Abstract
Surrogateassisted optimization algorithms are a commonly used technique to solve expensiveevaluation problems, in which a regression model is built to replace an expensive function. In some acquisition functions, the only requirement for a regression model is the predictions. However, some other acquisition functions also require a regression model to estimate the “uncertainty” of the prediction, instead of merely providing predictions. Unfortunately, very few statistical modeling techniques can achieve this, such as Kriging/Gaussian processes, and recently proposed genetic programmingbased (GPbased) symbolic regression with Kriging (GP2). Another method is to use a bootstrapping technique in GPbased symbolic regression to estimate prediction and its corresponding uncertainty. This paper proposes to use GPbased symbolic regression and its variants to solve multiobjective optimization problems (MOPs), which are under the framework of a surrogateassisted multiobjective optimization algorithm (SMOA). Kriging and random forest are also compared with GPbased symbolic regression and GP2. Experiment results demonstrate that the surrogate models using the GP2 strategy can improve SMOA’s performance.
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1 Introduction
A common remedy to expensive optimization problems is to replace exact evaluations with approximations learned from past evaluations. Theoretically, any supervised learning techniques can be used for building up surrogate models, for instance, Gaussian processes or Kriging [16], random forest [7], supported vector machine [4], genetic programming based (GPbased) symbolic regression [9], to name a few. Among these techniques, the Kriging/Gaussian process is appealing both in academic studies and in applications due to its estimation of both prediction and the corresponding uncertainty as outputs. The utilization of these two outputs allows a balancing property of exploitation and exploration in an acquisition function (a.k.a infill criteria in some papers) during the optimization processes.
Compared with the blackbox modeling technique of Kriging, the transparentbox modeling technique has been a hot topic recently due to its ability to interpret the relationship between inputs and predictions. Among many techniques of transparentbox modeling, GPbased symbolic regression (SR) attracts many researchers’ attention, as it searches the space of all possible mathematical formulas for concise or closedform mathematical expressions that best fit a dataset [1]. To quantify prediction uncertainty in GPbased symbolic regression, a simple technique is to utilize bootstrapping, which is widely used in the machine learning field. Bootstrapping is usually used in ensemble methods to reduce the bias of a (simple) predictor by using several predictors on different samples of a dataset and averaging their predictions. The set of predictions from each predictor can be used to estimate the properties of a predictor (such as prediction mean and its variance) by measuring those properties when sampling from an approximating distribution. For example, this technique is incorporated into a random forest (RF) in Sequential ModelBased Algorithm Configuration (SMAC) [12], where the variance is calculated using the predictions of the trees of the random forest. Following the same idea, the prediction variance of GPbased symbolic regression can also be estimated by the method of bootstrapping. Such similar works can be found in [2, 8, 10]. Another technique to quantify the uncertainty of the prediction’s residual is achieved in [19], in which a socalled GPbased symbolic regression with Kriging (GP2) was proposed by summing a GPbased symbolic expression and one additional residual term to regulate the predictions of the symbolic expression, where the residual term follows a normal distribution and is estimated by Kriging.
The techniques mentioned above allow GPbased symbolic regression to employ acquisition functions that balance exploration and exploitation during the optimization process. Under the framework of surrogateassisted multiobjective optimization algorithm (SMOA), this paper utilizes the most recently modeling technique GP2 [19] and GPbased symbolic regression incorporated with a bootstrapping technique as the surrogate models to solve multiobjective optimization problems (MOPs). These modeling techniques are compared with two stateoftheart modeling techniques, Kriging, and random forest.
The main contribution of this paper is extending the surrogate models that allow an acquisition function’s ability to balance exploration and exploitation in the field of SMOA. This paper is structured as follows: Sect. 2 describes the preliminaries of multiobjective optimization problems, and introduces the framework of surrogateassisted multiobjective optimization algorithm; Sect. 3 explains three different techniques for surrogate models; Sect. 4 introduces the definition of upper confidence bound (UCB) [6, 17] and its relating concepts; Sect. 5 shows the parameter settings and discusses the experiment results.
2 SurrogateAssisted Multiobjective Optimization
This section introduces the preliminaries of the SMOA, including the MOP’s definition in Sect. 2.1, and the framework of SMOA in Sect. 2.2.
2.1 Multiobjective Optimization Problem
A continuous MOP^{Footnote 1} is defined as minimizing multiple objective functions simultaneously and can be formulated as:
where m is the number of objective functions, \({f}_i\) stands for the ith objective functions \(f_i: \mathcal {X} \rightarrow \mathbb {R}\), \(i = 1, \dots , m\), and \(\mathcal {X}\) is a decision vector subset.
2.2 Framework of SurrogateAssisted Multiobjective Optimization
The fundamental concept of SMOA is to firstly build a surrogate model \(\mathcal {M}\) to reflect the relationship between a decision vector \(\textbf{x} = (x_1, \cdots , x_d)\) and its each corresponding objective value \(y_i = f_i(\textbf{x}), i\in \{1,\cdots ,m\}\) for each objective. In SMOA, it is usually assumed that objective functions are mutually independent in an objective space^{Footnote 2}.
SMOA starts with sampling an initial design of experiment (DoE) with a size of \(\eta \) (line 2 in Algorithm 1), \(\textrm{X}=\{\textbf{x}^{(1)},\textbf{x}^{(2)},\) \(\ldots , \textbf{x}^{(\eta )}\} \subseteq \mathcal {X}\). By using the initial DoE, \(\textrm{X}\) and its corresponding objective values, \(\textbf{Y}=\{\textbf{f}(\textbf{x}^{(1)}),\textbf{f}(\textbf{x}^{(2)}),\cdots ,\textbf{f}(\textbf{x}^{(\eta )})\} \subseteq \mathbb {R}^{m\times \eta }\) (line 3 in Algorithm 1), can be then utilized to construct surrogate models \(\mathcal {M}_i\). Then, an SMOA starts with searching for a decision vector set \( \textbf{x} '\) in the search space \(\mathcal {X}\) by maximizing the acquisition function \(\mathscr {A}\) with parameters of \(\gamma \) and surrogate models (line 7 in Algorithm 1). Here, an acquisition function quantifies how good or bad a point \( \textbf{x} \) is in objective space. In this paper, \(\mathscr {A}\) is the upper confidence bound (described in Sec. 4) due to its ability to balance exploration and exploitation, low computational complexity, and popularity in the deep learning fields.
By maximizing the acquisition function, a singleobjective optimization algorithm is deployed to search for the optimal decision vector \( \textbf{x} ^*\). In this paper, we use BIpopulation CMAES to search for the optimal \( \textbf{x} ^*\) due to the favorable performance on BBOB function testbed [11]. The optimal decision vector \( \textbf{x} ^*\) will then be evaluated by the ‘true’ objective functions \( \textbf{f} \). The surrogate models will be retrained by using the updated \(\textrm{X}\) and \(\textbf{Y}\). The main loop (as shown in Algorithm 1 from line 6 to line 12) will not stop until a stopping criterion is satisfied. Common stopping criteria include a number of iterations, convergence velocity, etc. In this paper, we specify the stopping criterion (\(T_c\)) as a number of function evaluations.
3 Surrogate Models
This section introduces a common surrogate model, Kriging in Sect. 3.1; a canonical GPbased symbolic regression in Sect. 3.2; and the GPbased symbolic regression with Kriging (GP2) in Sect. 3.3. Additionally, the method of quantifying prediction uncertainty is introduced in each subsection.
3.1 Kriging
Kriging is a statistical interpolation method and has been proven to be a popular surrogate model to approximate noisefree data in computer experiments. Considering a realization of y at n locations, expressed as the following vector \(\textbf{f}(\textbf{X}) =( {f}(\textbf{x}^{(1)}) \cdots , {f}(\textbf{x}^{(n)}) )\) and \(\textbf{X}=\{\textbf{x}^{(1)}, \cdots , \textbf{x}^{(n)}\} \subseteq \mathcal {X}\), Kriging assumes \(\textbf{f}(\textbf{X})\) to be a realization of a random process f of the following form [5, 13]:
where \(\mu (\textbf{x})\) is the estimated mean value and \(\epsilon (\textbf{x})\) is a realization of a Gaussian process with zero mean and variance \(\sigma ^2\).
The regression part \(\mu (\textbf{x})\) approximates the function f(.) globally. The Gaussian process \(\epsilon (\textbf{x})\) takes local variations into account and estimates the uncertainty quantification. The correlation between the deviations at two decision vectors (\(\textbf{x}\) and \(\mathbf {x'}\)) is defined as:
where R(., .) is the correlation function that decreases with the distance between two points (\( \textbf{x} \text { and } \textbf{x} '\)).
It is common practice to use a Gaussian correlation function (a.k.a., a squared exponential kernel):
where \(\theta _i \ge 0\) are parameters of the correlation model to reflect the variables’ importance [21]. The covariance matrix can then be expressed through the correlation function: \(Cov({\epsilon (\textbf{x})}) = \sigma ^2 \mathbf {\Sigma }\), where \( \mathbf {\Sigma }_{i,j}=R(\textbf{x}_i,\textbf{x}_j)\).
When \(\mu (\textbf{x})\) is assumed to be an unknown constant, the unbiased prediction is called ordinary Kriging (OK). In OK, the Kriging model determines the hyperparameters \(\mathbf {\theta } = (\theta _1, \theta _2, \cdots , \theta _n)\) by maximizing the likelihood over the observed dataset. The expression of the likelihood function is: \(L = \frac{n}{2}\ln ({\sigma }^2)  \frac{1}{2}\ln (\mathbf {\Sigma }) \).
Uncertainty Quantification: The maximum likelihood estimations of the mean \(\hat{\mu }\) and the variance \(\hat{\sigma }^2\) can be expressed by: \( \hat{\mu } = \frac{\textbf{1}^{\top }_n \mathbf {\Sigma }^{1} \textbf{y}}{\textbf{1}^{\top }_n \mathbf {\Sigma }^{1} \textbf{1}_n} \) and \( \hat{\sigma }^2 = \frac{1}{n}(\textbf{y}  \textbf{1}_n\hat{\mu })^{\top }\mathbf {\Sigma }^{1}(\textbf{y}\textbf{1}_n\hat{\mu }) \). Then the prediction of the mean and the variance of an OK’s prediction at a target point \(\textbf{x}^t\) can be derived as follows [13, 20]:
where \(\textbf{c} = (Corr(y(x^t),y(x_1)), \cdots , Corr(y(x^t),y(x_n)))^{\top }\).
3.2 Genetic Programming Based Symbolic Regression
Genetic programming (GP) [15] is a typical evolutionary algorithm to solve optimization problems that can be formulated as: \(\mathop {\mathrm {arg\,min}}\limits _{\textbf{x}} f(\textbf{x}), \textbf{x} \in \mathcal {X}\), where \(\textbf{x} = (x_1, \cdots , x_n)\) represents a decision vector (also known as individual) in evolutionary algorithms (EAs). Similar to other EAs, GP evolves a population of solution candidates Pop(.) by following the principle of the survival of the fittest and utilizing biologicallyinspired operators. Algorithm 2 is the pseudocode of GP, where Variation operator includes crossover/recombination and mutation operators.
The feature that distinguishes GP from other evolutionary algorithms is the variablelength representation for \(\textbf{x}\). The search space of GPbased symbolic regression problems is a union of a function space \(\mathcal {F}\) and a terminal space \(\mathcal {T} = \{\mathcal {S} \cup \mathbb {R} \} \), where \(\mathcal {S}\) represents a variable symbol space (e.g., \(x_1\)) and \(c \in \mathbb {R}\) represents a constant number. The function set \(\mathcal {F}\) includes available symbolic functions (e.g., \(\{+, , \times , \div , \exp , \log \}\)).
To distinguish the objective function and decision variables in MOPs, \(f_{sr}(\cdot )\) and \(\textbf{t}\) are used to represent the objective function in GPbased symbolic regression and a GP individual for symbolic regression problems, respectively. The objective function \(f_{sr}(\cdot )\) can be any performance metrics for supervised learning problems. For instance, mean squared error (MSE), the Pearson correlation coefficient (PCC), Coefficient of determination (R2), etc. In this paper, MSE, PCC, and R2 are used as the objective functions in GPbased symbolic regression, and they are defined as:
\(\text{ MSE } = \frac{1}{n}\sum _{i=1}^{n}(y_i  \hat{y}_i)^2\), \(\text{ PCC } = \frac{ \sum _{i=1}^{n}(y_i\bar{y})(\hat{y}_i\bar{\hat{y}}) }{\sqrt{\sum _{i=1}^{n}(y_i\bar{y})^2}\sqrt{\sum _{i=1}^{n}(\hat{y}_i\bar{\hat{y}})^2}}\), \(\text{ R2 } = 1  \frac{\sum _{i=1}^{n}(y_i  \hat{y}_i)^2}{\sum _{i=1}^{n}(y_i  \bar{y})^2}\),
where n is the number of samples, \(y_i\) and \(\hat{y_i}\) represent the real response and predicted response of the ith sample, \(\bar{y} = \frac{1}{n}\sum _{i=1}^n y_i\), and \(\bar{\hat{y}} = \frac{1}{n}\sum _{i=1}^n \hat{y}_i\).
Therefore, a GPbased symbolic regression problem in this paper can be well formulated as:
Uncertainty Quantification: Similar to other bootstrapping techniques, each individual in the population of the GPbased symbolic regression is trained on a list of bootstrap samples. The variance error is estimated according to the error on the bootstrap samples. In this paper, the detailed procedure of bootstrapping for GPbased symbolic regression is as follows:

1.
Resampling \(n_b \le n\) samples (to form a new dataset \(D'\)) from original training dataset D of n samples;

2.
Train the GPbased symbolic regression model (\( \textbf{t} \)) based on dataset \(D'\);

3.
Repeat (1) to (2) for \(n_{pop}\) times;

4.
The bagging prediction and variance at a query vector \( \textbf{x} ^t\) are then computed by the mean and variance over all \(n_{pop}\) predictions:
$$\begin{aligned}&\mu _{sr}( \textbf{x} ^t) = \frac{1}{n_{pop}}\sum _{i=1}^{n_{pop}} \textbf{t} _i( \textbf{x} ^t) \end{aligned}$$(5)$$\begin{aligned}&\sigma _{sr}^2( \textbf{x} ^t) = \frac{1}{n_{pop}} \sum _{i=1}^{n_{pop}} \big ( \textbf{t} _i( \textbf{x} ^t)  \mu _{sr}( \textbf{x} ^t ) \big ) ^2 \end{aligned}$$(6)
3.3 GPBased Symbolic Regression with Kriging
GPbased symbolic regression with Kriging (GP2) integrates GPbased symbolic regression and Kriging by summing a bestfitted expression (\(symreg( \textbf{x} )\)) on a training dataset and a residual term. The prediction of GP2 can be expressed as:
where \(res(\textbf{x})\) is the residualregulating term that follows a normal distribution and is obtained by Kriging.
The pseudocode of the GP2 is shown in Algorithm 3. The algorithm starts from generating the training dataset D by computing the real responses \(\textbf{Y}\) of the decision set X. The bestfitted symbolic expression \(symreg( \textbf{x} )\) on dataset D is generated by GPbased symbolic regression (at line 4 in Algorithm 3). The residuals of the bestfitted symbolic expression \(symreg( \textbf{x} )\) on \(\textbf{X}\) can be computed by \(\textbf{Y}  \hat{\textbf{Y}}\). \(\textbf{X}_{gp}\) (line 6 in Algorithm 3) represents the variables that affect the residuals of \(symreg( \textbf{x} )\). The most intuitive way is to set \(\textbf{X}_{gp}\) as \(\textbf{X}\). In addition, it is also reasonable to set \(\textbf{X}_{gp}\) as \(\hat{\textbf{Y}}\) and \(\{ \textbf{X} \cup \hat{\textbf{Y}}\}\), due to the fact that residual of \(symreg( \textbf{x} )\) highly depends on the the prediction \(\hat{\textbf{Y}}\). Then, Kriging models the relationship between the \(\textbf{X}_{gp}\) and the residuals of \(symreg( \textbf{x} )\) in the dataset D (at line 7 in Algorithm 3). The residual of \(symreg( \textbf{x} )\), noted as \(res( \textbf{x} )\), also depends on \(\textbf{X}\), as \(\textbf{X}_{gp}\) depends on \(\textbf{X}\). The final surrogate model built by the GP2, noted as \(\text {GP}^2_{symreg}( \textbf{x} )\), is the sum of \(symreg( \textbf{x} )\) and its corresponding residual distribution \(res( \textbf{x} )\).
Uncertainty Quantification: The prediction of the symbolic expression of GP2 is a sum of symbolic expression and residual term that follows a normal distribution. The prediction and variance at a query vector \( \textbf{x} ^t\) by using the GP2 strategy are:
where \(s_{sr}^2( \textbf{x} ^t)\) is the variance of the symbolic regression expression \(symreg( \textbf{x} )\) at \( \textbf{x} ^t\).
In this paper, we argued \(s_{sr}^2( \textbf{x} ) = 0\) due to the fact that \(symreg( \textbf{x} )\) is an explicit mathematical expression and there is no variance to \(symreg( \textbf{x} )\) at an arbitrary point \( \textbf{x} \). Therefore, Eq. (9) can be simplified as \(\sigma ^2_{GP2}( \textbf{x} ^t) = \sigma ^2_{ok}( \textbf{x} ^t)\).
4 Acquisition Functions
The Hypervolume Indicator (HV), introduced in [22], is an essentially unary indicator for evaluating the quality of a Paretofront approximation set in MOPs due to its Pareto compliant property^{Footnote 3} and no requirement of inadvance knowledge of the Pareto front. The maximization of HV leads to a highqualified and diverse Paretofront approximation set. The Hypervolume Indicator is defined as follows:
where \(\lambda _m\) is the Lebesgue measure and the reference point \( \textbf{r} \) clips the space to ensure a finite volume.
The Hypervolume improvement (HVI) measures the improvement of a new solution in terms of HV and is defined as:
The Upper confidence bound of the HVI (UCB) [6, 17] is an indicator based on the HVI and prediction uncertainty in a naïve way. Denoting \( \boldsymbol{\mu } ( \textbf{x} )=(\mu _1( \textbf{x} ), \ldots , \mu _m( \textbf{x} ))\) as the prediction and \({ \boldsymbol{\sigma } }( \textbf{x} )=({\sigma }_1( \textbf{x} ),\ldots , {\sigma }_m( \textbf{x} ))\) as the uncertainty of m independent surrogate models, the hypervolume improvement of upper confidence bound of a solution at \( \textbf{x} \), i.e., \({ \boldsymbol{\mu } }( \textbf{x} )  \omega { \boldsymbol{\sigma } }( \textbf{x} )\), \(\omega \in \mathbb {R}_{\ge 0}\) specifies the confidence level:
The parameter \(\omega \) controls the balancing weight between exploration and exploitation, and a positive \(\omega \) rewards the high prediction uncertainty.
Example 1
Two examples of the landscape of HVI and UCB are shown in Fig. 1. The Paretofront approximation set is denoted as \(\mathcal{P}\mathcal{F}= \{{\textbf{y}^{(1)}} = (1,3),\) \(\textbf{y}^{(2)}=(2, 2), \textbf{y}^{(3)} = (3, 1)\}\), the reference point is \(\textbf{r}=(4,4)\), and the same uncertainty (\(\boldsymbol{\sigma } = (1,1)\)) is used for every point in the objective space, and \(\omega \) is set as 1 for the UCB. The left and right figures show the landscape of HVI and UCB, respectively, in the objective space that ranges from −1 to 5 for both \(f_1\) and \(f_2\).
5 Experiments
In this paper, the algorithms are performed on ZDT series problems [23] with the parameters of \(d=5\), \(m=2\), and a fixed reference point \(\textbf{r}=[15,15]\). In this paper, four categories of surrogate models are compared, including Kriging (in Alg. I), five different GP2 variants by using different configurations (in Alg. II − VI), random forest (in Alg. VII), and GPbased symbolic regression (in Alg. VIII and IX). Each experiment consists of 15 independent runs.
5.1 Parameter Settings
The common parameters in all algorithms include DoE size \(\eta =30\) and the function evaluation budget \(T_c=200\). The stopping criteria of \(f_{sr}\) is 0.1, 0.99, 0.99 for MSE, PCC, and R2 metrics, respectively. The function evaluation of genetic programming in GPbased symbolic regression and GP2 is 4E3 [19] and 2E4 [18], respectively.
The parameter \(\omega \) of UCB is set as \(\sqrt{g /\log (g)}\) [14], where g is the number of function evaluation. The function space \(\mathcal {F}\) for symbolic regression is \(\{+, , \times , \div , \sin , \cos , \text{ pow } \}\). The acquisition function is the UCB and is optimized by CMAES in all algorithms. The hyperparameters of CMAES follow: the number of restarts is 3, the number of initialized population size is 7, the maximum generation number is 2000, and the remaining parameters are set in default.
Table 1 shows other parameter settings in each algorithm, where \(n_{pop}\) represents the number of estimators in a random forest. The other parameters (e.g., \(p_m\), \(p_c\), etc.) are set as default values in GPLearn [18].
5.2 Empirical Results
In this section, we evaluate the Paretofront approximation sets by using the HV indicator. Figure 2 shows the HV convergence curves of 15 independent runs on ZDT problems. To depict a slight difference in HV values, we discuss the average \(\log (\varDelta HV)\) convergence of HV relative to the reference HV value of 250 in this paper. Notice that the scale varies in different problems since all problems have different ranges of the Pareto fronts.
From Fig. 2, it is clear to see that Alg. VII of using the random forest as the surrogate model yields the worst experiment results w.r.t. mean HV and its corresponding standard deviation. Compared with Alg. I and other algorithms, the poor performance of Alg. VII is expected, because of the introduction of Kriging in other algorithms (except for Alg. VIII and IX), of which prediction is more accurate than that of the random forest when the training dataset has few samples. Compared with GPbased symbolic regression algorithms (Alg. VIII and IX), which use ‘plain’ GP, Alg. VII still can not outperform them, due to the efficient optimization mechanism in genetic programming.
On singlemodal convex Paretofront problem ZDT1, Fig. 2 shows that GPbased symbolic regression algorithms (Alg. VIII and IX) are only slightly better than random forest in Alg. VII, but are worse than the other algorithms of using Kriging. The performances of the algorithms (Alg. I – VI) are similar w.r.t. mean HV value over 15 runs. In addition, Alg. I, which merely uses Kriging as the surrogate model, converges the fastest at the early stage of optimization among all the test algorithms. Similarly, Alg. I also converges much faster than most of the test algorithms on the other two singlemodal concave Paretofront problems (ZDT2 and ZDT6). An explanation is the relatively simple landscape of these three test problems.
On discontinued Paretofront problem ZDT3 and a multimodal Paretofront problem ZDT4, the algorithms’ performance differs greatly. Firstly, the convergence of Alg. I is much slower than the other algorithms at the beginning stage of optimization. Since a relatively large^{Footnote 4} reference point is used in this paper, the HVI introduced by extreme solutions is definitely much larger than that of a knee point. Thus, we can conclude that the strategy of Kriging with UCB is difficult to search for extreme solutions at the early stage of optimization. Consequently, this will slow the convergence of Alg. I. Additionally, another intuitive explanation is that the landscape of the UCB based on Kriging models on ZDT3 and ZDT4 is not smooth enough to allow the optimizer CMAES to jump out of local optima. In other words, the UCB is hard to measure slight differences around the local optima based on Kriging’s prediction because of the landscape’s roughness.
All algorithms’ ranking (w.r.t. mean HV value over 15 runs) on each problem is shown in Table 2. From this table, it is easy to conclude that Alg. V, which utilizes GP2 method of \( \textbf{X} _{gp}=\{ \textbf{X} \cup \hat{ \textbf{Y} } \} \) and \(f_{sr}=\text{ R2 }\), outperforms the other algorithms, and Alg. IV (GP2 method of \( \textbf{X} _{gp}=\{ \textbf{X} \} \) and \(f_{sr}=\text{ R2 }\)) is the second best algorithm. Since the search space of \(\textbf{x}\) of surrogate models in Alg. IV is the same search space of the objective function \( \textbf{f} ( \textbf{x} )\) of MOPs, Alg. IV is more reasonable to interpret the residual regulation term \(res( \textbf{x} )\) (in Eq. (7)) when it is compared with Alg. V. Additionally, there is no significant difference between the performance of Alg. IV and Alg. V on the test problems, see Table 3 for details. Considering these two aspects, it is recommended to use Alg. IV when the acquisition function is the UCB.
Table 3 shows the performance of pairwise Wilcoxon’s RankSum test (\(+/\approx /\)) matrix among all test algorithms on the test problems. The sum of Wilcoxon’s RankSum test (sum of \(+/\approx /\)) confirms that Alg. V performs best, as it significantly outperforms 23 pairwise instances between algorithms and problems. Besides, compared with Algo. I, most of the GP2 methods (Alg. II – Alg. V) perform at least similar to or significantly better than Alg. I.
6 Conclusion and Future Work
This paper utilizes two different techniques to quantify the prediction uncertainty of GPbased symbolic regression, namely, a bootstrapping technique in GPbased symbolic regression and using the Kriging to estimate the residual uncertainty in GP2. Two surrogate modeling techniques, GPbased symbolic regression, and GP2 of different configurations, are compared with Kriging and random forest on five stateoftheart MOP benchmarks. The statistical results show the effectiveness of GP2 w.r.t. mean \(\varDelta \)HV convergence. Among five different variants of GP2, we recommend the usage of the GP2 by employing \( \textbf{X} \) or \(\{ \textbf{X} \cup \mathbf {\hat{Y}} \}\) to predict residuals by setting R2 as the fitness function in genetic programming, because of their good statistical performance.
For future work, it is recommended to compare the performance of bootstrapping and Jackknife to estimate the variance of GPbased symbolic regression. It is also worthwhile to investigate the performance of the proposed methods on realworld applications.
Notes
 1.
Constraints are not considered in this paper.
 2.
Recently, some work has considered the dependency in constructing the surrogate models, such as the socalled dependent Gaussian processes [3]. This paper does not consider a dependency between objectives for simplicity.
 3.
\(\forall A,B\subseteq \mathbb {R}^m: \text{ HV }(A, \textbf{r}) > \text{ HV }(B,\textbf{r})\implies A \prec B\).
 4.
The ideal Pareto fronts on ZDT problems hold: \(\max \big ({f_{1}( \textbf{x} )}\big ) \le 1, \max \big ( {f_{2}( \textbf{x} )} \big ) \le 1\).
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This work is supported by the Austrian Science Fund (FWF – Der Wissenschaftsfonds) under the project (I 5315, ‘ML Methods for Feature Identification Global Optimization).
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Yang, K., Affenzeller, M. (2023). Surrogateassisted Multiobjective Optimization via Genetic Programming Based Symbolic Regression. In: Emmerich, M., et al. Evolutionary MultiCriterion Optimization. EMO 2023. Lecture Notes in Computer Science, vol 13970. Springer, Cham. https://doi.org/10.1007/9783031272509_13
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