Multi-objective optimization via equivariant deep hypervolume approximation
Authors: Jim Boelrijk
Bernd Ensing
Patrick Forré
Venue: ICLR 2023 poster (2023)
Keywords: Multi-objective optimization
Hypervolume approximation
Geometric deep learning
Bayesian optimization
Evolutionary algorithms

Abstract: Optimizing multiple competing objectives is a common problem across science and industry. The inherent inextricable trade-off between those objectives leads one to the task of exploring their Pareto front. A meaningful quantity for the purpose of the latter is the hypervolume indicator, which is used in Bayesian Optimization (BO) and Evolutionary Algorithms (EAs). However, the computational complexity for the calculation of the hypervolume scales unfavorably with increasing number of objectives and data points, which restricts its use in those common multi-objective optimization frameworks. 
To overcome these restrictions, previous work has focused on approximating the hypervolume using deep learning. In this work, we propose a novel deep learning architecture to approximate the hypervolume function, which we call DeepHV. For better sample efficiency and generalization, we exploit the fact that the hypervolume is scale equivariant in each of the objectives as well as permutation invariant w.r.t. both the objectives and the samples, by using a deep neural network that is equivariant w.r.t. the combined group of scalings and permutations. We show through an ablation study that including these symmetries leads to significantly improved model accuracy. 
We evaluate our method against exact, and approximate hypervolume methods in terms of accuracy, computation time, and generalization. We also apply and compare our methods to state-of-the-art multi-objective BO methods and EAs on a range of synthetic and real-world benchmark test cases. The results show that our methods are promising for such multi-objective optimization tasks.

Published as a conference paper at ICLR 2023
MULTI-OBIECTIVE OPTIMIZATION VIA EQUIVARIANT DEEP HYPERVOLUME APPROXIMATION
Jim Boelrijk A14Science Lab, AMLab Informatics Institute, HIMS University of Amsterdam
Bernd Ensing
Patrick Forré A14Science Lab, AMLab Informatics Institute
A14Science Lab HIMS
University of Amsterdam b enn ensingeuva... nl
University of Amsterdam p.d. wwr.eeuva......
J. w.h.m.bbe.......... www.bberrrrikuee..
ABSTRACT
Optimizing multiple competing objectives is a common problem across science and industry. The inherent inextricable trade-off between those objectives leads one to the task of exploring their Pareto front. A meaningful quantity for the purpose of the latter is the hypervolume indicator, which is used in Bayesian Optimization (BO) and Evolutionary Algorithms (EAs). However, the computational complexity for the calculation of the hypervolume scales unfavorably with increasing number of objectives and data points, which restricts its use in those common multi-objeccivv optimization frameworks. To overcome these restrictions, previous work has focused on approximating the hypervolume using deep learning. In this work, we propose a novel deep learning architecture to approximate the hypervolume function, which we call DeepHV. For better sample efficiency and generalization, we exploit the fact that the hypervolume scale equivariant in each of the objectives as well as permutation invariant w.r.t. both the objectives and the samples, by using a deep neural network that equivariant w.r.t. the combined group of scalings and permutations. We show through an ablation study that including these symmetries leads to significantly improved model accuracy. We evaluate our method against exact, and approximate hypervolume methods in terms of accuracy, computation time, and generalization. We also apply and compare our methods to state-of-the multi-objective BO methods and EAs range of synthetic and real-world benchmark test cases. The results show that our methods are promising for such multi-objective optimization tasks.
INTRODUCTION
Imagine, while listening to a lecture you also quickly want check out the latest news your phone, so you can appear informed during lunch. As an experienced listener, who knows what lecture material is important, and excellent reader, who knows how confident in your abilities in each of those tasks. So you continue listening to the lecture, while scrolling through news. Suddenly you realize you need split focus. You face the unavoidable trade-off between properly listening to the lecture while slowly reading the news, or missing important lecture material while fully processing the news. Since this not your first rodeo, you learned over time how to transition between these competing objectives while still being optimal under those trade-off constraints. Since you don't want to stop listening to the lecture, you decide listen as closely as possible, while still making some progress in reading the news. Later, during lunch, you propose an Al that can read the news while listening to lecture. The question remains of how to train AI to Iearn to excel in different, possibly competing, tasks or objectives and make deliberate well-calibrated trade-offs between them, whenever necessary.
the headlines, you
Simultaneous optimization of multiple, possibly competing, objectives is not just challenge in our daily routines, it also finds widespread application in many fields of science. For instance, in machine learning (Wu et al. [2019] Snoek et al.] (2012). engineering (Liao et al.) [007] Oyama et al.] 2018). and chemistry (O Hagan et al. [ Koledina et al. 2019 MacLeod et al. 2022 Boelrijk et al. 2021
2023 Buglioni et al. 2022

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Published as a conference paper at ICLR 2023
The challenges involved in this setting are different from the ones in the single-objective case. If we are only confronted with a single objective function ff the task is to find a point x" that maximizes f: x" E argmaxxex* f fx), only by iteratively proposing imput points x, and evaluating f on x, and observing the output values f f(x)). Since the usual input space X' IR D is uncountably infinite can never be certain the finite number of points X1,.. XN contain a global maximizer x" of f. This is an instance of the exploration-Ve exploration-xs-expooiioee vs-exploitaation trade-off inherent to Bayesian optimization (BO) (nook et al.] 2012), active learning. (Burr) 2009) and reinforcement learning (aaebblig et al. 1996).
In the multi-objective (MO) setting, where we have M 2 2 objective functions, fi....: fM. it is desirable, but usually not possible to find single point x* that maximizes all objectives at the same time:
Tran Parko
Fstraited Pawkofoor
M x' E N argmaax (x). m=1
Riteneas pait
(I)
Hovport
Accilorai hpensae
XEX
A maximizer xi of Si might lead to a non-optimal value of fe. etc. So the best we can do in this setting is to find the set of Pareto points of F (MM) i.c. those points x E X' that cannot be improved in any of the objectives fm. m 1,.... .M, while not lowering the other values:
X = fxx E X 3x's E X. F(x) < F(x)),
(2) Figure 1: Illustration of Pareto front
and Hypervolume.
where y 3 means that ym S v'm for all m = ,,...,M .M and
y yxy y' that y y', but, y + yy'. In conclusion, in the multi-objective setting we rather concerned with the exploration of the Pareto front:
p³ = F(x") C RM =: V.
(3)
which often a (M - 11m-mminmii subspace of y = RMM Success in this setting is measured by how "close" the empirical Pareto front, based the previously chosen points XI....,NN:
PN= = (xx,,) n E [M], Dn'e E [N]. F(x,) - Fx C
(4)
is to the 'true' Pareto front P where [N] := NY. This is illustrated in Fig. where the blue points form the empirical Pareto front and where the black line depicts the true Pareto front.
Since the values F(x.) can never exceed the values P* w.r.t. - one way to quantify the closeness of Py to P is by measuring its hypervolume HV (Pw) inside y w.r.t. jointly dominated reference point r E y. This suggests the multi-objective optimization strategy of picking the next points XN+1 E X' in such way that it would lead to maximal improvement of the previously measured hypervolume HV (Px). This is illustrated in Fig. I where the hypervolume (i.e., the area in 2D) is shown for the empirical Pareto front (blue dots). Adding an additional point to the empirical Pareto front (1s) in Fig. D. increases the hypervolume indicated by the green area.
Unfortunately, known algorithms for computing the hypervolume scale unfavorably with the number of objective functions M and the number of data points N. Nonetheless, finding a fast and seal- able computational method that approximates the hypervolume reliably would have far-reaching consequences, and is an active field of research. Computation of O(2N NM) when computed in a naive way. however more efficient exact algorithms have been proposed such as WFG (O(N M/2 log N) While et al. (2012)) and HBDA (O(NM/2) Lacour et al. (2017)). However, these computational complexities are still deemed impractical for application in EAs (Tang et al. 2020), where computational overhead typically is required to be low. Also in BO, where one is typically less concerned with the computational overhead, faster hypervolume methods would be beneficial. For instance, the state-of-the-ar expected hypervolume improvement (qEHVT) (Daulton et al. 2020) is dependent on many hypervolume computations, greatly restricting its use to the setting of low M and N. In addition, the authors of the recently proposed MORBO (Daulton et al. 2022). which is current state-of-ther of-thheartt in terms of sample-efficiency and scalability to high M and identified the computational complexity of the hypervolume as a limitation.
the hypervolume has complexity
Therefore, efforts have been made to approximate the hypervolume. The method FPRAS (Bring) mann & Friedrich] 2010), provides an efficient Monte Carlo (MC) based method to approximate the hypervolume, of complexity O(N M/eM with the error of + In addition, at time of this work, a hypervolume approximation based on DeepSets (Zaheer et al., (217)) was proposed called HV-Net (Shang et al. 2022a). HV-Net uses a deep neural network with permutation invariance (i.e.
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DeepSets) w.r.t. to the order of the elements in the solution set to approximate the hypervolume of a non-dominated solution set. It showed favorable results on both the approximation error and the runtime on MC and tine-bsed hypervolume upproximation methods.
Other methods try to avoid hypervolume computation entirely and use other metrics to guide op- timization (Hernandez-Lobate et al., 2016). Such ParEgo (Knowles) 2006), which randomly scalarizes the objectives to generate diverse solutions. Other approaches focus on distance metrics regarding the empirical Pareto front (Rahat al. 2017, Deb al., 2002a). However, methods using the hypervolume to guide optimization, are typically superior. This is shown both in the setting of EAs and for BO. For example, a plethora of work shows that SMS-EMOA (Beume et al.) 2007) more sample-efficient compared to MO EAs using alternative infill criterion (Narukawa & Rodemann 2012 Tang et al. 2020). Also, qEHVI and MORBO have strong empirical performance compared to alternatives (Duulton ton et al. 2020 2022).
In this paper, we develop DeepHV, novel hypervolume approximation approach based on deep learning. As input the model requires non-dominated solution set after which the model outputs the predicted hypervolume of this solution set. By using specialized layers, DeepHV incorporates mathematical properties of the hypervolume, such scaling equivariance of the hypervolume w.r.t. the scaling of the objectives in the solution set. In addition, the DeepHV permutation invariant to the order of the elements and objectives in the solution set. Thus leveraging additional symmetry properties of the hypervolume compared to HV-Net. We show that our method allows for development
of single model that can be used for multiple objective cases up to M 10. We obtain improved performance compared to HV-Net on unseen test data. Moreover, we perform an ablation study to show that the additional symmetry properties improve model performance. We compare the wall time of our methods with exact methods and an approximate MC method. We show considerable speed- up that scales favorably with the number of objectives. In addition, to evaluate the useability of our approach, we use DeepHV in the setting of MO BO and EAs on range of synthetic and real-world test functions, and compare with state-of-the-ar methods and show competitive performance.
THE HYPERVOLUME
In this section, we introduce relevant information regarding the definition and symmetry properties of the hypervolume, which will later complement the definition of our method.
Hypervolume Definition For natural number N N. we put [N] == [I....,,N} Consider (column) vectors yn R in the M-dimensional Euclidean space. For a subset J C N, we abbreviate: y. := fy, J|T E R Mx (J), so Y(N] := [y,...., yN]] ] IR MN,, We then define the (M-dimensional) hypervolume y. w.r.t a fixed reference point E RM as:
N}.
Ulr.val jEJ
HV jy....... Vap M HV (yr) (vs) M
(5)
where A is the M-dimensional Lebesgue measure, which assigns to subset of R their M- dimensional (hyper) volume, and where we for y [vi,...., "M,....,AAe JM]]T E R abbreviated the M- dimensional cube as: yr 21] x... [FM yMr, if I' S ym for all E [M], and the empty-set otherwise. For fixed N and another point E R. we define its hypervolume improvement (over Y(N as:
HVT(y) == HVI (ylym)) = HM (y,y1.... yI.... .t......y yNN) HV (y1....,,y)
With help of the cettusion--xcllsiiin formula we can explicitely rewrite the hypervolume as:
M
(-1)SS++
II
min (ym,s rm)+.
SES
D#SCJ
m=1
The Symmetry Properties of the Hypervolume Function To state the symmetry properties of the hypervolume function, we will fix our reference point r == 0. which w.l.ogg always possible by subtracting from each y
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The hypervolume then scales with each of its components. for (e,..... exsl GR
HV (c @ Y(N)) Cι
HVM (y(N)),
(8)
where с O YJN] means that the m-th entry of each point Yn E E2 is multiplied by cm for all m E [M]. Furthermore, the order of the points does not change the hypervolume:
HV (Y)())..... (o()) HV (yI,...,yN),
for every permutation a E SN. where the latter denotes the symmetric group, the group of all permu- tations of N elements. But also the order the M M-components does not change the hypervolume:
HV M (r @ M(m) = НV M (y(v)),
(10)
for every permutation 7 E SM. where M[M] means that the row indices are permuted by
To jointly formalize these equivariance properties of HV together, define the following joint group as the semi-direct product of scalings and permutations:
G = RR- Sa x Sy,
(11)
where the group operation is defined as follows on elements (c1, T1, 01) and (c2, T2, 02) E G via: (C2, 72, 02) (C1, 01) (c2 ci, 72 TI, 02 o 01) E G,
(12)
where with C₁ = [cI,1..... C E R we use the abbreviation:
c = (er.())...........(
ER
(13)
This group then acts on the space of non-negative (M x -)-matrices as follows. For Y (Jm.n)mm E M and (c, E G we put:
(r(1).(1)
(N()
(C,T,0))Y
(14)
GM (+(M))
CM (M
The mentioned symmetries of the hypervolume can then be jointly summarized by the following G-equivariance property:
HV ((c, ((),T,o) -....... cas HV (Y).
(15)
This is the property that we will exploit for an efficient approximation of the hypervolume with G-equivariant deep neural network.
The hypervolume for different dimensions M are related follows:
HV = Hv ( YJ (
(16)
where we padded the (K x |J]) matrix with ones y., or, equivalently, K ones to each point y,, j E J. We leverage this property to combine datasets with different dimensions M and to train model that can generalize multiple dimensions.
DEEPHV EQUIVARIANT DEEP HYPERVOLUME APPROXIMATION
In this section, we first present the G-equivariant neural network layers used in DeepHV that incorporates the symmetry properties defined in Sec. / Following this, we discuss the general architecture of DeepHV. We provide intuitive explanation and mathematical proofs in App. A.I
3.1 THE EQUIVARIANT LAYERS
Before we introduce our G-equivariant neural network layers, we first recall how (SM x Sx)- equivariant layers are constructed.
Theorem 1 (See Hartford et al. (2018) Thm. 2.1). A fully connected layer of a neural network vec(Z) = a (W veo(Y)) with input matrix Y E MRN output matrix R x weight matrix W E RR. (MN) and strictly monotonic activation function (SM SN) equivvaiian if and only if it can be written as:
Z (Wi Y + W2 1M YM. + W3 Y IT W4 YMM 1 1xx + WA 1MxN), (17) where denotes the matrix of ones of the indicated shape, WT.... "y....,yy Wy E and where ML is the mean operation, applied to rows (Y ma), columns (Y 2z) both (Ym.s).
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We define the row scale of a matrix Y mm, ER MxN via:
YY) = max (ymmal m E [M]
E ERM,
(18)
1 (Y) TR ME[N]
and its row-wise inverse s
We then abbreviate the row-rescaled matrix Y s= (Y) @ Y E R RMN,
(19)
With these notations, are now able to construct a G-equivariant neural network layer with input channel indices i E I and output channel indices o O:
0) = Jo
Σ s iES
(
(x() + u2 w1
(0.1)
(0.0) -MM (x
(20)
M:
1MxN ug! 1MAx) )
+ w/ (o.1)
(1)
IN 10.1) (X())
M:M
where M applied the row-rescaled Y row-wise (Yı:)) column-wise (Y.) or both (YMM).
and where, again, M denotes the mean operation (but could also be max or min), and weights (o.5) (o) "k wi E R for k [4], 1.0,0E I. oa denotes any homogeneous activation function, i.e. require that for every y E IR, c Rzn:
v) = c.
(21)
This is. for instance, met by the leaky-ReLU activation function. An alternative way to achieve equivariance of the whole network w.r.t. to the scale of the whole network is by extracting the scales the input layer and then multiplying them all against the output layer, in contrast to propagating the scale through every layer individually.
3.2 ARCHITECTURE
All the DeepHV models presented here comprised of the same architecture, with the exception that we change the number of input and output channels in order to trade off the number of model parameters and the expressivity of the model. We denote models as DeepHV-c, c denoting the number of channels. Before the R N input is passed through the layers, we first extract and store the scale using eq. [8] and scale the inputs using eq. [9 We use five equivariant layers as in eq. 20) where the first layer has 1 input channel and output channels. The three intermediate layers have the same number e of input and output channels. The last layer then maps back to output channel, again resulting into R. x M output. Consequently, the mean taken of the resulting output to bring it back to permutation invariant quantity (IR ). This is then passed through a sigmoid function to map values between O and 1, as this is the value which the hypervolume can have in the scaled setting. Finally, we rescale the output by multiplying it with the product of the scaling factors. This step ensures that the model output is scale equivariant.
TRAINING
4.1 DATA GENERATION
To generate training data, adopt similar strategy proposed in [[Shag et al. (2022a), which is defined as follows: 1. Uniformly sample an integer E [1, 100]: 2. Uniformly sample 1000 solutions in [0, 1] M, 3. Obtain different fronts (P1, P2, ...) using non-dominated sorting, where P1 is the first Pareto front and Pg is the following front after all solutions of P1 are removed; 4. Identify all fronts P. with P. 2 n. If no front satisfies this conditions, go back to Step 2: 5. Randomly select one front P. with P. 2 n and randomly select n solutions from the front to construct one solution set. Using this procedure, we generate datasets consisting of million solution sets for each of the objective cases 3 < M < 10. We use this procedure as it generates a wide variety of non non-dominated solution sets (see Shang et al. (2022a)) and so that we can compare our model performance with HV-Net. We split our datasets into 800K training points and 100K validation and test points, respectively. We note that as we only sample solution sets with at maximum a hundred solutions, our model is likely restricted to this setting. In addition, using the relation of eq. [6] we also pad all datasets of M < 10 to M 10, by padding it with ones. As this does not change the hypervolume values of each solution set, create universal dataset that incorporates all generated objective cases.
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4.2 PERFORMANCE
We train models with 64, 90, 128. and 256 channels (See Sec. [3.2]
Table I: Test MAPE (lower is hetter) for models trained on each objective M separately. Bold numbers indicate the hest-performing model. Blue numbers indicate the best model between HV-Net (Shang et al. 2202aa) and our models with a comparable number of model parameters.
for architecture) on each objec- tive case separately. In addition,
we also train models on the com-
bined dataset (See Sec. |41] de- noted with -all) using 128 and 256 channels. All DeepHV mod- els have been trained with learn- ing rate of 10-5, using Adam and the Mean Absolute Percentage Er- 10
M
D-6
DeepHV-90
DeepHV-122
HV-Net
O. 0.00800
0.00744
0.00526
0,00488
0.015878
0.02065
0.01608
0.01689
0.01209
0.032067
0.04710
0.03381
0.01506
0.01093
0.042566
0.03200
0.02783
0.02395
0.011488
0.050867
ror (MAPE) loss function (de Myttenaere et al., 2016)) As the (scaled) hypervolume tends to become small in high-dimension cases, more common loss functions such as the RMSE become rather small rather quickly, despite having large relative errors. The MAPE is better suited for this setting. For the separate models, we use a batch size of 64 and train for 200 epochs. However, improvements where typically marginal after ~ 60 epochs. For the models trained on all objective cases simultaneously, we train for 100 epochs with a batch size of 128. We pick the best model within these epochs based on the lowest loss on the validation partition. We report MAPE test errors in Table [and compare with HV-Net (Shang et al., 2022a). Note that HV-Net uses million training datapoints, whereas we use 800K. When using 90 channels (DeepHV-90), (98.3K) w.r. HV-Net (99.7K) 7K), however we obtain roughly a twofold increase of performance on most objective cases. In fact, using 64 channels (49K parameters), we already outperform HV-Net on most objective cases. This boost in performance is likely due to the incorporation of scale equivariance and an additional permutation invariance over the order of the objectives, which are not incorporated into HV-Net. For larger models, we obtain even better results.
model has comparable number of parameters
We report results on DeepHV trained on all objective cases simultaneously in Table Here it is shown that the general models have significant increase in performance compared to the separate models. Interestingly, this shows that the model can generalize to multiple objective and can gain additional performance in each objective case separately by seeing instances of others. This potentially allows for the training of a universal model that can be trained on even higher objective cases.
Table 2: Test MAPE for models trained on each objective case separately, and for models trained on all objective cases simultaneously (denoted with additional -all). Bold numbers indicate the best model in their architecture.
M
V-22
DeepHIV-128-all
V-e6
DeepHVV-256aal
O. 00526
0.00673
0.00-183
0.00195
0. 0.906-
0. 0.00887
0. 0.00782
0.00714
O. 0.01699
0.00956
0.01209
0.00773
O. 0.01511
0.00954
0.01354
0.00761
0.00900
0.00967
0.00707
0.01506
0.008499
0.01093
0.00657
0.01506
0.00796
0.01216
0.00601
10
0.02395
0.00772
0.01148
0.00598
4.3 ABLATION STUDY
To further study the effect of the symmetries incorporated into DeepHV, we perform an ablation study. We train different models to address the cases of: a) no permutation invariances and no scaling equivariance; b) no permutation invariances, but with scaling equivariance; c) only permutation invariances but no scaling equivariance; and d) both permutation invariances and scaling equivariance. For the case a) we use standard multi-layer perceptron (MLP). For case b) we use an MLP with scaled inputs (using eq. [8]and and eq. 19). followed by rescaling of the model outputs. For case c), we use the DeepHV architecture without scale equivariance. Case d) corresponds to the original DeepHV architecture. All models had a similar number of parameters and were kept as comparable to each other as possible, details can be found in App. B Without incorporating scale equivariance, training with the MAPE loss was unstable and got stuck around a value of 1, meaning that the model always outputs hypervolumes close zero, this was also observed in Shang et al. (2022a). We therefore also

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performed training with the MSE loss and log MSE loss and report the best model found using these loss functions. We report MAPE test errors Tab. 3 where it is shown that each additional symmetry leads to inereased model performance. This becomes especially profound at higher objective cases M > 5, where the MLPs do not achieve lower errors than 19%.
Table 3: Test Mean Absolute Percentage Error. Bold numbers indicate the hest-perforning model.
M
MLP
MLP-wwit--saaling
DeepHV-128-no-caalee
DeepHV-128
0.03103
0.02397
0.01944
0.00526
0.06183
0.05820
0.03311
0.01689
0,2015
0.1928
0.04571
0.01506
10
0.3106
0.3031
0.05212
0.02395
EXPERIMENTS
5.1 TIME COMPARISON
We compare the time performance of DeepHV with the exact hypervolume methods of Lacour et al. (2017) (complexity O ((MM/+11)) implemented in BoTorch (Balandat et al., 2020) and the method of Fonseca et al. 2006) (complexity (O(N--1 log N)) as implemented in Pymoo (Blank
& Deb, 2020). In addition, we compare time performance with approximate Monte Carlo (MC) hypervolume method implemented in Pymoo. We perform the tests on CPU and GPU where applicable. Experiments shown in Fig. comprised of the computation of 100 solution sets randomly drawn from the test sets. The MC method used 10K samples. In all cases, DeepHV faster than the BoTorch and Pymoo MC implementation. Above M 5, both variants of DeepHV are also faster than the exact Pymoo implementation. We also compare the MAPE versus the computation time for DeepHV and the MC method with different number of samples in App. D.
5.2 BENCHMARKS
We empirically evaluate DeepHV on range synthetic test prob- lems from the DTLLZ (Deb et al. 2002b) problem suite, which contains standard test problems from the MO optimization lit- erature that are scalable in the number of input variables d and objectives M and contain vastly different types of Pareto fronts.
Beeek-9254
Dsuptri122
Computzzion 2ime 15)
Pymoo
In Sec. 5 we test DeepHV in the context of multi-objective evolutionary algorithms (MO EAs) and Sec. 5.2.2 in the context of Bayesian optimization (BO). In addition, we con-
Pyneo PC
BeTbrch
Bettech-aa
sider range of challenging real-world problems (Tanabe & Ishibuchi 2020): penicillin production (M = 3) problem, vehicle safety problem (M = 3), car side impact problem (M 4), water resource planning problem (M 6), and a conceptual marine design problem (M = 4). See App- [C.2 for more details regarding the problems and reference point specifications. We run all methods five times and report the means and two standard errors. Problems not reported in the consecutive sections are shown in in Sec. [D2.2] D.3.2
Number of objoctivss
Figure 2: Comparison of computation time, shown on log-scalc. We report the means and standard errors across trials.
5.2.1 MULTI-OBJECTIVE EVOLUTIONARY ALGORITTMS
To use DeepHV in the context of MO EAs, we replace the exact computation of the hypervolume in SMS-EMOA [Beume et al. 2007) with that of DeepHV, and compare performance. In addition, we compare performance with NSGA-II (Deb et al.) 2002a) and NSGA-III (Deb & Jain 2014). SMS-EMOA (Beume et al. 2007) is an EA that uses the hypervolume to guide optimization. In its original form. it uses staad--sttte (μ + 1) approach to evolve its population, where at each generation, one offspring is created and added to the population. Next, one solution removed from the population by first ranking the population into separate fronts, followed by the removal of the
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solution that has the least contribution to the hypervolume of its respective front. This is done by computing the hypervolume contribution of each solution, by computing the hypervolume difference between the respective front, and the front without the individual solution. Instead of steady-state approach, we use a (1 + (1) approach to evolve the population, effectively doubling the population at each generation, and then need to select half of the population using the selection strategy described above. We stress that this is an extensive test of the performance of DeepHV, as the model needs to predict both the correct hypervolume of the front, but also the contribution of each solution in the front (i.e. the front and all of its sub-fronts need to be predicted with high accuracy). In the case of M > 5, we use the MC approximation of the hypervolume instead of the exact hypervolume as its use becomes computationally prohibitive (See Fig. 2). We note that the DeepHV architecture could also be adopted and trained to directly predict the hypervolume contribution of each solution in the solution as in (Shang et al.] (022b), which would be more suitable for this task.
As another baseline, we compare with NSGA-II which is a renowned MO EA, due to its low computational overhead. Although it generally is less sample-efficient than SMS-EMOA, it is often the method of choice due to a lack of scalable alternatives. In NSGA-II, the population is evolved using a (μ + (4) approach, then individuals are first selected front wise using non-dominated sorting and then, remaining individuals are selected based on the Manhattan distance in the objective space, in order to maintain a diverse population. The task of the Manhattan distance corresponds to that of the hypervolume contribution in SMS- -MSA. NSGA-III replaces the Manhattan distance with a set of reference directions, where population is selected at each reference direction to maintain diversity, but is otherwise similar to NSGA-IL II. For each algorithm, we use a population size of 100 individuals and run for 100 generations, resulting into 10K function evaluations, and we record the exact hypervolume of each generation. Further algorithmic details can be found in the App. C.T
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DTLZ1, MaS
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Jel Nater Res, Flac., Merf
96500
Hygar
2.00
Vysernnume
05000
Hyperrnnumme
Hyperspume
4.4
4.6
2.50
0.58
07500
97020-
4,8
2500 5005 7505 16660 Function evalaations
5000
15030
2500 5000
7509
18030
2550
5000
7500
18509
Function evaluations
Function eval adions
Function avala
Deep
Figure 3: Optimization performance in terms of exact hypervolume (higher is better) versus function evaluations.
Fig. 3 presents results the DTLZI and DTLZ5 problems for M 5, and on the real-world marine design and water resource planning. We report wall times in Sec. [.2.1 and report results on additional synthetic test problems in App. [.2.2] The Pareto front of DTTLZ1 lies on the linear hyperplane EM f 0.3, and all methods find similar hypervolume, with a slightly worse performance of NSGA-II. DTLZ5's has beam-like Pareto front. On this problem, NSGA-II and SMS-EMOA perform best, followed by the DeepHV models. NSGA-III performs worst. On the Marine Design problem, the NSGA variations perform best, followed by SMS-EMOA and the DeepHV variations which perform similar. On the water resource planning problem, SMS-EMOA performs best, followed by DeepHV-256-all and the NSGA variations. Although the different DeepHV models perform similar on most problems, on this problem the separately trained models perform worst.
The closeness of the DeepHV variations to the exact SMS-EMOA provides an indication on the ability of the models to generalize the different encountered Pareto front shapes. We observe that on most problems, exact SMS-EMOA (blue line) typically obtains higher hypervolumes or similar than the DeepHV variations. On problems where SMS-EMOA performs better, it is observed that that DeepHV finds important points on the pareto front, but is not able to spread the population over the Pareto front as well as SMS-EMOA.
5.2.2 MULTI-OBJECTIVE BAYESIAN OPTIMIZATION
We use DeepHV in the context of MO BO by using DeepHV to compute the hypervolume improve- ment (HVI, eq. 6. We use the HVI in a Upper Confidence Bound (UCB) acquisition function, which
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either name UCB-HV when the exact hypervolume is used, UCB-DeepHV in case we DeepHV to approximate the hypervolume. It is defined as follows:
QUCB- DeeBB-DeepHv 3) = (xV 8)), where fUCB 3) = μ(x) + (x(
(22)
Here, pu(x and (x) are the mean and standard deviation of the GP posterior at input x, whereas / a trade-off parameter that controls exploration and exploration. In this work we use 0.3.
We compare with the oftte-of-the-ar methods qParEgo and qEHVI (Daulton et al. 2020). ParEgo
(Knowles) 2006) randomly scalarizes the objectives and uses the Expected Improvement acquisition function (Jones et al., 1998). qParego uses MC-based Expected Improvement acquisition function and uses exact gradients via auto cittoddfferentiation for acquisition optimization. Expected Hypervolume Improvement (EHVI) is the expectation of the HVI (Yang et al. 2019), thus integrating the HVI over the Gaussian process posterior. qEHVI is a parallel extension of EHVI and allows for acquistion optimization using auto-differennttiin and exact gradients, making it more computationally tractable than EHVI. gEHVI was empirically shown to outperform many other MO BO methods (including qParEgo) Daulton et al., 2020; 2022). We evaluate optimization performance in terms of the exact hypervolume on four test problems. Note that above the five objectives case, qEHVI becomes computationally prohibitive. UCB-HV becomes computationally prohibitive for M > 4. We show wall-clock runtime information in Sec. D and we provide additional experiments on other objective cases and additional problems in Sec. D.3 Fig. 4 shows results for a range of DTLZ synthetic test problems and the Car Side Impact problem. In general, it is expected that qEHVI is superior to UCB-HV and UCB-DeepHV, as it uses the exact hypervolume and integrates out the uncertainty of the GP posterior in more elaborate way. However, if these methods perform on par with qEHVI, then the acquisition function choice is reasonable. If UCB- CCB-DeepHV performs similarly to UCB-HV, it would that DeepHV approximated the hypervolume of the test problems well. qEHVI, ranks best on DTLZ2, DTL.Z7 and the Car Side Impact problem. On DTLLZ5, qEHVI performs worse than than the other methods. On most problems, UCB-HV and UCB-DeepHV have similar performance, and generally outperform qParEgo, which is a strong result.
Car Siow Impact, Mm4
DTL22, M-4
DTL25, M-5
DTL27, M-3
450
56000
2500-
0.8-
50.000
Heporot.me
Nyporeolmee
0.6
Hypereal km
67000
Hyporvolume
2000
96:00
7500
250
1000-
50
100
150
30
120
150
94000
100
250
50
200
150
Function evaluations
Function evaluations
Fanction etrakasii trakutiions
Function avaluations
UCB-HV
eBHV
UCB-DexpHV-126
UCB-Dxephh-226
UCBBDDeepHV2256-aa
gParEgo
Figure 4: Sequential optimization performance terms of hypervolume versus function evaluations.
6 DISCUSSION
We presented DeepHV, a novel hypervolume approximation based on deep learning. DeepHV is scale equivariant in each of the objectives as well permutation invariant w.r.t. both the objectives and the samples, thus incorporating important symmetry properties of the hypervolume. We showed that our method obtains lower approximation errors compared to HV-Net that does not incorporate all of these symmetries. In addition, we showed through an ablation study that incorporating these symmetries leads improved model performance. We showed that DeepHV achieved lower approximation errors than MC based approximation and is considerably faster than exact methods, becoming more pronounced at high objective cases M > 4. Therefore, we believe DeepHV promising as a replacement for computationally demanding exact hypervolume computations in of-te-of-the-ar (high-throughhut methods. A limitation that the DeepHV models are currently only trained on solution sets with at maximum 100 non-dominated solutions, restricting their use to this setting, However it can handle arbitrary amount of objectives and samples by design. We tested DeepHV in context of multi-objectivv BO and EAs and showed competitive performance compared with state- th- methods. We believe these results are promising, and with better training data generation methods, DeepHV could potentially be even more competitive with exact hypervolume methods.

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Published as a conference paper at ICLR 2023
7 ETHICS STATEMENT
The paper presents a method that can be used multi-objective optimization problems. These prob- lems find widespread use throughout society. For example in material design (catalysts, medicines, etc.), car design or finance. Although many such applications are for the general good, there are also applications which can have ethically debatable motivations, for instance, when used in the design of toxins, narcotics, or weaponry.
8 REPRODUCIBILITY STATEMENT
Code, models, and datasets used in this work can be found at: bttps://github.. com/ Jimbo994
L.mbo994/Geephv-ico/ deephv-iclz Where applicable, we have run experiments multiple times with different random seeds and have stated mean and standard error of the results. addition, we have described important implementation and experimental details in either the main body of the text or the Appendix.
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A PROOFS
Intuitive explanation We here provide an intuitive explanation which complements the mathemat- ics in S.c and the provided proof in the next paragraph. Consider a matrix Y E xN matrix to which we want to apply operations which are both permutation equivariant or invariant w.r.t. M and N and again result into an IR M output. The simplest operation that is invariant would be to apply mean operation both the columns (M) and rows (N) and multiplying by 1 1M (to maintain an R M N output). However, other invariant operations such as taking the min, max or sum would also be possible. Note that the resulting invariant output can also be multiplied with scalar values (i.e.,
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model weights) without violating the permutation invariance. This is the fourth term in eq. 20 (with w1). The simplest equivariant operation would be to multiply Y with a scalar value, which is the first term in eq. 20 (with w1). Shifting all RMxN values with some bias term, does not break this equivariance, this is the fifth term in eq. 20 (with ws). In similar fashion, we can take the mean over the .M rows and multiply with 1 X which would result in output that is equivariant w.r.t. to the columns and invariant w.r.t the rows, this can then also be multiplied with a scalar value (w2 in in eq. 20). Likewise, can do the same thing over the columns, which is the third term in eq. 20 (with Wz). In addition, we would like to do transformations which are scaling equivariant. This can be done by rescaling the inputs to some domain, and then rescaling the outputs after these operations.
A.1 PROOF OF SECTIONB.1
Proof To show that the layers in eq. 20 are really G-equivariant, let Y' = (Y and g(Y') the left hand side of eq. 20 and (c. a) G. Then (c,T,o) @ Y = ((c,T,0) o Y We can also see that:
,(()
= max |mm ()(m))a(m)) [E(N)
m [M]
(23)
Cm max (F(m).. m E [M]
= c s v.c.s(Y()) sYY
(24)
nE[N]
where acts trivially on the (M x vv-etr s (Y If we act on the defining equation for Y y
Y =s) Y!
(25)
with (c, 0) we then get:
7c) Y()) (c, o o
(26)
(27)
(s((c,T,0)
(28)
=s((C,T,,)
(29)
This shows that:
(c,,7,)) -
(30)
With this we get:
h((C,T,O) 3 Y())
(31)
:= ==w/o?? (0.0) ((c,
o x())
+ +wg 1o 1m ((1.7,0)
(32)
M.
w 1o.)
(11,700)010)
IN
(33)
:M
+ w (o,)
((.,7 ( Y(1)
1MxN - -agao) 1MxN
(34)
M:M
wi (oi))
(T, r
(0.0)
(T, @ (IN (x M
(35)
w (o.))
(T. ((x()) iM
IN
(36)
+ + (T,O)
1 1Mxx))
+ ug (T,,O) O 1
(37)
MMM
(T, (T,6))(((())))
(38)
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With these equations can finally get:
g(c.T,o)) 2Y) =a 0a
1
Σ
((.T,,0) on (
(39)
[
iET
0a
Σ
c-Y(( (T,OO) o (rl)))
(40)
[1]
IET
,T,) On (r
(41)
[1]
IET
Σ
(c,7,
o h (9())
(42)
[
iET
0a
(c. (c, ④
[
s s(((0)) @ (OIA)
(43)
iEl
= -c,,T,0) O
Σ iET
s(Y(0))
s
@ h (X1)))
(44)
[2|
= =
(45)
where we in the second to last step made use of the homogeneity of the activation function on and that it is applied element-wise. This shows that is G-equivariant.
B DETAILS ON ABLATION STUDY
All models were trained with learning rate of 10-5 using Adam for 100 epochs and in all layers we use the leaky-Rell activation function. DeepHV-128 (described in Sec. (.2 and DeepHV- 128- no-scaling were comprised of 198K parameters. The MLPs consisted of 6 layers. The first layer consisted of M x 100 input features and 180 output features. The intermediate layers consisted of 180 input and output features and the last layer consisted of 180 input features and 1 output feature. Similar as for DeepHV this output was then passed through a sigmoid function to map values between 0 and 1. This is also justified for the models without scaling equivariance, as all examples in the train, test, and validation set are between O and 1. As M influences the parameters of the first layer, the number of parameters in the model varied per objective case. The MLPs consisted of 184K parameters for M = 3, 220K parameters for M = 5, 274K parameters for M = 8 and 310K parameters for M 10. Even though the MI.ss were equipped with more parameters than DeepHV in cases M >> 5, this did not lead to improved performance.
C DETAILS ON EXPERIMENTS
C.I ALGORITHMIC DETAILS
Multi-objective Evolutionary Algorithmss: For, NSGA-II, NSGA-III, and SMS-EMOA we use
the default settings in the open-source source implementation of Pymoo ! replace the hypervolume computation with DeepHV or the MC hypervolume approximation and all other settings are kept default. For the MC hypervolume approximation we use 10, 000 samples.
where only for EMS-EMOA we
For NSGA-III, we create reference directions using the das-dennis approach (Das & Dennis 1998). Where we used the following number of partitions: (M=3:12, M=4:4, M=5:4, M=6:3, M=7:2, M=8:2, M=9:2, M=10:2)
Multi-objective Bayesian Optimization: For all methods and experiments we use an independent Gaussian Process (GP) with a constant mean function and Matérn-5/2 kernel with automatic relevance detection (ARD) and fit the GP hyperparameters by maximizing the marginal log- liceliocl
htttps://github..... http:://gggtuuuu.ooo/o//g/g///// mi zat in oo appmoo
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All experiments are initialized with 2d randomly drawn datapoints. We then allow for optimization budget of 150 function evaluations. For gEHVI and qParEgo we use 128 quasi-MC samples. All methods optimized using L-BFGS-B with 10 restarts and 64 raw samples (according to BoTorrh naming convention).
Methods using DeepHIV: We ensure that solution passed DeepHV only contain non- dominated solutions and are shifted so that the reference point lies at [0]* This is done by subtracting the specified reference point from the solution and applying non non-dominated sorting to only keep the non-dominated solutions.
C.2 SYNTHETIC & REAL-WORLD PROBLEMS
Information on the reference points used for hypervolume computation of each problem can be found in Table 4 and are based suggested values in Pymoo (Blank & Deb, 2020) and BoTorch (Balandat cet al. 2020) or specified later. The DTLZ problems are minimization problems. As BoTorch assumes maximization, multiply the objectives and reference points for all synthetic problems by -1 and maximize the resulting objectives.
In the multi-objective Bayesian optimization problems functions as provided in Botorch (Balandat et al., (020). Whereas in the multi-objective evolutionary algorithm experiments we use the implementations of Pymoo (Blank & Deb 2020).
the implementations of the synthetic
The DTLZ test suite contains a range of test functions which are considered standard test problems in the multi-objective optimization literature. Mathematical formulas of each test function are provided in Deb et al. (2005). The problems are scalable in both input dimension d and in the number of objectives M. We set the input dimension to = 2M for each problem. The shape of the Pareto front of each of these problems can be viewed in the Pymoo documentation ems eaellmmmmmmma /aay// dt 12.1.. html). ).
DTLZ:
Table 4: Specification of reference points used for all benchmark problems.
Problem
Reference Point Botorch
Reference Point Pymoo
DTLZI
-400 az
400 M
DTLZ2
1.1]As ar
[1]ar
DTLZ3
10000
[10000] M
DTLZ5
10) 10)
[00 M
DTLZ7
15
15] M
Vehicle Safety
1164.722
Not performed
Pennicilin
Not performed
[1885, 86.93, 514.7] Not performed
Car Side Impact
[-45.4872, -4.5114, 13.333, - 10..9442]
Water Resource Planning
Not performed
[84793, 1482, 3110300,
17141000, 381410, 103170] 493.7, 17126, 5113.6. 14.35904
Conceptual Marine Design
Not performed
The vehicle crash safety problem is described in Tanabe & Ishibuchi 2020) and is a 3-objective problem with d = 5 parameters describing the size of different components of the vehicle's frame. The goal is to minimize mass, toe-box intrusion (i.e. vehicle damage), and acceleration in a frontal collision. We use the BoTorch implementation.
Vehicle Crash Safety (RE3-5-4):
The Penicillin problem described Liang & Lai 2022 and is 3-objective problem with d = parameters. The goal is to maximize the penicillin yield while minimizing fermentatation time and CO2 byproduct. The input parameters are the culture volume, biomass concentration, temper- ature, glucose concentration, substrate feed rate, substrate feed concentration, and H+ concentration. We used the default BoTorch implementation of this problem.
Penicillin:
Car side impact (RE-4-7-1): The side impact problem is described in Tanabe & Ishibuchi (2020) and is a 4-objective problem with = parameters. The goal is to minimize the car weight, the force experienced by a passenger, and the average velocity of the V-pillar responsible for withstanding
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the impact load. The fourth objective is the of 10 constraint violations. We use the default implementation this problem BoTorch and Pymoo. Note this is objective problem in Pymoo.
Water Resource Planning (RE-6-3-1): The Water Resource Planning problem is described in Tanabe & Ishibuchi (2020) and is 6-objective problem with d = 3 parameters. The goal of the problem is to minimize the drainage network cost, the storage facility cost, the treatment facility cost, the expected flood damage cost, and the expected economic loss due to flood, of water resource planning. The sixth objective sum of seven contraint violations. The reference point was set using the infer reference__peee point heuristic in BoTorch on the Pareto frontier over large discrete set of random designs.
Conceptual Marine Design (RE4-6-2: The Conceptual Marine Design problem described in Tanabe & Ishibuchi (2020) and is a 4-objective problem with d 6 parameters. The goal is to minimize the transportation cost, the light ship weights and the annual cargo transport capacity. The fourth objecdtive is the sum of nine constraint violations. The reference point was set using the infer rearrrrffeeneeeeeeeeee pecieenceeppettttttt heuristic in BoTorch on the Pareto frontier over a large discrete set random designs.
D ADDITIONAL RESULTS
D.1 TIMING RESULTS
All computations shown in Fig. were performed Intel(R) Xeon(R) CPU B5-2640 CPU v4. and in the case of the GPU calculations on a NVIDIA TITAN X.
We now show the comparison of the Mean Absolute Percentage Error (MAPE) versus the computation for DeepHV and the approximate MC method (See Sec. [5. for details) for the cases of M = 3, M 5, M 8 and M 10, in Fig. [5] It is shown that in all cases, DeepHV obtains significantly lower MAPE at similar runtimes. The approximate MC method requires much more computation time obtain similar MAPE, which significantly increases at higher objective case M.
80000
Coaptii
Pymss MC
24:455
Pyrsoo MC
Coupity
Pyroo MC
go
g 2.32
6016
2.13
8004
2.09
0000
3.05
6000
2ozz
609
303
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D.2 MULTI-OBJECTIVE EVOLUTIONARY ALGORIHTMM
D.2.1 WALL-TIME COMPARISON
[ a comparison of the wall time for the experiments shown in Fig. 3 We observe that using the exact hypervolume implementation in Pytorch, is slowest, as is also supported by the results shwon Sec. [5.]] its use become prohibitively slow for M > 5. NSGA-II and NSGA-III are fastest. The MC approximation (used in Water Resource Planning); is, using the current implementation, faster than the DeepHV models. should be noted that the current implementation of the DeepHV models into Pymoo is not efficient it can be. Firstly, at the moment, the hypervolume of the front and all its sub-fronts now need to be predicted in an iterative fashion, which is not ideal, especially contributions of all solutions in the front in one go, as in (Shang et al. 2002b). which would likely give an order of magnitude speedup. This should be studies future research. The JCB-DeeppHV-2566 model is considerably slower than UCB-DeepHV-122 and UCBBDeepHHV-25) as it requires padding of the input shape 100 x 10 and thus requires additional computations. CPUs are not particularly suitable for these computations, and hence these wall times could further benefit from the use of GPUs.
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Figure 6: Cumullative wall time of experiments shown in Fig.
D.2.2 ADDITIONAL EXPERIMENTS
To study the performance of DeepHV in the context of multi-objective evolutionary algorithms on a broader range of problems and Pareto front shape, we ran additional experiments on a range of problems shown in Fig: 7 If M > 5 we use the MC approximation in SMS-EMOA When comparing DTLZI (M- 3) with DTLZI (M 5, in main text) and DTLZI (M = 8), it can be seen that in higher dimensions, the performance of NSGA-II drops significantly. The performance of the DeepHV models also deteriorates slightly, but they are still able to find a relatively good approximation of the Pareto front, albeit not diverse as for NSGA-III and MMM-E.MO. Of course, the 8 dimensional objective space is much larger than the 3 objective space, and perhaps more training data required to have better performance. This behavior is also observed for DTL75 but to a lesser extent. DTLZ2 has Pareto front that lies on the first octant of the unit hyper sphere, i.e. it has very idealistic concave shape. The DeepHV models find some points on the unit sphere, but fails to have nicely spread population over the sphere compared to SMS-EMOA and NSGA-III. In the case M = 5, it still outperforms NSGA-II. DTLZ3 has the same Pareto front shape as DTLZ2 but is parametrized differently, in addition we optimized this problem with differently set reference point [10000 J1 This effectively changes the shape of the Pareto front, and now all models perform similarly. DTL.7 (M = 3) has four disjoint Pareto-optimal regions, which generally is good test.
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Figure 7:
D.3 MULTI-OB/ECTIVE BAYESIAN OPTIMIZATION
D.3.1 WALL-TIME COMPARISON
Figure provides comparison of the wall time for the experiments shown in Fig. For the Car Side Impact and DTLZ2 (M = 4) problems, we show plots with and without UCB-HV for clearer comparison. addition, we show timings for DTT.Z2 (M = 6) to show scaling of qEHVI. We observe that UCB-HV, using the exact hypervolume implementation in Bo'Torc,, is slowest, as is also supported by the results shown in Sec. 5.1 Its use becomes prohibitively slow for M > 4. qParego generally is fastest and runtime scales well with increasing objectives. qEHVI, which has highly efficient parallel implementation, generally is in the same ballpark with CCB-DeepHV-128 and DeepHV-256 for problems up to M 5. However, above M 5, qEHVI is considerably slower than the other methods. The UCB-DeepHV-256- DeepHV-112 128 and UCB- it requires padding of the input shape 100 10 and thus
model is considerably slower than UCB-
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D.3.2 ADDITIONAL EXPERIMENTS
In order to further evaluate the performance of DeepHV in the context of multi-objective Bayesian optimization, we also report results for the DTLZI, DTLZ2, DTLZ3, DTLZ4 and DTLZ5 synthetic test problems for the objective cases M 3, and 5 if not otherwise shown in the main text. In addition, we show results on the real-world Penicillin and Vehicle Safety problems. In the case of UCB-HV, we do not show results for the case M and the DTLZ4 problems. These results are shown in Fig. I0 The other resuts are shown in Fig. 9 All experiments were performed 5 times, and we report the mean and standard errors. In general, DeepHV is competitive with UCB-HV, qParego and qEHVI, although qEHVI clearly performs best on DTLZ2, DTLZ4 and Vehicle Safety.
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Reviews:
[{"review_id": "VilI2xkqdv", "reviewer": "Reviewer_DvS5", "summary": "", "questions": "", "limitations": "", "rating": 6, "confidence": 3, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "main_review": "", "summary_of_the_paper": "This paper proposes a new method to approximate the hypervolume function with a deep neural network The network is built by using specialized layers to incorporate some symmetry properties of the hypervolume function, such as permutation and scale equivariance. ", "summary_of_the_review": "This paper proposes a new deep neural network-based hypervolume approximation method that is equipped with some symmetry properties. More discussions and more evaluations are needed to verify the advantages of the proposed method.", "strength_and_weaknesses": "Strengths\n1.\tA new deep neural network-based approximation with permutation and scale equivariance is proposed for the hypervolume function.\n2.\tThe organization is clear and easy to follow.\n\nWeaknesses\n1.\tThe advantages of the permutation and scale equivariance in the hypervolume approximation need more discussion.\n2.\tThe effectiveness of the proposed method needs more evaluation. First, when comparing the effectiveness of training a model on each objective case separately with that on all objective cases simultaneously, how is the experiment set to ensure fairness? Second, it is reported that the boost in performance of DeepHV compared to HV-Net is likely due to the incorporation of the symmetry properties of DeepHV. How? Third, the results shown in Section 5.2.1 do not appear to be consistent with that in Section 4.2. For example, SMS-EMOA that uses exact computation mostly achieves the best results, yet the approximation models (e.g., DeepHV-256-all) that are shown to achieve higher accuracy in Section 4 rarely show better results in Section 5. This raises the question of whether the metrics used in Section 4 are appropriate. Fourth, the experiments conducted in Section 5 should include more state-of-the-art (approximated) hypervolume-based methods.\n", "limitations_and_societal_impact": "", "review_text": "", "extra_scores": {"correctness": "3: Some of the paper’s claims have minor issues. A few statements are not well-supported, or require small changes to be made correct.", "technical_novelty_and_significance": "3: The contributions are significant and somewhat new. Aspects of the contributions exist in prior work.", "empirical_novelty_and_significance": "2: The contributions are only marginally significant or novel."}, "extra_text": {}}, {"review_id": "Uo4hGWlgQz", "reviewer": "Reviewer_9RfZ", "summary": "", "questions": "", "limitations": "", "rating": 6, "confidence": 4, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "main_review": "", "summary_of_the_paper": "The paper proposes a new method leveraging equivariant deep learning to approximate hypervolume calculation in multi-objective optimization (DeepHV). The authors first introduce important concepts in multi-objective optimization, such as problem formulation and the Pareto optimality. The authors then describe current hypervolume calculation methods (WFG, HBDA, FPRAS, HV-Net) and their scaling limitations in terms of number of objectives (M) and number of data points (N) and how hypervolume are used or avoided in current multi-objective optimization methods, such as multi-objective Bayesian optimization and multi-objective evolutionary algorithms. Following the introduction, the authors formally define the hypervolume calculation and it's important symmetric properties (permutation invariance and G-equivariance) which are then used to propose a G-equivariant variation of a fully-connected neural network layer that make up the building block of DeepHV. The Deep HV network used in the experiments includes five layers of G-equivariant layers proposed by the authors.\n\nFor their experiments, the authors generate $\\sim$ 1M training data points by sampling solutions for different dimension of objectives (3-10) that are then used to train the DeepHV approximator. The first set of experiments presented show that DeepHV achieves a lower MAPE compared to HVNet and generally performs better with a greater set of parameters. Subsequent experiments show runtime benefits of DeepHV compared to open-source hypervolume calculations in Pymoo and Botorch. The final set of experiments of experiments involve using DeepHV in active multi-objective EA and BO settings. In the multi-objective EA settings, DeepHV performs better than NSGA2 but fails to beat SMS-EMOA with a traditional Monte-Carlo calculator across various multi-objective optimization benchmarks. For multi-objective BO, the results are mixed with traditional methods sometimes outperforming DeepHV methods and vice versa.", "summary_of_the_review": "Overall, the paper is well-written and proposes a novel in a relevant problem settings which provide good reasons to argue for acceptance. Nevertheless, I think the paper can still be improved by addressing current weaknesses, such as providing a more detailed description related to generalization and compute costs of DeepHV compared to other methods. ", "strength_and_weaknesses": "**Strengths**\n\n* The paper proposes a novel methods in a relevant challenge in multi-objective optimization and show promising results for making hypervolume calculations computationally feasible in large-scale search with high numbers of objectives and data points.\n* The paper proposes a new G-equivariant layer based on first principles and provides results supporting their method compared to current methods (HVNet).\n* The paper shows experiments with DeepHV used in modern multi-objective optimization methods.\n\n**Weaknesses**\n\n* The paper lacks an ablation on the equivariant layers to clearly show what benefits they provide. It might be possible that the HVNet performance provides that ablation in an indirect sense, so it would be good for the authors to clarify this.\n* The generalization claim of the authors related to DeepHV does not appear to well-tested. Right now it seems that DeepHV trains on data for the same problems that it is applied on and one could argue that it somewhat generalizes to the distribution of hypervolume in those design problems. It would be nice to explore this notion further as greater generalization would provide a stronger case for DeepHV.\n* SMS-EMOA with Monte-Carlo estimation still outperforms DeepHV in terms of task performance. It might be interesting to see if the compute cost of SMS-EMOA is higher compared to the DeepHV methods. That might showcase an interesting trade-off between the methods worth exploring.\n\n**Additional Questions**\n\n* Could you provide an intuitive explanation for the mathematics in Section 3.1. This would complement the current section and proof in the appendix (which would be good to reference in the main text).\n* What is the importance of generating data only non-dominated solutions? Wouldn't hypervolume calculations be equally valid with dominated solutions?\n* It would nice to see to potentially have a measure of compute cost on the x-axis instead of function evaluations. Assuming you can do that analysis, does it show that DeepHV consistently saves on compute? If so, in what cases?\n* Any intuition why training with multiple dimensions of objectives does not help the 3-objective case in Table 2?\n* What is the primary source of noise in the multi-objective BO settings?", "limitations_and_societal_impact": "", "review_text": "", "extra_scores": {"correctness": "4: All of the claims and statements are well-supported and correct.", "technical_novelty_and_significance": "3: The contributions are significant and somewhat new. Aspects of the contributions exist in prior work.", "empirical_novelty_and_significance": "3: The contributions are significant and somewhat new. Aspects of the contributions exist in prior work."}, "extra_text": {}}, {"review_id": "KmAk1m_lSM", "reviewer": "Reviewer_favf", "summary": "", "questions": "", "limitations": "", "rating": 6, "confidence": 4, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "main_review": "", "summary_of_the_paper": "This work proposes DeepHV, a deep neural network based model, to approximate hypervolume for multi-objective optimization. The idea of model-based hypervolume approximation has been proposed in the previous work (HVNet). This work's contribution is to design a more powerful equivariant layer to further exploit hypervolume's scale-equivariant and permutation invariant properties for better modeling. Experimental results show that DeepHV can better approximate the hypervolume, and can be well combined with EA and BO methods for multi-objective optimization. ", "summary_of_the_review": "This work is a timely contribution to an important research problem for multi-objective optimization. However, due to the concerns on novelty, unclear comparison with closely-related work, and experimental settings, I cannot vote to accept the current submission.\n\n", "strength_and_weaknesses": "**Strengths:**\n\n+ This paper is generally well-organized and easy to follow.\n\n+ Efficiently calculating the (approximate) hypervolume, especially for problems with many objectives (e.g., >= 5), is an important while challenging problem for multi-objective optimization. This work is a timely contribution to an important research topic for multi-objective optimization.\n\n+ The proposed model can achieve good performance for hypervolume approximation, and can be easily used in the current EA/BO methods for multi-objective optimization.\n\n**Weaknesses:**\n\n**1. Novelty and Contribution** \n\nThe original idea of model-based hypervolume approximation, problem formulation (e.g., problem definition and training data generation), and model framework are already proposed in the previous work HV-Net [1]. This work's main contribution is the new equivariant layer for better approximation performance, which is a natural extension from the equivariant layer proposed in [2]. Although it is glad to see the proposed model can achieve better performance (as listed in the strengths), the contribution is not very novel given the previous works.\n\nIn addition, the close relation to the previous work HV-Net should be clearly discussed and emphasized early in this paper. I did believe this work has proposed the idea of model-based hypervolume approximation until the end of page 2. The claimed contribution in the abstract, \"To overcome these restrictions, we propose to\napproximate the hypervolume function with a deep neural network, which we call DeepHV\", is also a bit misleading. \n\n**2. Unclear Comparison with HV-Net**\n\nSince the main contribution is the improvement over HV-Net, their difference should be carefully discussed in detail, which is missing in the current paper.\n\n*a) Symmetry Properties* \n\nFor example, this work claims DeepHV can \"leverage additional symmetry properties of hypervolume compared to HV-Net\". It seems that both DeepHV and HV-Net have the permutation invariant property, while DeepHV additionally has the scale-equivariance of the objective values. Do the additional symmetry properties mean the scale-equivariance to objective values? Why and how this additional property can lead to better Hypervolume approximation? \n\nIn addition, it could be beneficial to have an ablation study and detailed analysis for the layer design (21) to clearly show the effect of a) no permutation invariance, no scale equivariance; b) only permutation invariance, but no scale equivariance; c) no permutation invariance, but only scale equivariance; d) both permutation invarince, and scale equivariance for hypervolume approximation. Is HV-Net corresponding to the case of b), and the proposed DeepHV is the case of d)?\n\nCould a simple normalization of the objective values (e.g., all to [0,1]) significantly improve the approximation performance of HV-Net? It seems that the related order of HV could be more important than the approximation accuracy for optimization. \n\n*b) Comparision Setting*\n\nSome comparison settings with HV-Net might need further clarification.\n\nThe DeepHV model is both trained and tested with the MAPE loss, while the HV-Net is trained with the log MSE loss and tested on the MAPE loss. Will it lead to unfair comparison?\n\nFor DeepHV model, the training data is split into train-validation-test sets, so the hyperparameters and settings can be carefully fine-tuned on the validation set. It seems that the HV-Net is directly trained on the whole dataset without fine-tune. Will it lead to unfair comparison?\n\n**3. Test Problems and Reported Results**\n\nIt is glad to see DeepHV can have promising performance with both EA and BO approaches to solve multi-objective optimization problems. However, the current reported results are not solid enough to truly show DeepHV's advantages.\n\nThis work uses the DTLZ benchmarks for most experiments, and only one real-world problem is reported at the very end of the Appendix (e.g., page 18). The DTLZ benchmarks have a regular shape of PS, which might be very different from real-world problems. It could be much better to report the results on the real-world problems suite in [3] with 2-9 objectives, which is now widely used in both the EA and BO community.\n\nWhy only the results of DTLZ 1,2,5,7 are reported? How about DTLZ 3, 4, 6?\n\nFor all experiments, please report the value of the log hypervolume difference (e.g., as in the qEHVI paper [4]) rather than the hypervolume value. The found Pareto set with \"similar\" hypervolume value could perform very differently and have quite different values of log hypervolume difference. \n\n**4. Experiments with EA**\n\nFor all EA experiments, only 1,000 function evaluation is far from the ideal setting. In the standard EA setting, all algorithms should be run with a large number of evaluations (e.g., 100,000) such that the found solutions can converge to the true Pareto front. Will the approximation nature of DeepHV make it hard to find the true Pareto solutions, and lead to poor performance in the standard EA setting?\n\nFor the baseline algorithm, it is well-known that the domination-based algorithm such as NSGA-II will perform poorly on problems with more than 3 objectives. Outperforming NSGA-II is not enough to show the advantage of DeepHV for many-objective optimization. NSGA-III [5] could be a much more suitable baseline for comparison (which is also available in Pymoo).\n\nIt is also interesting to know the wall-clock runtime of these algorithms for problems with different numbers of objectives.\n\n**5. Experiments with BO**\n\nFor all BO experiments, the important results of UCB-Exact HV and UCB-MC HV are missing. Will UCB-DeepHV have similar performances with these baselines?  \n\nOn the other hand, since UCB-DeepHV with different model sizes perform similarly to each other, will HV-Net also have similar BO performance?  \n\nIt is interesting to know the performance of qEHVI with UCB acquisition function as in UCB-DeepHV.\n\n**6. The Opening Story**\n\nI personally enjoy reading the opening story in the introduction for multi-objective optimization. However, reading this paper while driving could be dangerous and indeed against the law in many countries. It is better to replace it with a harmless story.\n\n**Other Comments**\n\n1. In Figure 2, it is interesting to know the computation time on GPU for DeepHV and the exact hypervolume calculation in BoTorch.\n\n2. If the goal of hypervolume approximation is to be used with EA or BO for optimization, it seems directly approximating the hypervolume contribution (HVC) could be a more reasonable choice. There is a concurrent work on HVC approximation (HVC-Net) [6], where a comparison is not needed, but a brief discussion could be beneficial.\n\n[1] HV-Net: Hypervolume Approximation based on DeepSets. IEEE Transactions on Evolutionary Computation 2022.\n\n[2] Deep models of interactions across sets. ICML 2018.\n\n[3] An easy-to-use real-world multi-objective optimization problem suite. Applied Soft Computing 2020.\n\n[4] Differentiable expected hypervolume improvement for parallel multi-objective Bayesian optimization. NeurIPS 2020.\n\n[5] An evolutionary many-objective optimization algorithm using reference-point-based nondominated sorting approach, part I: solving problems with box constraints. IEEE Transactions on Evolutionary Computation 2014.\n\n[6] HVC-Net: Deep Learning Based Hypervolume Contribution Approximation. PPSN 2022.\n", "limitations_and_societal_impact": "", "review_text": "", "extra_scores": {"correctness": "3: Some of the paper’s claims have minor issues. A few statements are not well-supported, or require small changes to be made correct.", "technical_novelty_and_significance": "3: The contributions are significant and somewhat new. Aspects of the contributions exist in prior work.", "empirical_novelty_and_significance": "3: The contributions are significant and somewhat new. Aspects of the contributions exist in prior work."}, "extra_text": {}}, {"review_id": "IaRlbejvsK", "reviewer": "Reviewer_cbGc", "summary": "", "questions": "", "limitations": "", "rating": 6, "confidence": 3, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "main_review": "", "summary_of_the_paper": "        This paper proposes a deep neural network to approximate the computation of the hyper-volume, which becomes expensive for a large number of objectives or points. The network has a special architecture to take into account the particularities of the hyper-volume computations such as invariances to permutations. The model is compared with state-of-the-art approaches, showing lower prediction error. It also compares favorably to the exact computation and other approximations in the context of evolutionary algorithms and Bayesian optimization. The computation speed is better also for a larger number of objectives.\n", "summary_of_the_review": "Overall I think that this is an interesting paper that proposes a new method that will receive the attention of the community. The paper is well written and the experimental section is strong, with experiments that clearly show the benefits of the propose approach. My only concern is that only synthetic experiments seem to have been carried out.\n\nMinor:\n\n        There are other methods that for multi-objective BO that do not use the hyper-volume directly, but that need to\n        solve a multi-objective problem. For example, see:\n\n        Predictive entropy search for multi-objective Bayesian optimization\n        D Hernández-Lobato, J Hernandez-Lobato, A Shah, R Adams\n        International conference on machine learning, 1492-1501\n", "strength_and_weaknesses": "Strengths:\n\n        - Well written paper.\n\n        - Original architecture proposed to compute the hyper-volume.\n\n        - Extensive experiments evaluating performance and computational time.\n\nWeaknesses:\n\n        - It seems only synthetic problems are considered in the experiments.\n\n        - The authors claim in the discussion that their approach outperforms MC approximations in terms of performance. However, the experiments only seem to indicate that the proposed approach is faster. MC approximations seem to perform better.\n\n        - No error curves are given in Table 1 and 2. However, since the test set is large. E.g. 100k points it may not be a problem.\n\n", "limitations_and_societal_impact": "", "review_text": "", "extra_scores": {"correctness": "4: All of the claims and statements are well-supported and correct.", "technical_novelty_and_significance": "3: The contributions are significant and somewhat new. Aspects of the contributions exist in prior work.", "empirical_novelty_and_significance": "3: The contributions are significant and somewhat new. Aspects of the contributions exist in prior work."}, "extra_text": {}}]

Author rebuttal:

Decision: Accept: poster