A Two-Stage Neural-Filter Pareto Front Extractor and the need for Benchmarking
Authors: Soumyajit Gupta
Gurpreet Singh
Matthew Lease
Venue: ICLR 2022 Submitted (2022)
Keywords: Pareto Optimality
Neural nets
Pareto Filter
Interpretability
Benchmarking

Abstract: Pareto solutions are optimal trade-offs between multiple competing objectives over the feasible set satisfying imposed constraints. Fixed-point iterative strategies do not always converge and might only return one solution point per run. Consequently, multiple runs of a scalarization problem are required to retrieve a Pareto front, where all instances converge. Recently proposed Multi-Task Learning (MTL) solvers claim to achieve Pareto solutions combining Linear Scalarization and domain decomposition. We demonstrate key shortcomings of MTL solvers, that limit their usability for real-world applications. Issues include  unjustified convexity assumptions on practical problems, incomplete and often wrong inferences on datasets that violate Pareto definition, and lack of proper benchmarking and verification. We propose a two stage Pareto framework: Hybrid Neural Pareto Front (HNPF) that is accurate and handles non-convex functions and constraints. The Stage-1 neural network efficiently extracts the \textit{weak} Pareto front, using Fritz-John Conditions (FJC) as the discriminator, with no assumptions of convexity on the objectives or constraints. An FJC guided diffusive manifold is used to bound the error between the true and the Stage-1 extracted \textit{weak} Pareto front. The Stage-2, low-cost Pareto filter then extracts the strong Pareto subset from this weak front.  Numerical experiments demonstrates the accuracy and efficiency of our approach.

Under review as a conference paper at ICLR 2022
A TWO-STAGE NEURAL-FILTER PARETO FRONT EXTRACTOR AND THE NEED FOR BENCHMARKING
Anonymous authors Paper under double-bling review
ABSTRACT
Pareto solutions are optimal trade-offs between multiple competing objectives over the feasible set satisfying imposed constraints. Fixed-point iterative strate- gies do not always converge and might only return one solution point per run. Consequently, multiple runs of a scalarization problem are required to retrieve a Pareto front, where all instances converge. Recently proposed Multi-TTask Learning (MTL) solvers claim to achieve Pareto solutions combining Linear Scalarization and domain decomposition. We demonstrate key shorteomings of MTL solvers, that limit their usability for real-world applications. Issues include unjustified con- vexity assumptions on practical problems, incomplete and often wrong inferences on datasets that violate Pareto definition, and lack of proper benchmarking and verification. We propose two stage Pareto framework: Hybrid Neural Pareto Front (HNPF) that is accurate and handles non-convex functions and constraints. The Stage-1 neural network efficiently extracts the weak Pareto front, using Fritz-John Conditions (FJC) as the discriminator, with no assumptions of convexity on the objectives or constraints. error between the true and the Stage-1 I extracted weak Pareto front. The Stage-2, low-cost Pareto filter then extracts the strong Pareto subset from this weak front. Numerical experiments demonstrates the accuracy and efficiency of our approach.
FJC guided diffusive manifold is used to bound the
INTRODUCTION
Multi-Objective Optimization (MOO) problems arise frequently across diverse fields such as engi- neering (Marler Arora 2004), finance (Tapia & Coello, 2007), and supply chain management (Trisna et al. 2016). Such problems share common requirement to satisfy multiple competing objectives under a set of constraints imposed by physical or economic limits. A Pareto optimal solution Pareto (1966) for an MOO problem is defined as the solution point away from which no single objective can be improved without diminishing at least one other objective. A Pareto front is then defined the set of all such optimal points that satisfy this definition.
Since practical MOO problems contain competing objectives and additional domain specific con- straints, Pareto solver should be able handle both functions and constraints. Moreover, problems in classification and recommendation are non-convex (Hsieh et al. 2013), necessitating solvers that are robust for non-convex problems. Although many saddle point solvers have been proposed in the literature (Benzi et al. 2005 Benzi & Wathen 2008), they cannot be generalized to handle constraints without specialized pre-conditioners designed to handle non-linear constraints. These existing solvers are based upon fixed-point iterations, returning one solution per run with specialized local initialization generate an even spread of Pareto points across the feasible set of saddle points.
Recently proposed Multi-Tass Learning (MTL) solvers (Sener & Koltun 2018: Lin et al. 2019a
Mahapatra Rajan 2020 Ma al.) 2020 Navon al. 2021) criticize Linear Scalarization (LS) since it retrieves a subset of the Pareto optimal set. MTI. solvers claim to resolve this issue by decomposing the functional/variable domain (into cones or rays) while using LS in each the sub- domains. However, the stationary points of the LS problem (now decomposed) remain unchanged (see Appendix B). and therefore any fixed-point iterative strategy will still retrieve only a subset of the Pareto optimal set. Additionally, the theorems presented in MTL works rely upon convexity assumption that are never justified through numerical experiments. Furthermore, a notable absence of benchmarks against known analytical forms makes it difficult assess reported results, verify

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optimality, and A/B test alternative methods. This is in contrast to studies in Operations Research (OR) (Das & Dennis 1998; Ghane-Kanath & Khorram 2013: Pirouz& & Khorram 2016) in which such comparative benchmarking and verification is well established. Although accurate and verifiable, existing OR methods tend to generate Pareto points with low density and limited scalability, with compute times ranging from hours to days as variable dimensionality increases.
While lack of formal guarantees for non-convex cases certainly does not preclude use of MTL methods for such problems, rigorous numerical evaluation becomes particularly important. However, the true Pareto front (closed-form analytical solution) is often unknown for real MOO problems, hence it is challenging to compare the accuracy of a Pareto solver on such problems. We follow the OR literature in advocating that the correctness of any proposed Pareto solver should first be tested on constructed benchmark problems with known analytic solutions. This is also consistent with broader MI. community practice of first evaluating proposed methods in controlled synthetic or simulated conditions to verify correctness, before applying these systems on real datasets. In general, rigorously evaluating new method provide valuable insights prior to venturing into the unknown and builds user trust in the model.
diverse set of complicated benchmarking scenarios can
Table HNPF vs. existing OR and MTL methods, OR methods are accurate and can handle constraints, with limited scalability for high-dimensional Neural MOO problems. MTL approaches scale well but do not support constraints or provide benchmark comparisons against known analytic solutions for non-conves problems.
Method
Pareto points only
Even Spread
Handle Constraints
Neural Scalable
Manifold Solution
mCHIM Kilume-Kanaf Kherram 2015
PK Fireuz Khonan 2010
NET Das Dennis 1998
HNPF
/
/
/
x
/
MOOMTI
Sener & Keltun 2018
X
X
X
X
PMTL en tie y0
EPO
Malappana
PLIN Navva ai 20-1
We propose two-stage framework: Hybrid Neural Pareto Front (HNPF) for extracting Pareto optimal solution sets. Stage 1 consists of an interpretable neural network that extracts weak Pareto solution manifold as the output, given dataset as input. Following this, Stage is a low-cost Pareto filter to remove dominated points from the weak Pareto set. The network loss function uses discriminator based on Fritz-John Conditions (FJC) to account for multiple objectives and constraints. An approximate weak Pareto manifold is extracted as weighted output of the softmax function from the last layer of the network. The softmax activation classifies weak Pareto vs. non-Pareto data points. HNPF extracts this weak Pareto front continuous manifold approximated by the Stage-1 neural network. Numerical experiments show the computational efficiency of HNPF vs. OR approaches and accuracy comparisons against MTL methods. HNPF produces only Pareto points (no false positives) with an even spread and high density. Furthermore. HNPF is scalable (compared to OR methods) with both increasing dimensions of the variable domain, and the number of functions and constraints. Table I summarizes key properties of HNPF vs. existing methods. Our key contributions are:
A neural manifold extraction strategy for weak Pareto front identification based on Fritz-John conditions as the discriminator, for both convex and non-convex scenarios, supporting constraints.
The final neural net layer interpretable as a continuous approximation of the weak Pareto mani- fold. The extracted manifold error is bounded below w.r.t. the true manifold upon convergence.
The necessity of Pareto filter to remove dominated points from the weak Pareto set. Design of a computationally efficient Pareto filter to extract the non-dominated Pareto optimal set.
4. Raising community awareness about existence of benchmarks from literature in OR field.
PROBLEM STATEMENT
general MOO problem can formulated as:
optimize F(x) =
(1)
s.t. I E = (F))))
M (x) S 0}
in n variables (xI,..., ..)). l objective functions (f1...., fk). and m constraint functions (g...... Im.) Here, H. is the feasible set i.e. the set of input values x that satisfy the constraints G(z).
For a MOO problem there is typically no single global solution, and it is often necessary to determine set of points that all fit predetermined definition for an optimum. See Appendix A for definitions.
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2.1 FRITZ JOHN CONDITIONS (FIC)
Let the objective and constraint function in Eq. be differentiable once at decision vector 2* H. The Fritz-John necessary conditions (FJC) for x to be weak Pareto optimal is that vectors must exists for 0 S X
R 0 S 1 E Ih and (A, p)) A (0, 0) (not identically zero) s.f. the following holds:
m
A V
Σ My V = O, ووg = 0, Vj = I,...,
j=1
Gobbi et al. 2013 presented an L matrix form of FJC as follows:
F
VG
L
[nn + m) x (k + m)]
(3)
0
G
Fnnkk= = Vf......
Vfl, VGnxm = Vig..... Vgnl,
Gmxm = diag(li....... ,mm)
In L matrix, for x" to be Pareto optimal, is to show the existence of 8 E R in Eq.
2 such that
P s.t. L( 0,8 -,
(4)
The non-trivial solution (6 not identically zero) for Eq. 4 is:
del(L.") L)
(5)
Remark. Note that (L) = 0 is equivalent to det T L.) (see Appendix G for derivation).
The weak Pareto front is characterized by the set of points such that matrix L is low rank. This ensures that points identified are either inside the feasible set or at boundaries dictated by the constraints. For e.g. if #4 = 0 for any A, then E; A.fi 0 must be satisfied for the corresponding internal point x to be Pareto. Similarly if j1 7 0, 1ight = 0 in the aforementioned case, then 91 = 0 holds true for the corresponding boundary point x= to be Pareto. All Pareto points satisfy V 0 for least one i whether they lie inside the feasible set or on the boundary, i.e. all points x" need to be local optimizers (stationary points for least one ft. The rank of the matrix L determines the dimension of the Pareto manifold. FJC written as det(L" L) is independent of the preference parameters X; M.j- Thus, Eq. 5 an oracle serving as discriminator in HNPF to identify a weak Pareto front.
Remark. For malti-objectivve optimization (MOO) problem scalarization results in single- objective optimization (SOO) problem. However, the stationary points of this resulting SOO problem can only be small subset of the weak Pareto solution set. The reader is referred to Appendix Bor a visual explanation of stationarity w.r.f. an SOO problem arising from an MOO problem.
RELATED WORKS
Generic and Enhanced Scalarization: One common approach is to convert an MOO problem
into a Single Objective Optimization (SOO) problem via Linear Scalarization (LSS) These include: Balashankar et al. 2019); Lin et al. (2019)): Martinez et al. (2020); Valdivia et al. (2021) Wei & Niethammer (2020). Enhanced scalarization approaches (Das & Dennis 1998; Ghane-Kann
& Khorram 2015 Pirouz& & Khorram 2016) rely upon localization of the objective space to handle non- convex functions and constraints. Although accurate and complete, these approaches suffer from low computational scalability and low density of Pareto points on the solution manifold. For example, the 30 dimensional benchmark (Case V in Section 5) shows enhanced scalarization methods (mCHIM and PK) generating a Pareto set in approximately 18 hours.
Multi Task Learning: MTL methods rely upon a utopia/ideal point (similar to Das & Dennis (1998) but shown to be scalable for high-dimensional MOO problems. MOOMTL (Sener & Koltun 2018) uses a multi-gradient descent approach, but does not guarantee an even spread of solution points along the Pareto front. PMTL (Lin al., 2019a) attempts address this issue by dividing the functional domain into equal spaced cones, with increasing computational cost as the number of cones increase. EPO Mahapatra & Rajan 2020) uses preference rays along specified weights find Pareto points in the vicinity of the rays. EPSE [Ma et al. 2020) uses a combination Hessian of the functions and Krylov subspace to find Pareto solutions. PHN (Navon et al. 202T) is initialized either
by LS or EPO and is the first neural framework that can approximate the front as neural manifold.
A stationary point w.E.t gradient descent fixed point iteration is the set of points corresponding to the local glohal minima, maxima saddle points of objective function.
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The reader is further referred to Appendix 0 for detailed review of these methods including other from the Bayesian, Genetic and Fairness literature.
Current Limitations: Firstly, these aforementioned point-based approaches extract solution point at a time, given the optimization problem converges. However, for practical applications, with non-convex objectives and constraints, ensuring optimality is non-trivial. Secondly, multiple runs with different trade-off parameters must be performed in order to extract the weak Pareto solution set, resulting in substantial computational overhead. Finally, the Pareto solution set can still form a non-convex manifold even when the objectives are convex due to the presence of non-convex constraints (Case III in Section These challenges prove to be major obstacles in the deployment of MOO solution methods as practical tools for Pareto set extraction.
4 HNPF FRAMEWORK
We draw inspiration from three seminal works: 1) Das & Dennis (998) proposed break the functional domain boundary into uniform and evenly spaced segments to identify weak Pareto points with guarantees. Motivated by this, we first identify the weak Pareto front using a neural network (Stage 1). 2) Messac et al. (2003) proposed the first Pareto filter to obtain the set of strong Pareto points from the aforementioned weak Pareto set. The filter uses an all-pair comparison criterion to reject dominated points from the weak Pareto set. This motivates our low-cost Pareto filter (Stage 2) using a plane search strategy to avoid an expensive all-pair comparison. 3) Gobbi et al. (015)) presented matrix form of the Fritz-John conditions satisfying the existence of weak Pareto points.
tanh
tanh
tarth
4.1 STAGE 1: NEURAL NET FOR WEAK PARETO FRONT
Given FJC, one can brute force classify the data points as weak Pareto or not. However,
intent is to extract an approximate Pareto front as indicator function (M E I)) parameter- ized by the neural network. The Stage-1 neural network (Fig. [1 smoothly approximates the true weak Pareto manifold (X*)
Las Binney Croas Esaeoge (de (10)))(()))....... Mix)
Figure 1: HNPF Stage-1 1 with FJC guided binary cross entropy loss to extract weak Pareto front.
as M(X) (see Appendix F for details on notation). The last layer has two neurons with softmax activation for binary clas- sification of Pareto vs. non-Pareto points, distinguishes weak Pareto vs. sub-optimal points, in the feasible set H. Our network loss is representation driven, since the FJC guided discriminator (Eq. 5 described by the objectives and constraints, explicitly classifies the input data points X points.
The Pareto front is a lower dimensional manifold (< (n - 1)) in an ne-dimnnsional variable domain. The Pareto vs. non-Pareto classes are not comparable since the measure of the Pareto front is zero. In other words, the volume of surface (or anything lower) is zero. To volumize the Pareto front and to make a Binary Cross Entropy measure computable, we use diffusive indicator function. The volume can then be made tending to zero by choosing a smaller approximating the lower dimensional weak-Pareto manifold. We use a slightly relaxed criterion with user-tunable threshold 0 OSc < 1 as the classification margin. Any point below this value will be classified weak Pareto. The FJC for weak Pareto optimality then requires that the D = det( L) = O. Therefore, |D] < e (True or False) naturally provides us with binary labels for the softmax activated output layer. A binary eross entropy loss ensures that the distribution of the extracted manifold (X) matches the distribution of the weak Pareto front satisfying FJC within the specified e tolerance. This is similar to recently popularized Physics Informed Neural Networks (PINNs) (Raissi al. 2019) where known analytical form of the regression function is supplied as loss term, instead of explicit labels.
Error bound and Network Interpretability. For user-preseribed relaxation margin 0 S S the approximation error between the network extracted manifold M (X) and the true solution MX*) is bounded below by (X) - (X")ll2 I c, upon convergence. See Appendix F for proof and Appendix H for a discussion on interpretability as a mathematical model.
Remark. For a n-dimensional variable domain, HNPF (Stage 1) uses the Fritz-Johns conditions in the determinant form to extract n-dimensional, diffusive, indicator function using binary cross entropy loss (weak Pareto vs. non- -an-Preeo)) The indicator function regresses to I at the weak Pareto front and 0 everywhere else with values between O and I in the neighborhood of this weak Pareto front (diffusive indicator). In this respect, Fritz-Johns condition is an oracle to gather the weak Pareto set where the determinant form is used instead of user provided labels identify the front.
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4.2 STAGE 2: PARETO FILTER FOR STRONG PARETO SET
A Pareto filter is algorithm that, given a set of weak Pareto points P of cardinality |P| = P in objective space, retains a subset of non non-dominated points. This corresponds to the strong Pareto set s.l. none of the points are dominated, i.e. the filter eliminates all dominated points from P. The state of the art Pareto filter (Messac et al. 2003) requires an all-pairs comparison (O(P!) time complexity), that becomes computationally expensive as the set P grows in size. However, since Stage-1 of HNPF generates weak Pareto points with high density, the filter proves to be quite expensive.
Remark. A fixed point iterative approach might converge to the stationary points (local criterion) of an MOO problem. However, it is not necessary that all such stationary points are non-dominated A filter is therefore necessary to remove the dominated points based upon a global criterion. See Section 5.7 and 5.4 for numerical experiments where filter is necessary.
We present an efficient Pareto filter for finding the strong Pareto set which is computationally scalable
1: Data P = (2') E RR weak Pareto points
2:
Imput f:(min), f: fi(mar);; Vi c
Input k #functions, h discretization level
to arbitrary dimensions. The algorithm is based on a plane search strategy, inspired by Kd-Trees (Benteyy 1990), well known for efficient data partitioning and storage. The compute cost is guided by the number of functions while being linearly proportional to the number of points. The inputs to the algorithm (Fig.
4:
for k do
5: level = f.(min)
for J E (fi(maz) fan--f(mmsiii fimmii)))/h)) do
temp
8:
for p P do if v f((p) < level + then temp temp Up
9:
10:
2 are the number of functions lt and their respec-
1I:
if (tmm > then
12:
Zy min fa(z), temp, i+1
tive global minima and maxima. These are known a priori for benchmark problems, or can be computed from the data itself in (kP)) time i.e. a linear search over the k functions for P points. fmin), f (maz), discretization level h of the function space and the weak Pareto points P. The output is the strong Pareto set {a") 3. Refer to Appendix NN for algorithmic details and usage illustration of the filter.
13:
P Pl (templ)7))
14:
level +evel+ h
15:
Output: Non-dominuted Pureto P (a")
Figure 2: Pareto Filter Algorithm
Time complexity: The three nested for loops carry the load. In the worst case that the points in the weak Pareto set are all strong Pareto, then the cardinality P of the set P remains unchanged. Let z = (fi(maz)= - fmmin)))/ / denote the number of chunks into which the function space is divided. The worst case complexity of the proposed Pareto filter is (kzP). For scenarios, where the strong Pareto set is subset of the original weak set P. the complexity reduces in the factor guided by P.
4.3 COMPARISON BETWEEN HNPF AND PHN
HNPF (Stage-1) 1) extracts the weak Pareto manifold as an n-dimensional diffusive indicator function as opposed to a (n 1) dimeennsia manifold itself. This results in an indicator function M R" - R that with an n-dimensional support in the variable space. There are two explicit advantages in describing this indicator function: (1) The regressed Pareto manifold is not only guided by the weak Pareto points (indicator value 1) but also the sub-optimal points (indicator value 0) for more robust and accurate extraction. This is comparable to classification under a balanced class density wherein the class densities were increasingly imbalanced, the extracted classification boundary becomes more inaccurate. (2) The Pareto optimal set can be extracted and represented by neural network (function approximator) even when the manifold an implicit surface. For eg., consider representing a circle (equation as opposed to function) as network extracted manifold in a 2D variable domain.
In comparison, PHN Navon et al. 202 uses a neural network to regress over the (n -i-mennsioa manifold in the variable domain directly, given solution points obtained from EPO or L.S. However, i) neither EPO nor LS are guaranteed to work for non-convex scenarios (see Section : ii) the non- dominated Pareto set is discontinuous, then PHN will wrongly identify non-Pareto points as Pareto optimal; iii) even when the Pareto optimal set is a continuous manifold, PHN inherently assumes that the point spread obtained from EPO or LS is uniformly distributed over the true manifold for the neural regressor to not overfit; and iv) if the manifold is an implicit surface then a direct regression using a neural network is not feasible. Further numerical justifications can be found in Appendix D
5 RESULTS
In this section, we present four numerical experiments for benchmarking and analysis, with five more in [ppendix These address analytical forms, with increasing complexity and scale in the number
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of functions (k), constraints (m) and variable dimension (n). The benchmark cases are chosen to test various aspects desired from a general Pareto solver. We compare HNPF vs. two OR methods: mCHIM and PK and four recent MTL methods: MOOMTL, PMTL, EPO, and PHN. For HNPF's setup and error tolerance specification, refer to Appendix J
Additional experiments are presented as an ablation study (Appendix K) with loss profiles (Appendix and runtimes (Appendix
Data Sampling: Since the location of the Pareto front is not know a priori, samples are drawn uniformly random from feasible variable domain. Objective functions evaluated at these points generate a quantized, topographic map of the function domain that is then used to identify optimal points. For each benchmark test case below, we generate 11k points from random uniform distribution in the feasible variable domain, to serve as training data. The training-validation validation split is 90-10%. Once the manifold M (X) is approximated by the network, we use 90k points in the feasible domain as test data to visualize the Pareto set.
Remark. Although MTL methods do not explicitly handle constraints, the Pareto solution set is only reported for the benchmark cases (I, II, IV) where the linear construints form a bounding box.
5.1 CHALLENGES IN PARETO OPTIMALITY & THE NEED FOR A PARETO FILTER (n = 2, 1,m = 0)
We now consider a two objective, unconstrained, MOO problem in single variable domain where the two, non-convex objective functions are given by:
(1) (s 1) 2)(x - 3) (1(x - 4)
f2(x) - 1.5) 1- 2.5)(zz 3.6) - 4.5)
We design this benchmark MOO problem to specifically test out HNPF's robustness when multiple optimal manifolds exist for the corresponding SOO problem. The LS single-objective function is:
a() + (1-)
off a0660)
arra Shxp
Un pi 60
1.055 2.05 30
(a) LS Single Objective (b) HNPF weak Pareto (c) HNPF strong Pareto
Figure 3: Pareto front extraction for the triple optimal manifold benchmark problem.
Fig. (a) shows the surface of S(a, x)) in (a, x)) (blue mesh) with the three optimal manifolds (red curves). Note that for any point x there exists at least one a on one of the three manifolds. Therefore. the final Pareto set can be formed by a union of the subset of each of these three manifolds. Fig. 3
(b) shows that the optimal manifolds (red curve) overlaps the entire f1 vs. f22 plot (cyan curve) in the functional domain, as expected. The Pareto optimal set is then obtained by applying filter based upon the global criterion of non non-dominated points in the functional domain as shown in Fig. 3 (c).
Note that for this particular benchmark case, the Pareto optimal set is a union of the subset of two optimal manifolds in Fig. 3 (a). All MTI. solvers fail in this case when the Pareto front in the functional domain is an implicit, self intersecting manifold. This numerical experiment also enunciates the importance of a Pareto filter to satisfy the global criterion non-dominated points inherent to the definition of Pareto optimality. A fixed point iteration converges locally by satisfying a local optimalityftermintiio criterion. The global criterion of non- con-dominated points can only be satisfied once the optimal set is gathered completely from the previous step.
Since the codes for mCHIM and PK are not publicly available, we snip results from their works to avoid any artifacts reproducibility. Hence the color discrepancy in plots.

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5.2 CASE I: CONVEX OBJECTIVES, LINEAR CONSTRAINTS (n = 2, = 2,
This problem was originally proposed in Fonseca & Fleming 1998). Jointly minimize:
(1(11 = 1 exp( (1((1 - 12) + (x2 - 1/v(2))))
22) = exp(-1(x1 1/ 1/V(2))2 + (x2 + 1/
s.t. 92 -1/V2 51, x1, S 11/V2
This is a common benchmarking problem in both OR and machine learning literature. Since the objectives and constraints are convex, HNPF (see Appendix [.1] OR and MTL methods all retrieve the Pareto optimal set accurately. The reader is referred to Appendix [ for detailed discussion and visual explanation of Pareto optimality using variant of this problem.
5.3 CASE II: NON-CONVEX OBJECTIVES, LINEAR CONSTRAINTS (n 2, k = 2,m = 2)
This problem was proposed in Ghane-Kanat & Khorram 2015).
Jointly minimize:
fi (T1, 02) 21, f2(21, 22) = x
21 0.1sin33T2
s.t. 91. 92 0 < 21 S 1, -2 22 <
10
0.5
0.5
04-
02-
02
Purata Front
0g
82
24
66
68
5.
0b:
(a) HNPF
(b) LS
(c) NBI
(d) mCHIM
1.5
10
0.5
D2-
02
0F:
0.4
0.0
0.2
02-
02-
000
0.2
0.4
00
D. 000
0.
200
(e) MOOMTL
(1) PMTL
(g) EPO
(h) PHI
Figure 4: Pareto Front for Case Note the even spread point HNPF produces. NBI produce an even spread while mCHIM cannot (uneven spread in red circle). PMTL and EPO produces sub-optimal points (blue circle).
Since PHN relies on EPO as a solver, it correspondligly fails in regions where EPO fails.
HNPF, LS. NBI, mCHIM, and MOOMTL methods identify Pareto points accurately while PMTL, EPO, and PHN also generate sub-optimal points (marked blue circle). Fig. 4 shows the results from our model with high point density. It also satisfies closely the true Pareto manifold MX*) given by
S 11 S
1, .2 in Fig. lb).
5.4 CASE III: CONVEX OBJECTIVES, NON- CON-CONVEX CONSTRAINTS (n 2, 2,
This problem was proposed in Tanaka al. (1995). Jointly minimize:
(21, 21, fa x2) I2
s.t. 9n (1), (4) 0.5)² 2 (x22 0.5) < 0.5
02) oं 2 0.1 cos arctan)
(11) 2
.22
93, 94 0 31, T2 S
Here, f1, are convex but the constraints 91, 92 are non-convex in the variable domain and conse- quently the resulting Pareto front is non-convex. An important point to note here is that the Pareto front is dominated solely by the two constraints gi and g2, while linear functions fi and f2 do not contribute. Since MTL solvers do not account for constraints, comparison is only shown against OR methods that report results for this benchmark. Here, mCHIM extracts a sparse set of dominntdd Pareto points (~ 40) while HNPF Stage-1 extracts dense set of weak Pareto points (Fig. 5). To arrive at the non-dominated Pareto set, we post-pree post-process this result using our Stage-2 Pareto filter (see The updated discontinuous set of non-dominated Pareto points is shown in Fig. (b).
Section 4 2
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1.0
10
0.4
0.4
0.2
fix. yl vs. 61F.
0.2
f1x, vs. NX.M
Poreto Front
0.0
06
(a) Dominated
(b) Non non-Dominattd
(c) NBI
(d) mCHIM
Figure 5: Pareto Front for Case III. (a) HNPF weak front in function space. (b) All dominated points
removed from the set after applying the Pareto filter.
5.5 CASE IV: NON-CONVEX OBJECTIVES, LINEAR CONSTRAINTS (n 30, 2, 30)
This problem was proposed in Zhang 2008). Jointly minimize:
2
fi(z) = +
yj
2 f2(2) = 1 vx1 +
[11
J22
JEJ1
JEJ2
91..... 930 S 21 5 1,
S 1, 2,... ......
J1 = {jl is odd, 2 S S m) = {iljis even, 2 و - m}
y1 Z [0.3x] cos (24xx] + - + 0.6z 1 coo (6TTZ] +
im
j J1 j J2
Ø3 [0.3xr cos (4 + 4 0.6z1 cos (6mrz + 3
This form is non-convex in both f1, f2. The dimension of the design variable space is 30. The Pareto front the variable domain is non-convex (sinusoidal spiral), implicit surface. HNPF results (Fig. 6 are in good agreement with mCHIM and PK methods, but with significantly higher density. See Appendix for compute time scalability of HNPF vs. OR methods. All MTL methods fail with the inherent min-norm solver returning NaN.
30
hls,y) vs. hix.y!
25
Pareto Front
0.5
2.0
220
1.5
x3
0.2
0.0
1.0
22
03
Frant
0.
(a) Function Space
(b) Variable Space
(c) mCHIM PK Variable
Figure 6: Pareto Front for Case IV. (a, b) HNPF front for Function and Variable space respectively.
Note the density difference between HNPF and mCHIM PK the variable space.
5.6 DISCUSSION OF METHOSS
OR methods: NBI works for cases where the detected weak Pareto front consists of non-dominated points only. Therefore, NBI generates correct solution in Cases I, II, IV and V with even density of points on the Pareto front. In essence, applying the Pareto filter on the NBI generated solution set would resolve the discontinuous cases too. NBI, mCHIM, PK and HNPF produce only Pareto points. Additionally, HNPF generates Pareto points uniformly with high density, while OR methods including mCHIM and PK, although accurate, limited to low point density (~ 40) with large compute overhead as the variable dimension scales. Table 2 shows comparison of the extracted Pareto point density (#extracted Pareto points/Hfunction evaluations) for HNPF, OR, and MTL methods.
Table Extracted Pareto point density #extaccedd Pareto points/#function evaluations). HNPF's Pareto point density is higher. NR - Not Reported, NS Not Supported, F Fails. Remaining Cases in Appendix The Pareto point density are reported w.E.f an error threhold of e 5e 4 in the neighborhood of the true front.
Case
HNPF
mCHIM
PK
MOOMTI.
PMTL
EPO
PHN
II
183
4.39c-2
1.00
0.05
0.05
0.05
III
1.38
1.00l-2
NR
NS
NS
NS
NS
V
6.57
5.86e-
NS
NS
NS
NS
VI
0.20
NR
1.32x-2
NS
NS
NS
V1I
6.57
NR
9.29c-5
F
F
E
8

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Under review as a conference paper at ICLR 2022
MTL Approaches: One of the main concerns regarding MTL solvers is that termination criterion associated with the optimization approach is not described. For any method to be called a solver termination criterion is necessary since the number of iterations to reach solution point is not known a-priori. We suggest the reader avail themselves of the open-source codes provided by the respective MTL authors wherein the solver uses for loop with a pre-specified number of iterations. Furthermore, it is not clear as to how MTL solvers, relying upon linear scalarized form (SOO) of the original MOO problem, arrive at the non-stationary points of this SOO. Even if the domain is partitioned into sub-domains (cones, rays etc.) the stationary points of the SOO remain unchanged. The reader is referred to Appendix for detailed discussion using a benchmark problem.
HNPF: Since HNPF Stage-1 approximates the true Pareto manifold (X*), it has the advantage of learning (X) from only 10k training points. No other enhanced scalarization method (mCHIM, PK or MTL) can approximate the Pareto manifold itself, rather detect Pareto points only: hypothet- ically having just the Fritz-John discriminator in Stage-1 without the neural network preceding it. Additionally, HNPF explicitly uses the Stage-2 filter necessary to remove dominated point from the Stage-1 weak Pareto solution set that cannot be addressed by an optimization problem.
Results on Multi-MNIST:: If HNPF or OR machineries are correct, then why not show results on practical datasets? This is valid question since the core objective of any approach is resolve practical problem. We are currently faced with the following obstacles: i) HNPF and OR methods are not yet neural scalable, hence cannot handle variable dimensions at orders 10 or higher; ii) MTL methods do not have a termination criteria and during numerical evaluation either produce non non-Pareto points fail completely for some of the benchmark non-convex cases. This raises serious concerns regarding their veracity on practical datasets where the ground truth Pareto solution is unknown; and iii) Framing Multi-MNIST as a Pareto optimality problem raises serious concerns from first principles, as to how the altered network architecture still abides by the Pareto framing. Readers are referred to Appendix E for a detailed discussion on this argument.
5.7 NUMERICAL CORRECTNESS OF HNPF
Table 3: Mcan Squared Error of HNPF extracted manifold AM w.r.f. the true analytical manifold M.
We numerically verify the proposed error bounds for the benchmark cases cosnidered. Since the analyti- cal form of the Pareto solution manifold M M(X*) is
Cases
11 I1
VI
vn
Brreet
x10"")
12
12
37
4.1
25
32
177
known, we can numerically verify the network approximated manifold M()) Table shows the Le error between true and approximated manifolds. Note that for all the experiments since = 5 10 the errors are bounded above by even when the functions and constraints are non-convex.
Since we are considering non-convex problems in an effort to edge closer to practical applications. providing theoretical guarantees is not possible without making debilitating convexity assumptions on the functions and constraints. MTL methods are supported by several theorems that hold true only for convex functions wherein numerical results are supplied for non-convex problems. Further, no numerical experiments are presented in MTL works that confirm their theoretical results. On the other hand, HNPF is a practical Pareto optimal set extractor with numerically verifiable error bounds without introducing impractical assumptions.
6 CONCLUSION AND FUTURE WORK
two-stage, neural-filter (HNPF) based optimization framework presented for extracting the Pareto optimal solution set for MOO constrained optimization problems. HNPF is computationally efficient and scales well with increasing dimensionality of design variable space, objective functions, and constraints w.r.t. OR methods. Results verifiable benchmark problems show that our Pareto solution set accuracy compares well against known solutions for a wide variety of benchmark problems. The benchmark problems are chosen test different aspects of our HNPF framework that might arise in practical MOO problems. The proposed neural architecture is interpretable with an FJC guided discriminator for weak Pareto manifold classification with an efficient Pareto filter to extract an optimal Pareto set. We also show that the approximation error between the true and extracted Pareto manifold can be easily verified for analytical solutions. In our future work, we will develop an FJC guided solver for a scalable HNPF algorithm to address high-dimensional neural MOO problems.

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Under review as a conference paper at ICLR 2022
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12


Reviews:
[{"review_id": "pVesHRdDaMW", "reviewer": "Reviewer_pPPS", "summary": "", "questions": "", "limitations": "", "rating": 3, "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": "**Strengths:**\n\n+ It studies how to find the whole Pareto set for multi-objective optimization problems, which could be useful for different applications.    \n\n+ The call for new benchmarks for multi-objective optimization other than multi-task learning is valuable, also see a related work [1] (I am not involved in this paper). However, this discussion is in the appendix and not the main point of this work. \n\n**Weakness:**\n\n*Method:*\n\n**1. Verification and Optimization:** \n\nThe proposed HNPF method is for verification (e.g., check whether a given solution x is Pareto optimal), but not for optimization (e.g., find an (approximate) Pareto solution x). It needs an extra search method, such as random sampling in this work, to first generate a large number of feasible solutions to cover the whole search space. Therefore, the underlying optimization is indeed random sampling (independent from HNPF), which could be extremely inefficient for a non-trivial search space. It is not suitable to put and compare the proposed HNPF method with other optimization methods that can directly find the (approximate) Pareto solution. \n\nSince HNPF depends on random sampling, it is not surprising that it can only work for small scale problems.\n\n**2. The Reason to Build the Model:**\n\nHNPF needs to first build a neural network to check whether a given solution x satisfies the Fritz-John Condition (FJC), which requires a large number of training samples (e.g., 11K for a two-dimensional problem). The learned model is mainly used to classify whether the extra randomly sampled solutions (e.g., 9K) are weak Pareto optimal or not. The reason for model building, such as the advantage over the simple FJC rule-based classification, is not well motivated and justified in this work.    \n\nThe proposed Pareto filter in stage 2 is not discussed and compared with other related nondominated sorting algorithms (e.g., [2]).\n\n**3. Necessary Condition for Pareto Optimality:**\n\nThe KKT[3] and FJC[4] are two types of first order necessary conditions for (local) Pareto efficiency (Pareto optimality). In my understanding, the multi-objective optimization based MTL algorithms mentioned in this work (Sener & Koltun, 2018; Lin et al., 2019a; Mahapatra & Rajan, 2020; Ma et al., 2020; Navon et al., 2021) mainly use the gradient-based multi-objective optimization methods (e.g., MGDA) [5-7], which is based on the KKT condition. For these methods, in each update step, the gradient can be written as a linear combination of the gradient for each objective with adaptive weights derived from the KKT condition. Therefore, they are all different from the simple linear scalarization with fixed weights. All the claims and analyses in this work that the previous works use simple linear scalarization is not correct.\n\nThe FJ condition is also for local Pareto convergence, similar to the KKT condition. The global convergence property is solely due to random sampling that only works for extremely low-dimensional problems. It is unfair to say the proposed algorithm can overcome the local convergence of other gradient-based methods. In addition, the proposed algorithm heavily depends on the Fritz-John condition, but the original work [4] is not cited.\n\n**4. Linear Scalarization and Convex Pareto Front:**\n\nIt is well-known that the simple linear scalarization cannot find the non-convex part of the Pareto front [8]. This finding leads to the seminal work on NBI scalarization (Das & Dennis.,1998), which is indeed one fundamental work that inspires the proposed method in this work (section 4, first sentence). The claim \"it is incorrect to state that LS itself fails if the Pareto front is non-convex\" (appendix, page 15) is questionable. \n\nSince the proposed HNPF can only verify whether a given solution is weak Pareto optimal, its ability to find the whole Pareto front totally depends on the extra sampling method (e.g., random sampling) to generate all the Pareto solutions (might be infinite). It is misleading to indicate the proposed HNPF method itself can find the whole Pareto front. In addition, since the Pareto set has measure zero and infinite cardinality, the random sampling + HNPF method can at most find a dense approximation to the Pareto set.     \n\n*Experiment:*\n\n**5. Algorithms for Comparison:**\n\nAll the multi-objective optimization based MTL algorithms are designed for optimizing a deep neural network with millions of parameters. They implicitly depend on the assumption that the deep neural network has good properties (e.g., no bad local optimum [9][10]) on its loss functions, which is consistent with other gradient-based single-objective optimization methods. They are not designed to find the global Pareto front for low-dimensional problems. \n\nFor low-dimensional problems, it is more suitable to compare with the model-free multi-objective optimization methods such as the multi-objective evolutionary algorithm [11,12] and multi-objective CMA-ES [13]. If the model building is allowed, Multi-Objective Bayesian Optimization (MOBO) algorithms can have a very good sampling efficiency for the low-dimensional problems [14,15]. It is also very common to conduct non-dominated filtering at the end of those model-free algorithms or MOBOs (e.g., only keeping the current non-dominated solutions). \n\n**6. Training + Sampling:**\n\nThe proposed method needs to first sample 11k solutions to train the neural network model, then randomly generate extra 9K solutions for filtering. Is there any advantage over simply using FJC to filter 9K (or 11k + 9K) randomly sampling solutions?\n\n**7. Figure from Other Works:**\n\nMany figures in the main paper and the appendix are directly borrowed from other works. I think this is not appropriate even the credits are given to the original works. \n\n**Reference:**\n\n[1] Ruchte, Michael, and Josif Grabocka. Multi-task problems are not multi-objective. arXiv preprint arXiv:2110.07301, 2021.\n\n[2] Roy, Proteek Chandan, Kalyanmoy Deb, and Md Monirul Islam. An efficient nondominated sorting algorithm for large number of fronts. IEEE transactions on cybernetics 49, no. 3: 859-869, 2018.\n\n[3] Kuhn, H. W., and A. W. Tucker. Nonlinear Programming. In Proceedings of the Second Berkeley Symposium on Mathematical Statistics and Probability, pp. 481-492. University of California Press, 1951.\n\n[4] Da Cunha, N. O., and E. Polak. Constrained minimization under vectorvalued criteria in finite dimensional spaces. Journal of Mathematical Analysis and Applications, 19(1), 103–124 ,1967.\n\n[5] Fliege, Jorg, and Benar Fux Svaiter. Steepest descent methods for multicriteria optimization. Mathematical methods of operations research 51, no. 3: 479-494, 2000.\n\n[6] Fliege, Jorg, and A. Ismael F. Vaz. A method for constrained multiobjective optimization based on SQP techniques. SIAM Journal on Optimization 26, no. 4: 2091-2119, 2016.\n\n[7] Desideri, Jean-Antoine. Multiple-gradient descent algorithm (MGDA) for multiobjective optimization. Comptes Rendus Mathematique 350, no. 5-6: 313-318, 2012.\n\n[8] Das, Indraneel, and John E. Dennis. A closer look at drawbacks of minimizing weighted sums of objectives for Pareto set generation in multicriteria optimization problems. Structural optimization 14, no. 1: 63-69, 1997.\n\n[9] Kawaguchi, Kenji. Deep learning without poor local minima. NeurIPS 2016.\n\n[10] Kawaguchi, Kenji, and Leslie Kaelbling. Elimination of all bad local minima in deep learning. AISTATS 2020.\n\n[11] Deb, Kalyanmoy, Amrit Pratap, Sameer Agarwal, and T. A. M. T. Meyarivan. A fast and elitist multiobjective genetic algorithm: NSGA-II. IEEE transactions on evolutionary computation 6, no. 2: 182-197, 2002.\n\n[12] Zhang, Qingfu, and Hui Li. \"MOEA/D: A multiobjective evolutionary algorithm based on decomposition.\" IEEE Transactions on evolutionary computation 11, no. 6: 712-731, 2007.\n\n[13] Igel, Christian, Nikolaus Hansen, and Stefan Roth. Covariance matrix adaptation for multi-objective optimization. Evolutionary computation 15, no. 1: 1-28, 2007.\n\n[14] Daulton, Samuel, Maximilian Balandat, and Eytan Bakshy. Differentiable Expected Hypervolume Improvement for Parallel Multi-Objective Bayesian Optimization. NeurIPS 2020.\n\n[15] Konakovic Lukovic, Mina, Yunsheng Tian, and Wojciech Matusik. Diversity-Guided Multi-Objective Bayesian Optimization With Batch Evaluations. NeurIPS 2020.\n\n", "summary_of_the_paper": "This work proposes a Hybrid Neural Pareto Front (HNPF) framework to solve multi-objective optimization problems. The proposed method needs to first generate a huge number of feasible solutions to cover the whole decision space (e.g., random sampling in this work), and then use a two-stage approach to select the Pareto optimal solutions. In the first stage, it builds a Fritz-John Condition (FJC) based neural network to verify the weak Pareto optimality for given solutions. In the second stage, it further filters the truly strong Pareto optimal subset from the obtained solutions. Experiments have been conducted on several low-dimensional problems to validate the proposed method's performance.\n", "summary_of_the_review": "This work aims at finding the Pareto front for a multi-objective optimization problem, which is important for many applications. However, there are many major concerns on both the method (verification rather than optimization, the reason for model building, questionable claims) and experiments (suitable comparison, figures from other works) that make this work unacceptable in the current form.\n\n============================= post rebuttal response ============================= \n\nI have read the other reviewers' comments with the authors' feedback, but still think the proposed verification method (FJC discriminator) should not be directly compared with other optimization methods (not limited to MTL). Therefore, I keep my initial score (3) and lean toward rejection.", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "extra_scores": {"correctness": "2: Several of the paper’s claims are incorrect or not well-supported.", "technical_novelty_and_significance": "2: The contributions are only marginally significant or novel.", "empirical_novelty_and_significance": "1: The contributions are neither significant nor novel."}, "extra_text": {}}, {"review_id": "lJ7Et-f_Rlt", "reviewer": "Reviewer_gpGn", "summary": "", "questions": "", "limitations": "", "rating": 3, "confidence": 5, "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": "Although they are claiming that their proposed algorithm solves the issues mentioned in the paper, I think the literature is not represented fairly to show novelty here. For instance, they mentioned that \"the theorems presented in MTL works rely upon convexity assumption that are never justified through numerical experiments.\" This claim is not true. Just for one case of [A], they explicitly discuss that they have no convexity assumptions. There are other proposals, mostly with multiple gradients descent algorithm (MGDA) variations that have convergence guarantees for both convex and non-convex objectives such as [B]. Besides, there is confusion regarding the convexity, whether they are discussing the convexity of objective functions or the Pareto frontier. Since even with convex objective functions, the Pareto frontier can be still non-convex.\n\nIn addition to the lack of fair representation of the literature to justify the claims, I have several other concerns regarding the approaches in this paper as follows:\n1. In most parts, they are using algorithms and proposals from other papers, hence it is not clear what is their own novelty added in this paper. For instance, the FJC for the optimality and Kd-tree algorithm seems to be borrowed from other papers. Hence, it is not clear, what is the novelty in this paper since these are the main parts of this proposal.  They use the optimality condition derived from FJC to use as a loss function for an MLP in order to decide whether a point is a Pareto or not.\n\n2. In addition, as they mentioned themselves, their proposal has some similarities with PHN [C]. In the case of PHN, the algorithm and training procedure is an end-to-end solution and very clear. However, in this paper, the training procedure is not clear at all. The main concern is that how they are getting the $X$ parameters in the first place to feed to their MLP model for prediction of Pareto and not Pareto. Are they using some existing methods like EPO to find $X$? The training procedure should be stated clearly. \n\n3. The filtering of Pareto points is not novel in my opinion. This is the procedure most of the proposed algorithms in this domain are doing one way or another. Since this is a search space over candidates for dominated points on the weak Pareto set is not that big, the computational complexity is not that big a concern to be claimed as the novelty here. See this [link](https://stackoverflow.com/a/40239615) for several solutions for this problem in the community.\n\nIn addition to the aforementioned concerns, one of the main concerns, I reflected upon several times, is regarding the clarity of the manuscript. I believe many parts of this paper require more clarification and in-depth discussion while some parts can be omitted or moved to the appendix. The distinction between the contribution of this paper and other proposals should be clear, and the contribution of the literature should addressed fairly and clearly.\n\n\n\n[A] Debabrata Mahapatra and Vaibhav Rajan. Multi-task learning with user preferences: Gradient descent with controlled ascent in pareto optimization. In International Conference on Machine Learning (ICML), pp. 6597–6607, 2020.\n\n[B] Kamani, Mohammad Mahdi, et al. \"Pareto Efficient Fairness in Supervised Learning: From Extraction to Tracing.\" arXiv preprint arXiv:2104.01634 (2021).\n\n[C] Aviv Navon, Aviv Shamsian, Gal Chechik, and Ethan Fetaya. Learning the pareto front with hypernetworks. In International Conference on Learning Representations (ICLR), 2021. URL https://openreview.net/forum?id=NjF772F4ZZR.", "summary_of_the_paper": "This paper intends to find points on the Pareto frontier of a multiobjective optimization problem. Despite other baseline methods for this problem, they claim that their proposed algorithm is suitable for non-convex optimization, and their solutions are evenly spread across the frontier. Also, their algorithm can handle constraints for the MOO problems similar to solutions proposed in existing operation research methods. Empirical evaluations show that the proposed algorithm can find Pareto points and omit non-dominated points in the final stage.", "summary_of_the_review": "Overall, I believe the novelty of the proposed algorithms is marginal and the clarity of the paper is very low at this stage. ", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "extra_scores": {"correctness": "2: Several of the paper’s claims are incorrect or not well-supported.", "technical_novelty_and_significance": "2: The contributions are only marginally significant or novel.", "empirical_novelty_and_significance": "1: The contributions are neither significant nor novel."}, "extra_text": {}}, {"review_id": "hWvdbOU3xo_", "reviewer": "Reviewer_GeVa", "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": "Pros:\n- I resonate with the call for analytical benchmarks. Multi-objective optimization as a field may be growing within ML, but I feel many work are not evaluated in a way that is conducive to a holistic understanding of the methods capabilities and limitations. Using these benchmarks, the paper establishes clear cases where recent method fails. \n- The paper helps bridge some of the gap between multi-objective work in ML and Operations Research.  I think an ML audience will learn much from the paper, just like I did. \n- The proposed two-stage method is reasonable, and shows good performance on the proposed benchmarks.\n\nCons: \n- The proposed two-stage method is a somewhat orthogonal contribution, and I am not sure if it really connects well with the benchmark contribution. \n- I wonder if the neural net for weak Pareto front will be practical in applications where it is difficult to compute the value in objective space. For example, if we were to optimize the hyperparameters of deep neural nets like [1] and [2], it may be expensive to obtain samples for this neural net. Some discussion would be good.\n- Pareto density (Table 2) is not the only way to evaluate, and it does penalize methods like MTL. There are other evaluation metrics, like hyper volume, which may result in a different ranking of methods. Some discussion would be desired. \n\n[1] Zhang & Duh. Reproducible and Efficient Benchmarks for Hyperparameter Optimization of NMT Systems\n. TACL2020 https://aclanthology.org/2020.tacl-1.26/\n[2] Klein & Hutter. Tabular benchmarks for joint architecture and hyper-parameter optimization. https://arxiv.org/abs/1905.04970\n\nAdditional comments: \n- It will be great if the authors release the data or generator code (and evaluation code) for the benchmarks in an easy-to-use fashion. I understand it is relatively easy to implement, but it can help encourage more people to work on the benchmark.", "summary_of_the_paper": "This paper has two contributions. First it argues for the need to benchmark multi-objective problems  using analytic functions for convex and non-convex conditions. Second, it presents a two-stage method for finding the Pareto front. ", "summary_of_the_review": "I think this paper presents a strong argument for benchmarks in multi-objective problems. More discussion that ties the benchmark and the evaluation of methods back to various practical considerations (e.g. feasibility of sampling in the objective space, desirability of Pareto density vs. hypervolume) would make this paper even stronger.  ", "strength_and_weaknesses": "", "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": "2: The contributions are only marginally significant or novel.", "empirical_novelty_and_significance": "3: The contributions are significant and somewhat new. Aspects of the contributions exist in prior work."}, "extra_text": {}}, {"review_id": "K85nRfhrVkq", "reviewer": "Reviewer_yaU2", "summary": "", "questions": "", "limitations": "", "rating": 3, "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": "My main concern about this work is its scalability. At first phase of finding the weak pareto front HNPF needs to sample the solution space in uniform manner. Covering the solution space for real life solution is not feasible (in terms of memory and runtime) as such models contain hundreds of millions of parameters. Next, the authors propose a Neural Net (NN) to approximate the weak pareto front, since the NN optimization process is depend on labels there is additional labeling process. Although, this labeling process is automatically (i.e. without human intervention) it requires the calculation of $det(L^TL)$ where $L\\in\\mathbb{R}^{\\(n+m\\)\\times\\(k+m\\)}$ per sampled solution. The NN network is working on the solution/model parameter space which is not scalable as NNs contain much more parameters than their input space to ensure degrees of freedom.\n\nFor the second step HNPF utilizes variant of Kd trees to filter our dominated solutions. In my perspective the novelty of this section is limited, it can be applied to any MTL/MOO approach. It will be interesting to see how the compared methods' results change after applying this filter.\n\nThe experiment part is a bit weak. Due to the scalability problem the authors did not test HNPF on more challenging datasets (not even MNIST).\n\n\n", "summary_of_the_paper": "The authors present key shortcomings of MTL solvers in addition to Hybrid Neural Pareto Front (HNPF) that aims to handle non-convex functions and constraints.\n\nThe authors claim the following contributions:\n- New strategy for weak Pareto front identification based on Fritz-John conditions.\n- New Pareto filter to remove dominated points from the weak Pareto front.\n", "summary_of_the_review": "Some questions were raised during the review process.", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "extra_scores": {"correctness": "2: Several of the paper’s claims are incorrect or not well-supported.", "technical_novelty_and_significance": "2: The contributions are only marginally significant or novel.", "empirical_novelty_and_significance": "2: The contributions are only marginally significant or novel."}, "extra_text": {}}]

Author rebuttal:

Meta-review: The paper makes two contributions: (1) Multi-task benchmarks where the pareto solution is known analytically; and (2) a verification method for testing if solutions are on the pareto front. The authors make the point that MTL methods are applied to large-scale problems, but fail to find the pareto front in problems where it is known. 

Reviewers appreciated the discussion and insights by the approach, and the idea that correctness of scalable methods should be evaluated with problems that have analytic solutions, but they also had grave concerns. The primary concern is that without an efficient search, a verification method that builds on filters randomly generated solutions cannot scale to high dimensional problems. There were also disagreement about the role of LS and comparison with previous literature. 

As a result, the contribution of the submission is not sufficient for acceptance to ICLR
Decision: Reject