Title: FSEO: A Few-Shot Evolutionary Optimization Framework for Expensive Multi-Objective Optimization and Constrained Optimization
Venue: NeurIPS 2024
Accepted: no
Keywords: Expensive optimization

few-shot optimization

multi-objective optimization

constrained optimization

surrogate-assisted evolutionary optimization
Abstract: Meta-learning has been demonstrated to be useful to improve the sampling efficiency of Bayesian optimization (BO) and surrogate-assisted evolutionary algorithms (SAEAs) when solving expensive optimization problems (EOPs). However, existing studies focuses on only single-objective optimization, leaving other expensive optimization scenarios unconsidered. We propose a generalized few-shot evolutionary optimization (FSEO) framework and focus on its performance on two common expensive optimization scenarios: multi-objective EOPs (EMOPs) and constrained EOPs (ECOPs). We develop a novel meta-learning modeling approach to train surrogates for our FSEO framework, an accuracy-based update strategy is designed to adapt surrogates during the optimization process. The surrogates in FSEO framework combines neural network with Gaussian Processes (GPs), their network parameters and some parameters of GPs represent useful experience and are meta-learned across related optimization tasks, the remaining GPs parameters are task-specific parameters that represent unique features of the target task. We demonstrate that our FSEO framework is able to improve sampling efficiency on both EMOP and ECOP. Empirical conclusions are made to guide the application of our FSEO framework.

Abstract

Meta-learning has been demonstrated to be useful to improve the sampling efficiency of Bayesian optimization (BO) and surrogate-assisted evolutionary algorithms (SAEAs) when solving expensive optimization problems (EOPs). However, existing studies focuses on only single-objective optimization, leaving other expensive optimization scenarios unconsidered. We propose a generalized few-shot evolutionary optimization (FSEO) framework and focus on its performance on two common expensive optimization scenarios: multi-objective EOPs (EMOPs) and constrained EOPs (ECOPs). We develop a novel meta-learning modeling approach to train surrogates for our FSEO framework, an accuracy-based update strategy is designed to adapt surrogates during the optimization process. The surrogates in FSEO framework combines neural network with Gaussian Processes (GPs), their network parameters and some parameters of GPs represent useful experience and are meta-learned across related optimization tasks, the remaining GPs parameters are task-specific parameters that represent unique features of the target task. We demonstrate that our FSEO framework is able to improve sampling efficiency on both EMOP and ECOP. Empirical conclusions are made to guide the application of our FSEO framework.

1 Introduction

Expensive optimization problems (EOPs) aim to find as good as possible solutions within a budget of limited solution evaluations. Conventional Bayesian optimization (BO) and surrogate-assisted evolutionary algorithms (SAEAs) have been widely used to solve EOPs, but they train surrogate models from the scratch. To further improve the sampling efficiency and optimization performance, many efforts have been made to pre-train surrogates with the prior experience gain from related optimization tasks, resulting in experience-based optimization algorithms [1, 21, 36, 35].

This work considers solving EOPs on the context of few-shot problems [5, 41], where plenty of 25 expensive related tasks are available and each of them can provide a small dataset for experience 26 learning. Therefore, many experience-based optimization approaches such as multi-tasking optimiza27 tion [43, 2, 47], transfer optimization [35, 17, 16] are not considered as they cannot learn experience 28 from small related tasks (A discussion is available in Appendix A). In comparison, meta-learning 29 [14] has been proved to be powerful in solving few-shot problems, leading to a new subcategory of 30 experience-based optimization, namely few-shot optimization (FSO) [46]. 31

Existing studies on FSO are mainly few-shot Bayesian optimization (FSBO) where meta-learning 32

approaches are combined with BO to solve EOPs with only one objective. In this paper, we propose 33

a generalized few-shot evolutionary optimization (FSEO) framework to address EOPs from the 34

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perspective of SAEAs and consider two expensive optimization scenarios which have been limited studied: multi-objective EOPs (EMOPs) and constrained EOPs (ECOPs). Major contributions are summarized as follows.

A novel meta-learning method, namely Meta Deep Kernel Learning (MDKL), is developed to gain prior experience from related expensive tasks. Our model architecture and parameter designs make it possible to generate a regression-based surrogate on the prior experience and then continually adapt the surrogate to approximate the target task.

We propose a FSEO framework to solve EOPs from the perspective of SAEAs. FSEO framework is applicable to regression-based SAEAs since FSEO embed our meta-learning models in these SAEAs as their surrogates. In addition, an update strategy is designed to adapt surrogates constantly during the optimization. Note that our FSEO framework is a general framework but we focus on its performance on EMOPs and ECOPs in this paper.

Experiments are conducted on EMOPs and ECOPs to show our FSEO framework is effective. Our comprehensive ablation studies discover the influence of some factors on FSEO performance and provide empirical guidance to the application of FSEO framework.

2 Related Work

Experience-based optimization can be divided into several subcategories according to the techniques of learning prior experience from related tasks. A detailed classification and discussion on these subcategories is available in Appendix A. This subsection focuses on related work on FSO.

FSO studies in the literature can be classified based on their model architectures. Most studies metalearn parameters for Gaussian Processes (GPs) [44], namely FSBO or Meta Bayesian Optimization (MBO) [32, 42, 26, 38]. In addition, [23] meta-learns with transformer neural processes and [46, 6] meta-learn parameters for the architecture of deep kernel learning (DKL) [45]. The MDKL model in our FSEO belongs to the last category as its model architecture is relevant to DKL.

Our work is different from existing studies in three points: Firstly, many studies [46] use existing meta-learning models [27] as their surrogates. No further adaptations are made to these surrogates during optimization since they are not originally designed for optimization. In comparison, we try to develop a meta-learning model, MDKL, for optimization purpose. MDKL has explicit task-specific parameters, which allows continually model adaptations during the optimization. Secondly, existing work investigated only global optimization, leaving other optimization scenarios such as EMOP and ECOP still awaiting for investigation. As our MDKL is designed for optimization and is capable of continually adaptation, we pay attention on EMOPs and ECOPs which require more effective models than global optimization. Our work widens the scope of existing FSO research and it focuses on the perspective of SAEAs instead of BO. Lastly, in-depth ablation studies are lacking in the literature, making it unclear which factors affect the performance of FSO. Our extensive ablation studies fill this gap and we conclude some empirical rules to improve the performance of FSO.

3 Background

This section gives preliminaries about meta-learning and DKL. The former is the method of experience learning, the latter is the underlying structure of experience representation.

3.1 Meta-Learning in Few-Shot Problems

In the context of few-shot problems, we have plenty of related tasks, each task T contributes a couple of small datasets D = { ( S, Q ) } , namely support dataset S and query dataset Q , respectively. After learning from datasets of random related tasks, a support set S ∗ from new unseen task T ∗ is given and one is asked to estimate the labels or values of a query set Q ∗ . The problem is called 1-shot or 5-shot when only 1 data point or 5 data points are provided in S ∗ . A comprehensive definition of few-shot problems is available in [5, 41].

Meta-learning methods have been widely used to solve few-shot problems [41]. They learn domain81 specific features that are shared among related tasks as experience, such experience is used to 82 understand and interpret the data collected from new tasks encountered in the future. 83

3.2 Deep Kernel Learning (DKL) 84

DKL aims at constructing kernels that encapsulate the expressive power of deep architectures for GPs. 85 To create expressive and scalable closed form covariance kernels, DKL combines the non-parametric 86 flexibility of kernel methods and the structural properties of deep neural networks. In practice, a deep 87 kernel k ( x i , x j | γ ) transforms the inputs x of a base kernel k ( x i , x j | θ ) through a non-linear mapping 88 given by a deep architecture φ ( x | w , b ) : 89

where θ and ( w , b ) are parameter vectors of the base kernel and the deep architecture, respectively. γ = { θ , w , b } is the set of all parameters in this deep kernel. Note that in DKL, all parameters γ of a deep kernel k ( x i , x j | γ ) are learned jointly by using the log marginal likelihood function of GPs as a loss function. Such a jointly learning strategy has been shown to make a DKL algorithm outperform a combination of a deep neural network and a GP model, where a trained GP model is applied to the output layer of a trained deep neural network [45].

3.3 Meta-Learning on DKL

An important distinction between DKL algorithms and the applications of meta-learning to DKL is that DKL algorithms learn their deep kernels from single tasks instead of collections of related tasks. Such a difference alleviates two drawbacks of single task DKL [39]: First, the scalability of deep kernels is no longer an issue as datasets in meta-learning are small. Second, the risk of overfitting is decreased since diverse data points are sampled across tasks.

4 Few-Shot Evolutionary Optimization (FSEO) Framework

In this paper, T ∗ denotes the target optimization task, and plenty of small datasets D i sampled from related tasks T i are available for experience learning. A complete list of notations is available at the beginning of Appendix.

4.1 Overall Working Mechanism

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As illustrated in Fig. 1, all modules covering the optimization of target task T ∗ are included in a grey block. The modules beyond the grey block are associated with related tasks T i and experience learning, which distinguishes our FSEO framework from conventional SAEAs and BO. The MDKL surrogate modeling method consists of two procedures: meta-learning procedure and adaptation procedure. The former learns prior experience from T i , the latter uses experience to adapt surrogates to fit T ∗ . The framework of FSEO is depicted in Alg. 1, it consists of the following major steps.

Experience learning : Before expensive optimization starts, a meta-learning procedure is conducted to train task-independent parameters γ e for MDKL surrogates (line 2). N m datasets { D m 1 , . . . , D mN m } collected from N related tasks { T 1 , . . . , T N } are used to train γ e . γ e is the experience that represents the domain-specific features of related tasks.

Initialize surrogates with experience : Optimization starts when a target optimization task T ∗ is given. An initial dataset S ∗ is sampled (line 3) to adapt task-specific parameters γ ∗ on the basis of experience γ e . After that, MDKL surrogates are updated (line 4).

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Reproduction : MDKL surrogates h ( γ ∗ ) are combined with a SAEA optimizer Opt to search for optimal solution(s) x ∗ on h ( γ ∗ ) (line 7). This is implemented by replacing the original (regression-based) surrogates in a SAEA with h ( γ ∗ ) .

Update archive and surrogates : New optimal solution(s) x ∗ is evaluated on target task T ∗ (line 8). The evaluated solutions will be added to dataset S ∗ (line 9) which serves as an archive. Then, surrogate adaptation is triggered, surrogates h ( γ ∗ ) are updated (line 10).

Stop criterion : Once the evaluation budget has run out, the evolutionary optimization will be terminated and the optimal solutions in dataset S ∗ will be outputted. Otherwise, the algorithm goes back to step 3.

Algorithm 1 FSEO Framework.

1: Input: D i : Datasets collected from related tasks T i , i= { 1 , . . . , N } ; N m : Number of subsets D m for meta-learning; | D m | : Size of subsets D m . | D m | ≤ | D i | due to D m ⊆ D i ; Batch size B ; Surrogate learning rates α, β ; Target task T ∗ ; A SAEA optimizer Opt ; Fitness evaluation budget FE max .

2: Experience γ e ← Meta-learning( D i , N m , | D m | , B, α ). / ∗ Alg. 2. ∗ /

3: S ∗ ← Sampling 1 d solutions from T ∗ .

4: h ( γ ∗ ) ← Adaptation( γ e , S ∗ , β ). / ∗ Initialize surrogate. ∗ /

5: Set evaluation counter FE = | S ∗ | .

6: while FE < FE max do

7: Candidate solution(s) x ∗ ← Surrogate-assisted optimization ( Opt, h ( γ ∗ ) ).

8: f ( x ∗ ) ← Evaluate x ∗ on T ∗ .

10: h ( γ ∗ ) ← Update( γ ∗ , S ∗ , β ). / ∗ Alg. 4. ∗ /

9: S ∗ ← S ∗ ∪ { ( x ∗ , f ( x ∗ )) } .

11: Update FE .

12: end while

13: Output: Optimal solutions in S ∗ .

4.2 Learning and Using Experience by MDKL

In MDKL, the domain-specific features of related tasks are used as experience, which are represented by the task-independent parameters γ e learned across related tasks. To make MDKL more capable of expressing complex domain-specific features, the base kernel k ( x i , x j | θ ) in GP is combined with a neural network φ ( w , b ) to construct a deep kernel (see Eq.(1)). The modeling of a MDKL model consists of two procedures: meta-learning procedure and adaptation procedure. To make a clear illustration, we introduce frameworks of two procedures and then explain them in detail.

Meta-learning procedure: Learning experience

Our MDKL model uses the kernel in [18] as its base kernel:

Therefore, the deep kernel will be: 138

where γ = { w , b , θ , p } is a set of deep kernel parameters. φ, w and b are neural network and its parameters (see Eq.(1)). Details about alternative base kernels are available in [44].

The aim of meta-learning procedure is to learn experience γ e from related tasks { T 1 , . . . , T N } , 141 including neural network parameters w , b , and task-independent base kernel parameters θ e , p e . The 142 pseudo-code of meta-learning procedure is given in Alg. 2. Ideally, the experience γ e is learned from 143 plenty of ( N m ) small datasets D m collected from different related tasks. However, in practice, the 144 number of available related tasks N may be much smaller than N m . Hence, the meta-learning is 145 conducted gradually over U update iterations (line 3). During each update iteration, a small batch of 146

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related tasks contribute B small datasets { D m 1 , . . . , D mB } for meta-learning purpose (lines 5 and 7). Note that if N < N m , a related task T i can be used multiple times in the meta-learning procedure.

Algorithm 2 Meta-learning( D i , N m , | D m | , B, α )

1: Input: D i : Datasets collected from related tasks T i , i= { 1 , . . . , N } ; N m : Number of subsets D m for meta-learning; | D m | : Size of subsets D m . | D m | ≤ | D i | due to D m ⊆ D i ; Batch size B ; Learning rate for priors α .

2: Randomly initialize w , b , θ e , p e .

3: Set the number of update iterations U = N m /B .

4: for j = 1 to U do

5: { D ′ 1 , . . . , D ′ B } ← Randomly select a batch of datasets from { D 1 , . . . , D N } . for all D ′ i in the batch do D mi ← A subset of size | D m | from D ′ i . Initialize task-specific increment ∆ θ i , ∆ p i . Compute task-specific parameters: θ i = θ e +∆ θ i , p i = p e +∆ p i . Obtain deep kernel k ( x i , x j | γ ) based GP: h ( γ ) , where γ = { w , b , θ i , p i } (Eq.(3)). Compute the loss function L ( D mi , h ( γ )) (Eq.(4)). end for Update w , b , θ e , p e via gradient descent: α glyph[triangleinv] L ( D mi , h ( γ )) (Eq.(5)). end for Output: Task-independent parameters: γ e = { w , b , θ e , p e } .

For a given dataset D mi , we denote θ i = θ e +∆ θ i and p i = p e + ∆ p i as the task-specific kernel parameters, where ∆ θ i , ∆ p i are the distance we need to move from the taskindependent parameters to the task-specific parameters (line 9). The loss function L of MDKL is the likelihood function defined as follows [18]: 6: 7: 8: 9: 10: 11: 12: 13: 14: 15:

where | R | is the determinant of the correlation matrix R , each element in the matrix is computed 148 through Eq.(3). y is the fitness vector of D mi . µ and σ 2 are the mean and the variance of the prior 149 distribution, respectively. Experience γ e = { w , b , θ e , p e } is updated by gradient descent (line 13), 150 take θ e as an example: 151

After U iterations, γ e has been trained sufficiently by N m small datasets D m and will be used in target task T ∗ later.

Adaptation procedure: Using experience

The meta-learning of experience γ e enables MDKL to handle a family of related tasks in general. To approximate a specific task T ∗ well, surrogate h ( γ e ) needs to adapt task-specific increments ∆ θ ∗ and ∆ p ∗ in the way described in Alg. 3. A diagram of the deep kernel implemented in our MDKL model is illustrated in Fig. 2.

Algorithm 3 Adaptation( γ ∗ , S ∗ , β )

1: Input: Current surrogate parameters γ ∗ ; A dataset S ∗ sampled from target task T ∗ (Archive); Learning rate for adaptation β .

2: if γ ∗ == γ e then

3: Initialize task-specific increments ∆ θ ∗ , ∆ p ∗ .

4: Compute task-specific parameters: θ ∗ = θ e + ∆ θ ∗ , p ∗ = p e +∆ p ∗ .

5: Obtain deep kernel k ( x i , x j | γ ∗ ) based GP: h ( γ ∗ ) , where γ ∗ = { w , b , θ ∗ , p ∗ } (Eq.(3)).

6: end if

7: Compute the loss function L ( S ∗ , h ( γ ∗ )) (Eq.(4)).

9: Output: Adapted MDKL h ( γ ∗ ) .

8: Update ∆ θ ∗ , ∆ p ∗ using gradient descent: β glyph[triangleinv] L ( S ∗ , h ( γ ∗ )) .

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Surrogate prediction. Due to the nature of a GP, when predicting the fitness of a solution x ∗ , a MDKL surrogate produces a predictive Gaussian distribution N (ˆ y ( x ∗ ) , ˆ s 2 ( x ∗ )) , the predicted mean ˆ y ( x ∗ ) and covariance ˆ s 2 ( x ∗ ) are specified as [18]:

where r is a correlation vector consisting of covariances between x ∗ and S ∗ , other variables are explained in Eq.(4).

4.3 Surrogate Update Strategy

In this subsection, we describe the update strategy in our FSEO framework. To properly integrate experience and data from T ∗ , our update strategy is designed to determine whether a MDKL surrogate should be adapted in the current iteration or not, ensuring an optimal update frequency of surrogates.

As illustrated in Alg. 4, the surrogate update starts when a new optimal solution(s) has been evaluated on expensive functions and an updated archive S ∗ is available. For a given surrogate h ( γ ∗ ) , its mean squared error (MSE) on S ∗ is selected as the update criterion: If the MSE after an adaptation e 1 (line 4) is larger than the MSE without an adaptation e 0 (line 2), then the surrogate will roll back to the status before the adaptation. This indicates the surrogate update has been refused and h ( γ ∗ ) will not be adapted in the current iteration. Otherwise, the adapted surrogate will be chosen (line 6). Note that no matter whether surrogate adaptations are accepted or refused, the resulting surrogates will be treated as updated surrogates, which are employed to assist the SAEA optimizer in the next iteration.

5 Computational Studies

Our computational studies can be divided into three parts: (1). Appendix D evaluates the effectiveness of learning experience through a synthetic problem and a real-world engine modeling problem. (2). Sections 5.1 to 5.2 use EMOPs as examples to investigate the performance of our FSEO framework in depth. Empirical evidence is provided to guide the use of our FSEO framework. (3). Section 5.3 investigates the performance of our FSEO framework on a real-world ECOP. The source code is available online 1 For all meta-learning methods used in our experiments, their basic setups are listed in Table 1. The neural network structure is suggested by [10, 27], and the learning rates are the default values that have been widely used in many meta-learning methods [13, 27].

5.1 Performance on EMOPs

In the following subsections, we aim to demonstrate the effectiveness of our FSEO framework. The experiment in this subsection is designed to answer the question below: With the experience learned from related tasks, can our FSEO framework helps a SAEA to save 9 d solutions without a loss of optimization performance?

The computational study is conducted on the DTLZ test problems [8]. All the DTLZ problems have d = 10 decision variables and 3 objectives, as the setups that have been widely used in [25, 33]. The details of generating DTLZ variants (related tasks) are provided in Appendix C. We test our FSEO framework using an instantiation on MOEA/D-EGO, resulting MOEA/D-FS. Details of the comparison algorithms are given in Appendix E.1.

Algorithm 4 Update( γ ∗ , S ∗ , β )

1: Input: Current surrogate parameters γ ∗ ; Updated archive S ∗ ; Learning rate for further adaptations β . 2: e 0 ← MSE ( h ( γ ∗ ) , S ∗ ). 3: h ( γ ′ ) ← Adaptation ( γ ∗ , S ∗ , β ) . / ∗ Temporary surrogate, Alg. 3. ∗ / 4: e 1 ← MSE ( h ( γ ′ ) , S ∗ ) . 5: if e 0 > e 1 then 6: update γ ∗ = γ ′ , obtain new h ( γ ∗ ) . 7: end if 8: Output: Surrogate h ( γ ∗ ) .

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5.1.1 Experimental setups

The parameter setups for this multi-objective optimization experiment are listed in Table 2. During the optimization process, an initial dataset S ∗ is sampled using Latin-Hypercube Sampling (LHS) method [24], then extra evaluations are conducted until the evaluation budget has run out. Note that we aim to use related tasks to save 9 d evaluations without a loss of SAEA optimization performance. Hence, the total evaluation budgets for MOEA/D-FS and comparison algorithms are different.

Since the test problems have 3 objectives, we employ inverted generational distance plus (IGD+) [15] as our performance indicator, where smaller IGD+ values indicate better optimization results. 5000 reference points are generated for computing IGD+ values, as suggested in [25]. More results in IGD [4] and HV [55] metrics are reported in Appendix E.3.

5.1.2 Results and analysis

The statistical test results are reported in Fig. 3 and Appendix E.2 (Table 5). It can be seen from Fig. 3 that, although 90 fewer evaluations are used in surrogate initialization, MOEA/D-FS can still achieve competitive or even smaller IGD+ values than MOEA/D-EGO on all DTLZ problems except for DTLZ7. In addition, the IGD+ values obtained by MOEA/D-FS drop rapidly, especially during the first few evaluations, implying the experience learned from DTLZ variants are effective. Therefore, in most situations, our FSEO framework is able to assist MOEA/D-EGO in reaching competitive or even better optimization results, with the number of evaluations used for surrogate initialization reduced from 10 d to only 1 d .

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MOEA/D-FS is less effective on DTLZ7 than on other DTLZ problems, which might be attributed to the discontinuity of the Pareto front on DTLZ7. Note that MOEA/D-FS learns experience from small datasets such as D m and S ∗ . The solutions in these small datasets are sampled at random, hence, the

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probability of having optimal solutions being sampled is small. However, it is difficult to learn the discontinuity of the Pareto front from the sampled non-optimal solutions. As a result, the knowledge of 'there are four discrete optimal regions' cannot be learned from such small datasets ( | D m | = 20 ) collected from related tasks. The performance analysis between MOEA/D-FS and other comparison algorithms are available in Appendix E.2.

5.1.3 More comparison experiments

We also compared the performance of our FSEO framework when only 10 evaluations are used for surrogate initialization for comparison algorithms. The results are reported in Table 8 in Appendix E.4. In addition, the performance of our FSEO framework in the context of extremely expensive optimization has been investigated in Appendix H (Table 11 and Fig. 7).

The question raised at the beginning of this subsection can be answered by the results discussed so far. Due to the integration of the experience learned from related tasks (DTLZ variants), although the evaluation cost of surrogates initialization has been reduced from 10 d to 1 d , our FSEO framework is still capable of assisting regression-based SAEAs to achieve competitive or even better optimization results in most situations.

5.2 Ablation Studies

We conduct two ablation studies to investigate the influence of task similarity and that of the dataset size used in meta-learning, results and analysis are reported in Appendixes F and G, respectively.

5.3 Performance on Real-World ECOPs

The experiments on EMOPs have investigated the performance of our FSEO framework in depth. In this subsection, we evaluate our FSEO framework on a real-world gasoline motor engine calibration problem, which is an ECOP.

The calibration problem has 6 adjustable engine parameters, namely the throttle angle, waste gate orifice, ignition timing, valve timings, state of injection, and air-fuel-ratio. The calibration aims at minimizing the BSFC while satisfying 4 constraints in terms of temperature, pressure, CA50, and load simultaneously [53].

5.3.1 Comparison algorithms

Since the comparison algorithms in the DTLZ optimization experiments are not designed for handling constrained optimization, our comparison is conducted with 3 state-of-the-art constrained optimization algorithms used in industry: A variant of EGO designed to handle constrained optimization problems (cons_EGO) [53], a GA customized for this calibration problem (adaptiveGA) [53], and a bilevel constrained SAEA (SAB-DE) [50]. The settings of the comparison algorithms are the same as suggested in the literature. In this experiment, we apply our FSEO framework to cons_EGO and investigate its optimization performance. The GP surrogates in cons_EGO are replaced by our MDKL surrogates to conduct the comparison, and the resulting algorithm is denoted as cons_FS.

5.3.2 Experimental setups

The setup of related tasks ( N,D i ) is the same as described in Appendix D. In the meta-learning procedure, both the support set and the query set contain 6 data points, thus | D m | = 12 . The total evaluation budget for all algorithms is set to 60. For adaptiveGA, all evaluations are used in the optimization process as it is not a SAEA. For cons_EGO and SAB-DE, 40 samples are used to initialize the surrogates and 20 extra evaluations are used in the optimization process. For cons_FS, only 6 samples are used to initialize MDKL surrogates, and the remaining evaluations are used for further optimization.

5.3.3 Optimization results and analysis

The left side and right side of Fig. 4 plot the normalized BSFC results and the number of feasible solutions found over the number of used evaluations, respectively. Solid lines are mean lines, while shadows are error regions. From the left side of Fig. 4, it can be observed that the minimal BSFC

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obtained by cons_FS decreases drastically in the first few evaluations, implying that the experience learned from related tasks is effective. In comparison, the minimal BSFC obtained by adaptiveGA and cons_EGO drops in a relatively slow rate, even though cons_EGO has used 34 more samples to initialize its surrogates. The star marker denotes the point at which cons_FS has evaluated 20 samples after surrogate initialization. It is worth noting that when 20 samples have been evaluated in the optimization, cons_FS achieves a smaller BSFC value than cons_EGO. After the star marker, the decrease of BSFC becomes slow as cons_FS has reached the optimal region. Therefore, further improvement in the normalized BSFC value is not significant and thus hard to be observed. The advantages of our FSEO framework can also be observed in constraint handling. In the right side of Fig. 4, cons_FS finds more feasible solutions than the 3 comparison algorithms. These results indicate that our FSEO framework improves the performance of cons_EGO on both objective function and constraint functions. Meanwhile, only 1 d evaluations are used to initialize surrogates.

5.3.4 Discussion on runtime

It should be noted that real engine performance evaluations on engine facilities are very costly in terms of both time and financial budget [49]. Since a single real engine performance evaluation can cost several hours [22, 49], the time cost of the meta-learning procedure is negligible as it takes only a few minutes. Savings from reduced real engine performance evaluations on engine facilities and the reduced development cycle due to our FSEO framework could amount to millions of dollars [49]. our FSEO framework is an effective and efficient method to solve this real-world calibration problem.

6 Conclusion and further work

Conclusion. In this paper, we present a FSEO framework to address EMOPs and ECOPs from the perspective of SAEAs. A novel meta-learning approach MDKL is proposed to learn prior experience from related expensive tasks. Our MDKL model is designed for optimization and has explicit task-specific parameters, which allows continually update of task-specific parameters during the optimization process. Our empirical experiments show that the FSEO framework is able to improve the sampling efficiency and thus save expensive evaluations for existing regression-based SAEAs. Ablation studies reveal the influence between optimization performance and solutions similarity as well as the size of datasets for meta-learning.

Limitation and further work. The limitations of this work can be summarized as the following 287 two points: First, we do not have a mathematical definition of related tasks. Second, the proposed 288 framework is currently for regression-based SAEAs only. 289

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A Discussion on Experience-based Optimization 445

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In the past decade, experience-based optimization has attracted much attention as it uses the experience gained from other optimization problems to improve the optimization efficiency of target problems, which mimics human capabilities of cognitive and knowledge generalization [12]. The optimization problems that provide experience or knowledge are regarded as source tasks, while the target optimization problems are regarded as target tasks. To obtain useful experience, the tasks that are related to target tasks are chosen as source tasks since they usually share domain-specific features with target tasks. Diverse experience-based optimization methods have been proposed to use the experience gained from related tasks to tackle target tasks. They can be divided into two categories based on the direction of experience transformation.

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In the first category, experience is transformed mutually. Every considered optimization problem is a target task and also is one of the source tasks of other optimization problems. In other words, the roles of source task and target task are compatible. One representative tributary is EMTO that aims to solve multiple optimization tasks concurrently [9, 43, 20, 2, 47]. In EMTO, experience is learned, updated, and spontaneously shared among target tasks through multi-task learning techniques. A variant of EMTO is multiforms optimization [12, 52, 11]. In multiforms optimization, multi-task learning methods are employed to learn experience from distinct formulations of a single target task.

In the second category, experience is transformed unidirectionally. The roles of source task and target task are not compatible, an optimization problem cannot serve as a source task and a target task simultaneously. One popular tributary is transfer optimization which employs transfer learning techniques to transform experience from source tasks to target tasks [35, 17, 16, 40]. In transfer learning, experience can be transformed from a single source task, multiple source tasks, or even source tasks from a different domain [54]. However, these transfer learning techniques pay more attention to experience transformation instead of experience learning. Despite diverse and complex situations of experience transformation have been studied [29, 30], the difficult of learning experience from small (expensive) source tasks has not been well studied. Actually, a common scenario in transfer learning is that the source task(s) is/are large enough such that useful experience can be obtained easily through solving source task(s) [54]. In contrast to transfer optimization, recently, some experience-based optimization algorithms attempted to use meta-learning methods to learn experience from small source tasks, which are known as few-shot optimization (FSO)[46]. Since meta-learning only works for related tasks in the same domain, the situations of experience transformation are less complex than that of transfer learning. As a result, meta-learning pays more attention to experience learning instead of experience transformation. Domain-specific features are extracted as experience and no related task needs to be solved.

Our work belongs to the FSO in the second category discussed above since our experience is transformed unidirectionally. More importantly, our experience is learned across many related expensive tasks, rather than gained through solving more or less source tasks. Therefore, our work is different from transfer optimization.

B Discussion on Framework Compatibility and Limitation

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C Generation of DTLZ variants

Our DTLZ optimization experiments generate m -objective DTLZ variants in the following ways: DTLZ1 :

where z is a vector consisting of the last k = d -m + 1 variables in x . In other words, z = { z 1 , . . . , z k } = { x m , . . . , x d } . The variants of DTLZ1 introduce only one variable a ∈ [0 . 1 , 5 . 0] m in Eq.(8), Eq.(9), and Eq.(10), where a = 1 in the original DTLZ1. For out-of-range test, a ∈ [1 . 5 , 5 . 0] m .

DTLZ2 :

The variants of DTLZ2 introduce two variables a ∈ [0 . 1 , 5 . 0] m and b ∈ [0 . 5 , 2 . 0] m in Eq.(12), Eq.(13), and Eq.(14), where a = 1 and b = 2 in the original DTLZ2. For out-of-range test, a ∈ [1 . 5 , 5 . 0] m , b ∈ [0 . 5 , 1 . 5] m .

DTLZ3 : The variants of DTLZ3 are generated using the same way as described in DTLZ2, except the equation g from Eq.(15) is replaced by the one from Eq.(11).

DTLZ4 : The variants of DTLZ4 are generated using the same way as described in DTLZ2, except all x i are replaced by x 100 i .

DTLZ5 : The variants of DTLZ5 are generated using the same way as described in DTLZ2, except all x 2 , . . . , x m -1 are replaced by 1+2 gx i 2(1+ g ) .

DTLZ6 :

The variants of DTLZ6 are generated using the same way as described in DTLZ5, except the equation g from Eq.(15) is replaced by the one from Eq.(16).

DTLZ7 :

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The variants of DTLZ7 introduce one variable a ∈ [0 . 1 , 5 . 0] m in Eq.(17) and Eq.(19), where a j =1: m -1 = 0 and a m = 1 in the original DTLZ7. For out-of-range test, a ∈ [1 . 5 , 5 . 0] m .

D Effectiveness of Learning Experience

Evaluating the effectiveness of learning experiences aims to demonstrate that our MDKL model can learn experience from related tasks and outperforms other meta-learning models. For this reason, the experiment is designed to answer the following questions:

Given a small dataset S ∗ from target task T ∗ , can MDKL learn experience from related tasks and then generate a model that has the smallest MSE?

If yes, which components of MDKL contribute to the effectiveness of learning experience? Meta-learning or/and deep kernel learning? If not, why not?

To answer the two questions above, we consider two experiments to evaluate the effectiveness of learning experience: amplitude prediction for unknown periodic sinusoid functions, and fuel consumption prediction for a gasoline motor engine. The former is a few-shot regression problem that motivates many meta-learning studies [10, 13, 39, 27], while the latter is a real-world regression problem [53].

D.1 Effectiveness of Learning Experience: Sinusoid Function Regression

In the sinusoid regression experiment, we learn experience from a series of 1-dimensional sinusoid functions:

where the amplitude A and phase w of sine waves are varied between functions. The target is to approximate an unknown sinusoid function with a small support dataset S ∗ and the learned experience. Clearly, by integrating experience with S ∗ , we estimate parameters ( A,w,b ) for an unknown sinusoid function. As a result, the output y of the given sinusoid function can be predicted once a query data x is inputted.

D.1.1 Generation of Sinusoid Function Variants

As suggested in [10, 13], we set amplitude A ∈ [0.1, 5.0], frequency w ∈ [0 . 999 , 1 . 0] , phase b ∈ [0, π ], and Gaussian noise glyph[epsilon1] ∼ (0 , 0 . 1) . Therefore, a sinusoid function can be generated by sampling three parameters ( A,w,b ) from their ranges uniformly. In total, N m = N = 20000 related sinusoid functions are generated at random.

D.1.2 Experimental Setups

All data points x are sampled from the range ∈ [-5.0, 5.0]. In the meta-learning procedure, both support set and query set contain 5 data points. Hence, a dataset D i is sampled from each (related) sinusoid function T i , and | D i | = | D m | = 10 . Six experiments are conducted where | S ∗ | = { 2 , 3 , 5 , 10 , 20 , 30 } data points are sampled from the target function. Considering Gaussian noise glyph[epsilon1] could be relatively large when amplitude A is close to 0.1, normalized mean squared error (NMSE) is chosen as a performance indicator. NMSE is measured using a dataset that contains 100 data points sampled uniformly from the x range.

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D.1.3 Comparison methods

In this experiment, three families of modeling methods are compared with our MDKL model:

Meta-learning methods that were proposed for regression tasks: MAML [10], ALPaCA [13], and DKT [27]. The configurations of MAML, ALPaCA, and DKT are the same as suggested in the original literature.

Non-meta-learning method that is widely used for regression tasks: the GP model. We choose a GP as a baseline since it is effective and more relevant to MDKL than other non-meta-learning modeling methods. We set the range of base kernel parameters in the GP model as θ ∈ [10 -5 , 10] and p ∈ [1 , 2] .

MDKL related methods that are designed to investigate which components of MDKL contribute to the modeling performance: GP_Adam, DKL, and MDKL_NN. GP_Adam is a GP model fitted by Adam optimizer. The combination of GP_Adam and a neural network results in a kind of DKL algorithm. MDKL_NN is a meta-learning version of DKL, but it learns only neural network parameters through meta-learning and has no task-independent base kernel parameters.

D.1.4 Results and Analysis

Table 3 reports the statistical test results of the NMSE values achieved by comparison algorithms in sinusoid function regression experiments. Each row lists the results obtained when the same number of fitness evaluations | S ∗ | are used to train models. The results of Wilcoxon rank sum test between MDKL and other compared algorithms are listed in the last row. It can be observed that both MDKL and DKT have achieved the smallest NMSE values on all tests in the comparison with other meta-learning and non-meta-learning modeling methods.

Contributions of MDKL components are analyzed through statistical tests between MDKL related methods. The statistical test results between DKL and GP_Adam are 5/1/0, showing that DKL is preferable to GP_Adam when only a few data points are available for modeling. Hence, using a neural network to build a deep kernel for GP is able to enhance the performance of modeling. When meta-learning technique is applied to DKL, the statistical test results between MDKL_NN and DKL are 3/3/0. The meta-learning of neural network parameters is necessary since it contributes to the performance of MDKL. Further statistical test between MDKL and MDKL_NN gives results of 3/3/0, indicating that the meta-learning of base kernel parameters is effective on this regression problem.

D.2 Effectiveness of Learning Experience: Engine Performance Regression

In this subsection, we focus on a Brake Special Fuel Consumption (BSFC) regression task for a gasoline motor engine [53], where BSFC is evaluated on a gasoline engine simulation (denoted by T ∗ ).

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D.2.1 Experimental setups

The related tasks T i used in our experiment are N = 100 gasoline engine models. These engine models have different behaviors when compared with T ∗ , but they share the basic features of gasoline engines. All related tasks and the target task have the same six decision variables. Each related task T i provides only 60 solutions, forming a dataset D i . The size of datasets used for meta-learning | D m | is set to 40. Six tests are conducted where | S ∗ | = { 2, 3, 5, 10, 20, 40 } data points are sampled from the real engine simulation T ∗ . MSE is chosen as an indicator of modeling accuracy, which is measured using a dataset consisting of 12500 data points that are sampled uniformly from the engine decision space. The comparison algorithms are the same as described in Appendix D.1.

D.2.2 Results and analysis

The statistical test results of the MSE values achieved by comparison algorithms in BSFC regression experiments are summarized in Table 4. Each row lists the results obtained when the same number of fitness evaluations | S ∗ | are used to train models. The results of Wilcoxon rank sum test between MDKL and other compared algorithms are listed in the last row. It can be observed that MDKL and MDKL_NN outperform other comparison modeling methods since they have achieved the smallest MSE on all tests.

Additional Wilcoxon rank sum tests have been conducted between MDKL related algorithms to answer our second question (results are not reported in Table 4). The statistical test results between DKL and GP_Adam are 1/5/0, indicating that the neural network in DKL makes some contributions to the performance of MDKL. The statistical test results between MDKL_NN and DKL are 6/0/0, demonstrating that the meta-learning of neural network parameters constructs a useful deep kernel and contributes to the improvement of modeling accuracy. However, there is no significant difference between the performance of MDKL and that of MDKL_NN, the meta-learning on base kernel parameters does not play a critical role on this engine problem. In comparison, the meta-learning on base kernel parameters is effective in sinusoid function regression experiments (see Appendix D.1). In addition, the statistical test results between MDKL_NN and MAML are 4/2/0. Considering that MAMLis a neural network regressor learned through meta-learning, we can conclude that GP is an essential component of our MDKL. In summary, all components in MDKL are necessary, they all contribute to the effectiveness of learning experience.

The comparison experiments on sinusoid functions and the gasoline motor engine have demonstrated the effectiveness of our MDKL modeling method in the learning of experience. Given a small dataset of the target task, the model learned through MDKL method has the smallest MSE among all comparison models. Additionally, the investigation between MDKL and its variants shows that all components in MDKL have made their contributions to the effectiveness of learning experience. However, similar to other meta-learning studies [10, 13], we have not defined the similarity between tasks. In other words, the boundary between related tasks and unrelated tasks has not been defined. This should be a topic of further study on meta-learning. Moreover, the relationship between task similarity and modeling performance has not been investigated. Instead, we study the relationship between task similarity and SAEA optimization performance in Section F, since our main focus is the surrogate-assisted evolutionary optimization.

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E Additional Details on Expensive Multi-Objective Optimization

E.1 Comparison algorithms

As explained in Section B, our FSEO framework is compatible with regression-based SAEAs. Hence, we select MOEA/D-EGO [51] as an example and replace its GP surrogates by our MDKL surrogates. The resulting algorithm is denoted as MOEA/D-FS. Note that it is not necessary to specially select a newly proposed regression-based SAEA as our example, our main objective is to save evaluations with experience and observe if there is any damage to the optimization performance caused by the saving of evaluations. Therefore, it does not make any difference which regression-based SAEA or BO we choose as our example. Additionally, to demonstrate the improvement of optimization performance caused by using experience on DTLZ problems is significant, several state-of-the-art SAEAs and MOBO are also compared as baselines, including ParEGO [19], K-RVEA [7], CSEA [25], OREA [48], KTA2 [33], ESBCEO [3], and KMOEA-TIC [28]. Among these algorithms, ParEGO, K-RVEA, KTA2, KMOEA-TIC use regression-based surrogates, CSEA uses a classification-based surrogate, OREA employs an ordinal-regression-based surrogate, and ESBCEO is a recently proposed MOBO.

We implemented the FSEO framework, MOEA/D-EGO, ParEGO, and OREA, while the code of K-RVEA, CSEA, KTA2, and ESBCEO [3] is available on PlatEMO [37], an open source MATLAB platform for evolutionary multi-objective optimization. The code of KMOEA-TIC [28] is obtained from its authors. To make a fair comparison, all comparison algorithms share the same initial dataset S ∗ in an independent run. We also set θ ∈ [10 -5 , 100] d and p = 2 for all GP surrogates as suggested in [33], these GP surrogates are implemented through DACE [31]. Other configurations are the same as suggested in their original literature.

E.2 Result Table and Analysis of Expensive Multi-Objective Optimization

The experience learned from related tasks makes MOEA/D-EGO more competitive when compared to other SAEAs. The use of MDKL surrogates results in significantly smaller IGD+ values on DTLZ1, DTLZ2, DTLZ3, and DTLZ5 than before. As a result, MOEA/D-FS achieves the smallest IGD+ values on DTLZ2 and DTLZ3, and its optimization results on DTLZ1 and DTLZ5 are much closer to the best optimization results (e.g. results obtained by ParEGO and OREA) than MOEA/D-EGO. Although MOEA/D-FS does not achieve the smallest IGD+ values on all DTLZ problems, it should be noted that MOEA/D-FS is still the best algorithm among comparison SAEAs due to its overall performance. Table 5 shows that no comparison SAEA outperforms MOEA/D-FS on three DTLZ problems, but MOEA/D-FS outperforms all comparison SAEAs on at least three DTLZ problems. Furthermore, the IGD+ values of MOEA/D-FS are achieved with an evaluation budget of 60, while the IGD+ values of other SAEAs are reached with a cost of 150 evaluations (see Table 2).

E.3 Result Tables and Figures in IGD and HV Metrics

The performance of our method and the comparison algorithms are also evaluated on inverted generational distance (IGD) [4] and Hypervolume (HV) [55] metrics.

Results in IGD values are reported in Table 6 and Fig. 5. A smaller IGD value indicates a better optimization result.

Results in HV values are reported in Table 7 and Fig. 6. A larger HV value indicates a better optimization result.

679

680

681

682

683

684

685

686

Problems

MOEAD-EGO

MOEAD-FS

ParEGO

K-RVEA

KTA2

CSEA

OREA

ESBCEO

KMOEATIC

DTLZ1

1.07e+2(2.05e+1)

+

9.70e+1(1.87e+1)

7.82e+1(1.54e+1)

-

1.18e+2(2.41e+1)

+

1.01e+2(2.34e+1)

≈

1.10e+2(2.46e+1)

+

1.02e+2(1.97e+1)

≈

8.81e+1(1.18e+1)

≈

1.10e+2(2.29e+1)

+

DTLZ2

3.30e-1(7.23e-2)

+

1.72e-1(2.41e-2)

3.59e-1(2.82e-2)

+

3.08e-1(4.93e-2)

+

2.45e-1(3.57e-2)

+

3.36e-1(3.96e-2)

+

2.14e-1(4.10e-2)

+

3.64e-1(3.29e-2)

+

2.86e-1(6.31e-2)

+

3.15e+2(6.04e+1)

+

1.97e+2(1.64e+1)

2.30e+2(5.99e+1)

≈

3.24e+2(5.80e+1)

+

2.67e+2(6.58e+1)

+

2.82e+2(6.85e+1)

+

2.72e+2(6.88e+1)

+

2.09e+2(4.23e+1)

≈

2.98e+2(6.14e+1)

+

DTLZ4

7.51e-1(1.50e-1)

≈

7.96e-1(2.25e-1)

7.65e-1(1.14e-1)

≈

5.94e-1(1.28e-1)

-

6.30e-1(1.51e-1)

-

7.00e-1(1.48e-1)

-

5.64e-1(2.01e-1)

-

6.70e-1(8.05e-2)

-

5.23e-1(8.60e-2)

-

DTLZ5

2.47e-1(7.21e-2)

+

1.17e-1(2.08e-2)

2.83e-1(3.13e-2)

+

2.13e-1(5.55e-2)

+

1.61e-1(4.60e-2)

+

2.33e-1(3.65e-2)

+

8.64e-2(2.48e-2)

-

2.83e-1(3.00e-2)

+

1.18e-1(3.17e-2)

≈

DTLZ6

1.36e+0(4.10e-1)

≈

1.18e+0(5.35e-1)

1.78e+0(6.29e-1)

+

4.63e+0(6.26e-1)

+

3.37e+0(6.50e-1)

+

6.26e+0(3.28e-1)

+

4.61e+0(1.18e+0)

+

2.45e+0(7.92e-1)

+

2.92e+0(5.35e-1)

+

DTLZ7

4.22e-1(3.16e-1)

-

2.56e+0(1.86e+0)

5.34e-1(1.25e-1)

-

2.55e-1(4.36e-2)

-

5.54e-1(2.38e-1)

-

4.20e+0(1.11e+0)

+

2.21e+0(1.11e+0)

≈

6.21e-1(2.43e-1)

-

1.85e-1(1.81e-2)

-

+

/

≈

/

-

4/2/1

-/-/-

3/2/2

5/0/2

4/1/2

6/0/1

3/2/2

3/2/2

4/1/2

ParEGO

K-RVEA

CSEA

OREA

KTA2

ESBCEO

KMOEATIC

MOEAD-EGO

==- MOEAD-FS

.-MOEAD-FS(out)

0.7

700

1.2

200

0.6

009

1.0

Value

Value

0.5

IGD Value

500

IGD Value

150

0.8

IGD

IGD

0.4

400

100

0.3

300

0.6

0.2

200

0.4

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50

0

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IGD Value

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IGD Value

IGDValue

6-

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0.1

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50

E.4 Performance on Expensive Multi-Objective Optimization Under the Same Evaluation Budget

The statistical test results reported in the last row of Table 5 show that ParEGO [19] and OREA [48] are the best two comparison algorithms when compared with our MOEA/D-FS. In this subsection, we evaluate the performance of MOEA/D-FS when no extra evaluation is saved. For this purpose, we compare the optimization performance of these three SAEAs under the same evaluation budget: 10 evaluations (1 d ) for surrogate initialization and 50 evaluations for further optimization. The statistical test results are reported in Table 8. It can be seen that our MOEA/D-FS generally outperforms the

ParEGO

K-RVEA

CSEA

OREA

KTA2

ESBCEO

KMOEATIC

MOEAD-EGO

==- MOEAD-FS

.- MOEAD-FS(out)

0.5

0.04

0.04

FE'0

0.4

HV Value

0.02

HV Value

0.3

HV Value

0.02

HV Value

0.2

0.00

0.00

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0.1

-0.02

-0.02

0.1

0.04

-0.04

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0

20

40

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20

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0

20

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0

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0.06

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0.15

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HV Value

HV Value

HV Value

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0.10

0.02

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0.05

0.00

0.1

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-0.02

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0

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compared SAEAs when only 1 d evaluations are used to initialize their surrogates. The effectiveness of our FSEO framework has been demonstrated. Note that OREA is an evolutionary algorithm assisted by ordinal-regression-based surrogates. Currently, our FSEO framework is applicable to the SAEAs working with regression-based surrogates. The meta-learning of ordinal-regression models can be a topic of further research.

F Influence of Task Similarity

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In real-world applications, it is optimistic to assume that some related tasks are very similar to the 693 target task. A more common situation is that all related tasks have limited similarity to the target task. 694 To investigate the relationship between task similarity and FSEO optimization performance, we also 695 test the performance in an 'out-of-range' situation, where the original DTLZ is excluded from the 696 range of DTLZ variants during the MDKL meta-learning procedure. As a result, only the DTLZ 697 variants that are quite different from the original DTLZ problem can be used to learn experience. The 698 'out-of-range' situation eliminates the probability that MDKL surrogates benefit greatly from the 699

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DTLZ variants that are very similar to the original DTLZ problem. Detailed definitions of the related tasks used in the 'out-of-range' situation are given in Appendix C. Apart from the related tasks used, the remaining experimental setups are the same as the setups described in Section 5.1. For the sake of convenience, we denote the situation we tested in Section 5.1 as 'in-range' below.

The statistical test results reported in Table 9 show that the 'out-of-range' situation achieves competitive IGD+ values to the 'in-range' situation on all 7 test instances. This suggests that related tasks that are very similar to the target task have a limited impact on the optimization performance of our FSEO framework. Useful experience can be learned from related tasks that are not very similar to the target task. Crucially, when comparing the performance of the 'out-of-range' situation and that of MOEA/D-EGO, we can still observe competitive or improved optimization results on 6 DTLZ problems (see Table 9, the row titled by 'vs MOEA/D-EGO', or Fig. 3). Moreover, it can be seen from the last row of Table 9 that the 'out-of-range' situation achieves better/competitive/worse IGD+ values than all compared SAEAs on 27/7/8 test instances. In comparison, the corresponding statistical test results for the 'in-range' situation are 26/9/7. The difference between these statistical test results is not significant.

A study on the 'out-of-range' situation in the context of extremely expensive multi-objective optimization is presented in Appendix H.2. Consistent results can be observed from Table 12 and Fig. 7.

Consequently, related tasks that are very similar to the target task are not essential to the optimization performance of our FSEO framework. In the 'out-of-range' situation, our MOEA/D-FS can still achieve competitive or better optimization results than MOEA/D-EGO while using only 1 d samples for surrogate initialization.

G Influence of the Size of Datasets Used in Meta-Learning

We also investigated the performance of our FSEO framework when different sizes of datasets | D m | are used in the meta-learning procedure. The experimental setups are the same as the setups of MOEA/D-FS in Section 5.1 except for | D m | .

It is evident from Table 10 that when each DTLZ variant provides | D m | = 60 samples for the meta-learning of MDKL surrogates, the performance of both MOEA/D-FSs are improved on 2 or 3 DTLZ problems. Particularly, a significant improvement can be observed from the optimization results of DTLZ7. As we discussed in Section 5.1, the poor performance of our experience-based optimization on DTLZ7 is caused by the small size of D m . Optimal solutions have few chances to be included in a small D m , which makes D m fails to provide the experience about the discontinuity of optimal regions. In comparison, the experience of 'optimal regions' can be learned from large datasets D m and thus the optimization results are improved significantly.

In conclusion, for our FSEO framework, a large D m for the meta-learning procedure indicates 734 more useful experience can be learned from related tasks, which further improves the performance 735 of experience-based optimization. Therefore, when applying our FSEO framework to real-world 736

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optimization problems, it is preferable to collect more data from related tasks for experience learning.

H Experiments on Extremely Expensive Multi-Objective Optimization

In this section, we investigate the performance of our FSEO framework in the context of extremely expensive optimization, where the allowed fitness evaluations on target problems are fewer than that in the experiment carried out in Sections 5.1 of the main file and Appendix F.

H.1 Performance between Comparison Algorithms

We conduct the experiment described in Section 5.1 of the main file, but with a smaller evaluation budget than the budget listed in Table 2. The size of the initial dataset S ∗ is set to 10, 60 for our MOEA/D-FS and comparison algorithms, respectively. 30 extra evaluations for further optimization are allowed. The total evaluation budget is 40, 90 for our MOEA/D-FS and comparison algorithms, respectively.

The aim of this subsection is to answer the question below:

Is our FSEO framework more suitable for the optimization problems in which evaluations are extremely expensive? In other words, will the advantage of our FSEO framework become more prominent if the optimization problems allow a smaller evaluation budget?

The comparison results reported in Fig. 7 and Table 11 show that MOEA/D-FS has achieved competitive or smaller IGD+ values than MOEA/D-EGO on all DTLZ problems except for DTLZ7. Meanwhile, 5 d evaluations have been saved.

Consistent with the results discussed in Section 5.1 of the main file, MOEA/D-FS fails to achieve a 756 competitive result compared to MOEA/D-EGO on DTLZ7 since experience is learned from small 757 datasets collected from related tasks. Although we set a different evaluation budget for all SAEAs, 758 the size of datasets for meta-learning | D m | has not been modified. However, it can be observed from 759 the statistical test results (see the last row of Tables 5 and 11) that our MOEA/D-FS outperforms 760 the comparison algorithms on 26, 29 test instances when the total evaluation budget of comparison 761

ParEGO

K-RVEA

CSEA

OREA

KTA2

ESBCEO

KMOEATIC

：MOEAD-EGO

MOEAD-FS

.-MOEAD-FS(out)

700

60

200

0.6

600

0.8

IGD+Value

IGD+Value

0.5

IGD+ Value

500

IGD+Value

0.7

150

0.4

400

0.6

0.3

100

300

0.5

0.2

200

0.4

0

10

20

30

0

10

20

30

0

10

20

30

0

10

20

30

10

0.5

9

8

IGD+Value

0.4

IGD+Value

IGD+Value

6

0.3

4

4

0.2

2

2

0.1

F0

0

10

20

30

0

10

20

30

0

10

20

30

algorithms is set to 150, 90, respectively. This answers the question we raised before: The advantage 762 of our FSEO framework is more prominent in the extremely expensive problems where a smaller 763 evaluation budget is allowed. The comparison between the results obtained from Tables 5 and 11 has 764 demonstrated that our FSEO framework is preferable when solving optimization problems within a 765 very limited evaluation budget. 766

767

H.2 Out-Of-Range Analysis on Extremely Expensive Optimization

In Section F of the main file, we carried out an experiment to study the influence of task similarity 768 on the performance of experience-based expensive multi-objective optimization. The optimization 769 results obtained from the 'in-range' and the 'out-of-range' situations are compared. In this subsection, 770 we conduct an experiment to investigate the difference between the 'in-range' and the 'out-of-range' 771 situations for extremely expensive multi-objective optimization. The experimental setups are the 772 same as the setups described in Section F of the main file, except the allowed fitness evaluation budget 773 is the same as described in Appendix H.1. 774

Table 12 gives the statistical test results, it can be seen that the 'out-of-range' situation achieves 775 competitive IGD+ values to the 'in-range' situation on all 7 test instances. In comparison to MOEA/D776 EGO, the experience gained in the 'out-of-range' situation leads to competitive or smaller IGD+ 777 values on 6 DTLZ problems. Furthermore, similar results can be observed in the last row of Table 12, 778 the 'out-of-range' situation achieves better/competitive/worse IGD+ values than all compared SAEAs 779 on 28/9/5 test instances. In comparison, the 'in-range' situation achieves better/competitive/worse 780 IGD+ values than all compared SAEAs on 29/8/5 test instances. There is only a minor difference 781 between the optimization results obtained in two situations. These observations are consistent with 782 the conclusions we made in Section F of the main file. 783

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H.3 Result Tables and Figures in IGD and HV Metrics 784

Results in IGD values are reported in Table 13 and Fig. 8. A smaller IGD value indicates a better 785 optimization result.

Table 13: Mean IGD values and standard deviation (in parentheses) obtained from 30 runs on 7 DTLZ problems. MOEA/D-FS and comparison algorithms initialize their surrogates with 10, 60 samples, respectively. Extra 30 evaluations are allowed in the further optimization. ' + ', ' ≈ ', and ' -' denote MOEA/D-FS is statistically significantly superior to, almost equivalent to, and inferior to the compared algorithms in the Wilcoxon rank sum test (significance level is 0.05), respectively. The last row counts the total win/tie/loss results.

Results in HV values are reported in Table 14 and Fig. 9. A larger HV value indicates a better optimization result.

I Summary of Experiments

Our computational studies have demonstrated the following: First, we provide empirical evidence to show the effectiveness of learning experience: The meta-learning of neural network parameters and

ParEGO

K-RVEA

CSEA

OREA

KTA2

ESBCEO

KMOEATIC

—MOEAD-EGO

MOEAD-FS

.-MOEAD-FS(out)

0.7

700

1.2

200

0.6-

600

Value

IGD Value

0.5

Value

500

Value

1.0

150

IGD

0.4

IGD

400

IGD

0.8

100

0.3

300

0.6

0.2

200-

0

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IGD Value

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IGD Value

IGDValue

6

0.3

4

0.2

2

2

0.1

0

0

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30

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30

0

10

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30

ParEGO

K-RVEA

CSEA

OREA

KTA2

ESBCEO

KMOEATIC

MOEAD-EGO

==-MOEAD-FS

.-MOEAD-FS(out)

0.04

0.4

0.04

0.15

HV Value

0.02

HV Value

0.3

HV Value

0.02

0.10

0.00

0.00

0.2-

0.05

-0.02

-0.02

0.1

0.04

0.04

F000

0.0

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30

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0.04

0.15

0.3

HV Value

HV Value

0.02

HV Value

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0.2

0.05-

5

0.00

0.1

0.0

0.00

-0.02

0

10

20

30

0

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20

30

base kernel parameters are essential to the modeling accuracy of a MDKL model. As a result, our 792 MDKL model outperforms the compared meta-learning modeling and non-meta-learning modeling 793 methods on both the engine fuel consumption regression task and the sinusoid function regression 794 task. 795

Second, we demonstrate the main contribution of this work: In most situations, the proposed FSEO 796 framework can assist regression-based SAEAs to reach competitive or even better optimization 797

results, while the cost of surrogate initialization is only 1 d samples. Due to the effectiveness of 798 saving evaluations, our FSEO framework is preferable to other SAEAs when solving problems within 799 a very limited evaluation budget. Moreover, some empirical guidelines are concluded to help the 800 application of our FSEO framework. For the influence of task similarity, we find that related tasks that 801 are very similar to the target task are not necessary to the application of our approach. The influence 802 of these similar tasks on the optimization performance is limited. Our FSEO framework can achieve 803 competitive results without datasets from very similar related tasks. Besides, for the related tasks 804 used for meta-learning, we have demonstrated that more useful experience can be learned if more 805 data points are sampled from related tasks. 806

Third, the effectiveness of our FSEO framework is validated on a real-world engine calibration 807 problem. Competitive or better results are achieved on the objective and constraint functions, while 808 1 d samples are used to initialize surrogates. Therefore, our FSEO framework can also be applied to 809 optimization scenarios such as single-objective optimization and constrained optimization. 810

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5. Open access to data and code

Question: Does the paper provide open access to the data and code, with sufficient instructions to faithfully reproduce the main experimental results, as described in supplemental material?

Answer: [No]

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The answer NA means that paper does not include experiments requiring code.

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Providing as much information as possible in supplemental material (appended to the paper) is recommended, but including URLs to data and code is permitted.

6. Experimental Setting/Details

Question: Does the paper specify all the training and test details (e.g., data splits, hyperparameters, how they were chosen, type of optimizer, etc.) necessary to understand the results?

Answer: [Yes]

Justification: We have described all the details about of experiments.

Guidelines:

The answer NA means that the paper does not include experiments.

The experimental setting should be presented in the core of the paper to a level of detail that is necessary to appreciate the results and make sense of them.

The full details can be provided either with the code, in appendix, or as supplemental material.

7. Experiment Statistical Significance

Question: Does the paper report error bars suitably and correctly defined or other appropriate information about the statistical significance of the experiments?

Answer: [Yes]

Justification: We have conducted statistical tests in our experiments, error bars are plotted in figures.

Guidelines:

The answer NA means that the paper does not include experiments.

The authors should answer "Yes" if the results are accompanied by error bars, confidence intervals, or statistical significance tests, at least for the experiments that support the main claims of the paper.

The factors of variability that the error bars are capturing should be clearly stated (for example, train/test split, initialization, random drawing of some parameter, or overall run with given experimental conditions).

The method for calculating the error bars should be explained (closed form formula, call to a library function, bootstrap, etc.)

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The assumptions made should be given (e.g., Normally distributed errors).

It should be clear whether the error bar is the standard deviation or the standard error of the mean.

It is OK to report 1-sigma error bars, but one should state it. The authors should preferably report a 2-sigma error bar than state that they have a 96% CI, if the hypothesis of Normality of errors is not verified.

For asymmetric distributions, the authors should be careful not to show in tables or figures symmetric error bars that would yield results that are out of range (e.g. negative error rates).

If error bars are reported in tables or plots, The authors should explain in the text how they were calculated and reference the corresponding figures or tables in the text.

8. Experiments Compute Resources

Question: For each experiment, does the paper provide sufficient information on the computer resources (type of compute workers, memory, time of execution) needed to reproduce the experiments?

Answer: [No]

Justification: We did not provide information about compute workers and memory since our experiments do not have specific requirements on memory or other computation resource.

Guidelines:

The answer NA means that the paper does not include experiments.

The paper should indicate the type of compute workers CPU or GPU, internal cluster, or cloud provider, including relevant memory and storage.

The paper should provide the amount of compute required for each of the individual experimental runs as well as estimate the total compute.

The paper should disclose whether the full research project required more compute than the experiments reported in the paper (e.g., preliminary or failed experiments that didn't make it into the paper).

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Justification: Not applicable.

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The answer NA means that the authors have not reviewed the NeurIPS Code of Ethics.

If the authors answer No, they should explain the special circumstances that require a deviation from the Code of Ethics.

The authors should make sure to preserve anonymity (e.g., if there is a special consideration due to laws or regulations in their jurisdiction).

10. Broader Impacts

Question: Does the paper discuss both potential positive societal impacts and negative societal impacts of the work performed?

Answer: [No]

Justification: We do not think optimization algorithm can cause any negative social impacts. Guidelines:

The answer NA means that there is no societal impact of the work performed.

If the authors answer NA or No, they should explain why their work has no societal impact or why the paper does not address societal impact.

Examples of negative societal impacts include potential malicious or unintended uses (e.g., disinformation, generating fake profiles, surveillance), fairness considerations (e.g., deployment of technologies that could make decisions that unfairly impact specific groups), privacy considerations, and security considerations.

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The conference expects that many papers will be foundational research and not tied to particular applications, let alone deployments. However, if there is a direct path to any negative applications, the authors should point it out. For example, it is legitimate to point out that an improvement in the quality of generative models could be used to generate deepfakes for disinformation. On the other hand, it is not needed to point out that a generic algorithm for optimizing neural networks could enable people to train models that generate Deepfakes faster.

The authors should consider possible harms that could arise when the technology is being used as intended and functioning correctly, harms that could arise when the technology is being used as intended but gives incorrect results, and harms following from (intentional or unintentional) misuse of the technology.

If there are negative societal impacts, the authors could also discuss possible mitigation strategies (e.g., gated release of models, providing defenses in addition to attacks, mechanisms for monitoring misuse, mechanisms to monitor how a system learns from feedback over time, improving the efficiency and accessibility of ML).

11. Safeguards

Question: Does the paper describe safeguards that have been put in place for responsible release of data or models that have a high risk for misuse (e.g., pretrained language models, image generators, or scraped datasets)?

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Justification: Code will be released after acceptation, it would be open access, no safeguards are required.

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The answer NA means that the paper poses no such risks.

Released models that have a high risk for misuse or dual-use should be released with necessary safeguards to allow for controlled use of the model, for example by requiring that users adhere to usage guidelines or restrictions to access the model or implementing safety filters.

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If this information is not available online, the authors are encouraged to reach out to the asset's creators.

13. New Assets

Question: Are new assets introduced in the paper well documented and is the documentation provided alongside the assets?

Answer: [NA]

Justification: We did not introduce any new assets.

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The answer NA means that the paper does not release new assets.

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Peer reviews:
{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThis paper introduce a meta-learning framework into few-shot optimization to assist the surrogate modelling in expensive evaluation setting. The authors parameterize a mapping function to get the hidden feature of the solution space and then integrate such mapping into a gaussian kernel function as a deep kernel. They then facilitate meta-training of the proposed deep kernel over a group of related tasks to attain an experience model, by maximizing the posterior likelihood. During the online optimziation of the target task, the experience model is firstly adpated to the new task in the same way above and then updated acoording to its accuracy in terms of the predicted objective value. The experimental results show that the proposed framework achieves competitive performance against some strong baselines over EMOPs and ECOPs benchmarks.\n\nSTRENGTHS:\n1. The idea of integrating meta-learning into the kernel-learning based surrotgate methods is novel, and might improves the surrogate-based optimziation towards generalizable setting.\n\n2. The expriments result show that the proposed FSEO framework is at least competitive with the existing baselines, which is acceptable and should be encouraged for further development.\n\n3. The overall writing is clear.\n\nWEAKNESSES:\nBefore the next round of author-reviewer rebuttal, following concerns exist:\n1. Given that the likelihhod-based loss function (Eq. 4) should be maximized to fit the samples from all of the related tasks, why its update should follow a gradient descent rather a gradient ascent? Correct me if I was wrong.\n\n2.  line 144 ~ 146, the authors state that the U update interations roots from the smaller number of available related tasks. I can not understand the reason behind, can you explain it more?\n\n3. The neural network $\\phi$ used in the deep kernel function is a 2-layer MLP, which limits the FSEO to meta-learn surrogate function among the related tasks with the same slution dimension. However, in practice, related tasks might not share the same dimension. I would appreciate the authors to provide realistic scenarios where FSEO is eefective. Besides, the effectiveness of the FSEO on traditional single-objective tasks should also be verified to make it more convincing that FSEO is a general framework.\n\n4. Although the overall writing of this paper is not bad, it is still difficult for less-skilled readers to understand the whole picture. In particular, the content in Section 3.2 and Section 3.3 should be carefully refined to make sure the potential readers fully understand how the DKL and MDKL operate. For now, it is too simple and ambiguous.\n\nQUESTIONS:\nsee Weaknesses.\n\nLIMITATIONS:\nThe limitations listed in Conclusion are quite brief. I would appreciate the authors to explain more about the two limitations listed here.", "TITLE": "Official Review", "RECOMMENDATION": 4, "REVIEWER_CONFIDENCE": 3, "SOUNDNESS_CORRECTNESS": 2, "DATE": "2024-07-05"}

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThis paper proposes Meta Deep Kernel Learning (MDKL), a new surrogate for SAEAs. MDKL consists of a deep kernel with meta-learning. Empirical studies demonstrate its effectiveness in expensive multi-objective optimization and constrained optimization.\n\nSTRENGTHS:\n1. This paper is well-written and easy to follow. The technical details are well presented.\n2. This paper extends deep kernel and meta-learning-based surrogates into evolutionary algorithms.\n3. This paper investigated multi-objective optimization and constrained optimization.\n\nWEAKNESSES:\nMeta-learned deep kernel surrogates have already been well-studied in Bayesian Optimization [1]. The authors are also aware of this as they mentioned in Related Work. I think this paper does not present significant new advancements based on the previous work.\n\nFirst, the authors claim that MDKL is specially designed for optimization, while the previous work is not. In this regard, I do not see many differences between MDKL and previous meta-learned deep kernels. The authors claim that the advantage of MDKL lies in continuous adaptation; however, most models support parameter updates or fine-tuning. The authors also did not sufficiently explain the relationship between continuous adaptation and optimization problems, or what significance it has for optimization problems.\n\nSecond, the authors propose that one of the novelties of this paper is taking expensive multi-objective optimization problems (EMOPs) and expensive constrained optimization problems (ECOPs) into account. MDKL, as a surrogate, can be integrated into almost any expensive optimization algorithm. It seems to be able to cooperate with Bayesian optimization as well. The authors simply replaced the surrogate in a multi-objective optimization algorithm with MDKL and conducted some experiments, without providing any new analysis, insights, or proposing any new methods specifically for MOPs or COPs. Therefore, I believe this paper does not make a significant contribution to solving EMOPs and ECOPs.\n\n[1] Martin Wistuba and Josif Grabocka. Few-shot Bayesian optimization with deep kernel surrogates. ICLR 2021.\n\nQUESTIONS:\n1. Regarding MOPs, this paper only includes results on synthetic problems. I recommend using some real-world instances, such as NAS and Hyperparameter tuning.\n2. Table 14 and Fig. 9. The HV values are all 0, indicating that the reference point is set too low.\n3. P2, Line 64. MOPs and COPs can also be global optimization problems.\n4. P5, Algorithm 3, Line 3. How are the increments initialized?\n\nLIMITATIONS:\nNo concerns.", "TITLE": "Official Review", "RECOMMENDATION": 4, "REVIEWER_CONFIDENCE": 4, "SOUNDNESS_CORRECTNESS": 2, "DATE": "2024-07-08"}

comments: SUMMARY:
The authors developed a few-shot evolutionary optimization framework to effectively solve the multi-objective EOPs and constrained EOPs.

STRENGTHS:
The proposed method can solve the multi-objective EOPs and constrained EOPs with little data, especially for the engineering problems.

WEAKNESSES:
The learning results may rely on the relation degree of different tasks.

QUESTIONS:
(1)  How to define the relation degree of the related task T1-Tn. If the tasks are highly related, it is relatively easy to get a good result.
(2)  Though the performance can be improved compared to some basic algorithms such as MOEA/D-EGO and constrained-EGO, the computational cost of the proposed method is also increased. Give more details about the overhead of the algorithm.
(3)  The proposed method uses the same constraint handling method as con_FS. Why can the proposed method find more feasible solutions in the right figure of Figure 4.
(4)  The complexity of the proposed algorithm should be analyzed.

LIMITATIONS:
The overhead of the algorithm need to be further decreased.
TITLE: Official Review
SOUNDNESS_CORRECTNESS: 3
DATE: 2024-07-12

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThis paper proposes a new surrogate-assistant evolutionary algorithm that utilizes a Gaussian process with Deep Kernel Learning as the surrogate model. The method employs few-shot meta-learning to learn from multiple tasks to construct the surrogate. It is then integrated with the existing MOEA/D-EGO algorithm to create a new approach. Experiments are conducted on the DTLZ benchmark problems and a gasoline motor engine calibration problem to evaluate the performance of the proposed algorithm.\n\nSTRENGTHS:\n1. The proposed method is instantiated and tested in expensive multi-objective optimization and constrained optimization scenarios.\n2. A real-world problem is considered in the experiments.\n\nWEAKNESSES:\n1. Many important algorithmic details are unclear. For instance, the main distinction between the proposed MDKL and the existing DKL algorithms is its ability to learn from a set of related tasks, yet its implementation is not clearly explained. How parameters from different source tasks collectively form the experience, and how parameters from both source and target tasks jointly create this experience, are not clearly addressed.\n2. The effectiveness of the algorithm is primarily tested on expensive multi-objective optimization problems, but state-of-the-art algorithms in this field were not selected for comparison.\n\nQUESTIONS:\n1. Two variables, theta and p, are used to describe task-independent parameters. What are they, and what is the difference between them?\n2. According to Algorithm 1 and Algorithm 3, task-independent parameters are only used to determine whether the condition in line 2 is met. They have no other function. How do the source tasks improve the algorithm's performance?\n3. How is the initialization of increments done in Algorithm 2 and Algorithm 3?\n\nLIMITATIONS:\nEven if a clear definition of task similarity cannot be provided, it is recommended to offer some hints to help users apply the algorithm.", "TITLE": "Official Review", "RECOMMENDATION": 4, "REVIEWER_CONFIDENCE": 3, "SOUNDNESS_CORRECTNESS": 2, "DATE": "2024-07-14"}