to be promising by a RNN meta-model of the encoding space.
Dong et al.
(2018) extended this method to use a multiobjective approach, selected the K networks based on their Pareto optimality when compared to other networks.
Hsu et al.
(2018b) also present a meta-model approach that generates models with state of the art accuracy.
This approach may be ad hoc as no analysis is presented on how the progressive search affects the trade-off frontier.
Thomas Elsken (2018) use a simple hill climbing method along with a network morphism (Wei et al., 2016b) approach to optimize network architectures quickly on limited resources.
Chen et al.
(2018b) combine the ideas of RL and EA.
A population of networks is maintained and are selected for mutation with tournament selection (Goldberg & Deb, 1991).
A recurrent network is used as a controller to learn an effective strategy to apply mutations to networks.
Networks are then trained and the worst performing network in the population is replaced.
This approach generates state of the art results for the ImageNet classiﬁcation task.
Chen et al.
(2018a) show that an augmented random search approach to optimize networks for a semantic segmentation application.
Finally, Kandasamy et al.
(2018) present a Gaussian process based approach to optimize network architectures, viewing the process through a Bayesian optimization lens.
E E NCODING D ETAILS
The overall architecture comprises of different phases (computational blocks), within each phase, the resolution of the information is maintained.
Each phase comprises of a set of nodes (basic computational unit) which are operation or a sequence of operations to be performed on the inputs.
The maximum number of phases n p in an overall neural-network architecture is pre-speciﬁed.
Resolution of a phase is determined by path encoding x p , while, the set of operations to be executed within a given phase is encoded in x o vector.
In the following sections, we discuss the operation encoding x o and the path encoding x p that are combined to create an entire network genotype x = ( x p , x o ) .
E. O PERATION E NCODING : x o
We emphasize that this encoding is originally used in (Xie & Yuille, 2017).
We present a small variation on that method with a different notation here.
A phase is a computational block of the overall neural-network architecture.
Thus, each phase is a convolution-neural-network by itself.
We will ﬁrst explain the operation-encoding for a phase, x
( i ) o , where i is the phase number/phase-id.
The overall operation-encoding x o is then generated by concatenating these phase-encodings x ( i ) o , i.e. x o =
[ x ( 1 ) o , x ( 2 ) o , . . .
, x ( n p ) o ] , where n p is the number of phases.
The proposed encoding scheme is similar to the one presented in (Xie & Yuille, 2017), with a minor modiﬁcation, wherein we append an extra bit to represent a direct connection from the main input to the main output .
The resultant architecture is a directed graph where each node of the graph encapsulates following operations in sequence: convolution (3 × 3), batch-normalization and ReLu .
Nodes are assigned with a node-id , which can be assume an integer value from 1 to n .
Information in the neural network architecture is constrained to ﬂow from a node with lower node-id to a node with higher node-id.
Binary encoding is then generated as follows:
• Starting from node-2, each node has its corresponding substring of length j − 1 (where j is the node-id , and j ≥ 2 ).
[Table]
• Elements of the binary substring of j th node can be represented by b ( j ) i , where i = 1 , . . .
, j − 1 and b ( j ) i = (cid:40) 1 if node i is connected to node j 0 otherwise P P P
P
P
P      𝒙 𝒑 = [−2, −2,−2, −2,4,4] Phases R e s o l u t
i on s (
[Figure]
[Figure]
Thus, for a phase with n -nodes, n × ( n − 1) / 2 bits are required to generate the encoding.
It is possible for some nodes to have only outﬂow of the information and no input, while some nodes can have only inﬂow of the information and no output connection.
These types of nodes are connected to the main-input-node and the main-output-node , respectively.
Nodes which are devoid of any inputs and outputs (hanging nodes), are expelled from the ﬁnal architecture.
An extra bit is appended to represent a residue connection from the main-input-node to the main-output-node.
Finally, operation encoding x o of the overall-network is generated by concatenating encodings ( x ( j ) o ) of each phase.
If the maximum number of nodes a phase j can have is n ( j ) n , and the maximum number of phases allowed in the ﬁnal architecture is n p , then the length of x o will be
[Equation]
The search-space of x o thus comprises of 2 L s binary-strings.
This search-space however has redundancy as multiple sub-strings/genomes can decode to generate identical directed-graph.
Since the training of a convolution-neural-network is a computationally expensive task, it is necessary to avoid training of the same CNN architecture which is represented by a different encoding.
To achieve this, we have devised an approximate duplicate-check algorithm, described in Section E.3.
E. P ATH E NCODING x p
As mentioned before, the main neural-network is partitioned into different phases and each phase operates at a particular image-resolution.
A n p -dimensional vector, x p (where n p is the maximum number of phases a neural-network architecture can have), is used to store the information about the resolution of each phase.
In NSGA-Net, the path encoding, x p =
( x ( 1 ) p , x ( 2 ) p , . . .
, x ( n p ) p ) , is a n p -dimensional integer vector whose entries are in the range [ − r, r ] , where r is the original input information resolution (e.g. for
CIFAR-10, r = 32) and n p is the number of phases (computational blocks) in the network.
Each x p ( i ) indicates the stride value for the pooling operation after i th phase, where positive encodes up-sampling, negative encodes down-sampling and zero encodes no pooling operation.
See 16a for a pictorial representation of this.
See 16b for an example of path-based crossover.
E. D UPLICATE C HECKING AND R EMOVAL
Due to the directed acyclic nature of our encoding, redundancy exists in the search space deﬁned by our coding, meaning that there exist multiple encoding stings that decode to the same network architecture.
Empirically, we have witnessed the redundancy becomes more and more severe as the allowed number of nodes in each phase’s computational block increase, as shown in Figure 17. [Figure]
Since the training of a deep network is a computationally taxing task, it is essential to avoid the re-computation of the same architecture.
In this section, we will provide with an overview of an algorithm we developed to quickly and approximately do a duplicate-check on genomes.
The algorithm takes two genomes to be compared as an input, and outputs a ﬂag to indicate if the supplied genomes decode to same architecture.
In general, comparing two graphs is NP-hard, however, given that we are working with Directed
Acyclic Graphs with every node being the same in terms of operations, we were able to design an efﬁcient network architecture duplicate checking method to identify most of the duplicates if not all.
The method is built on top of simply intuition that under such circumstances, the duplicate network architectures should be identiﬁed by swapping the node numbers.
Examples are provided in
Figure 18.
Our duplicates checking method ﬁrst derive the connectivity matrix from the bit-string, which will have positive 1 indicating there is an input to that particular node and negative 1 indicating an output from that particular node.
Then a series row-and-column swapping operation takes place, which essentially try to shufﬂe the node number to check if two connectivity matrix can be exactly matched.
Empirically, we have found this method performs very efﬁciently in identifying duplicates.
An example of different operation encoding bit-strings decode to the same network phase is provided in Figure 18.
[Figure] [Figure]
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[Figure] [Figure]
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[Table]
F N ETWORK A RCHITECTURE C OMPLEXITY E STIMATION
We argue that the choice of inference time or number of parameters as proxies for computational complexity are sub-optimal and ineffective in practice.
In fact, we initially considered both of these objectives.
We concluded from extensive experimentation that inference time cannot be estimated reliably due differences and inconsistencies in computing environment, GPU manufacturer, and
GPU temperature etc.
Similarly, the number of parameters only relates one aspect of computational complexity.
Instead, we chose to use Multiply-Adds (FLOPs) for our second objective.
The following table compares the number of active nodes, the number of connections, the total number of parameters and the multiply-adds over a few sampled architecture building blocks.
See Table 3 for examples of these calculations.
G E XPERIMENTAL R ESULTS C ONTINUED
Due to space constraints in the main paper, we present more interesting results here to show how
NSGA-Net performs in comparison to other state-of-the-art methods.
G. R EGRESSION T ASK D ETAILS
In this section, we demonstrate another example of using NSGA-Net to ﬁnd a set of efﬁcient trade-off network architectures for object alignment task.
We use the CMU-Car dataset described in (Boddeti et al., 2013).
The CMU-Car dataset is an object alignment task containing around 10,000 car images in different orientations and environments.
Similarly to the classiﬁcation example, we again use an 80/20 train/validation split from the original training set and the testing set contains only images of occluded cars and the training does not.
For both tasks during the search the testing data is not touched until the search concludes .
For back-propagation in object alignment, we use RMSProp (Ruder, 2016) again with a cosine annealing learning rate (Loshchilov & Hutter, 2016).
Our initial learning rate is 0.00025 and we train forepochs which takes about 50 minutes on a single NVIDIA 1080Ti GPU.
[Table] .6M - 2.
P100 MONAS (Hsu et al., 2018a) - - 4. - - - 1080Ti E A GeNet v2 (Xie & Yuille, 2017) - - 7.
Titan X CoDeepNEAT (Miikkulainen et al., 2017) - - 7. - 7, - - Hierarchical(Liu et al., 2018) - - 3. P100 AmoebaNet-A (6, 36) (Real et al., 2018) 3.2M - 3. - 20,000 3, - AmoebaNet-B (6, 128) (Real et al., 2018)
34.9M - 2.
- 20,000 3, - PPP-Net (Dong et al., 2018)
11.39M 1364M 4. Titan X RSearch w/ our encoding 3.3M 1247M 3.86 21.
- - - NSGA-Net(#ﬁlters = 128) 3.3M 1290M 3.85 20. 1,  1080Ti NSGA-Net(#ﬁlters = 256) 11.6M 4534M 3.72 19. 1,  1080Ti
[Figure]
G. R ESULTS
Classiﬁcation task :
Firstly, for the classiﬁcation task on CIFAR-10 dataset, a more comprehensive comparison with state-of-art methods including both hand-crafted and search-generated is provided in Table 4.
And examples of the efﬁcient trade-off network architectures found by NSGA-Net on
CIFAR-10 are provided in Figure 20.
The reported NSGA-Net architecture in Table 4 with 128 ﬁlters is also trained on CIFAR-100, and the resulting performance compared to other methods is shown in
Figure 19a.
Regression task :
During the NSGA-Netsearch we ﬁx the operation x
o to be the residual unit described in the original Hourglass paper (Newell et al., 2016).
We then search over the path x p .
We stack the hourglass twice during training and train for 20 epochs.
We use a smaller population of 20 in the regression task.
Once the search is done, we increase the number of ﬁlters and train for more epochs.
Our best architecture obtains 8.64% error (Table 5).
This is 1% worse than the state-of-the-art method, however at the cost of half the parameters, which may be desirable in some applications.
The trade-off frontier achieved by NSGA-Net compared to uniform random sampling from our path encoding, x p , is provided in Figure 19b.
[Table]