## 2 Inductive Design Exploration Method with Active Learning
### IDEM with Active Learning  
Active learning is a subcategory of machine learning. The key idea behind active learning is that a learning algorithm can achieve greater accuracy with fewer training data [13]. The algorithm is allowed to choose the candidate training data, specifically, the more useful data to choose, from which it learns. For any supervised machine learning to perform well, it must often be trained with abounding given labeled instances (e.g., yes or no, success or failure, feasible or infeasible, and so on). The models also can be further updated if more labeled samples are provided. However, there are situations in which unlabeled data is abundant but manually labelling is very difficult, time-consuming, or expensive. Sometimes, the labeled data may be uninformative and the trained model may thus not be useful. In such scenarios, active learning algorithms can be employed to iteratively query which samples need to be labeled based upon past responses. As the learner chooses the samples, the sample size to learn a model can often be much smaller than that required in normal (passive) supervised learnings [13]. Figure 2 shows the general steps in active learning algorithms. The model for a certain task is first trained based on initially labeled data. The model has access to a pool of unlabeled data and can evaluate their informativeness. Given this informativeness, we ask the next query for which informativeness is highest. In other words, the most informative unlabeled data is selected as the new training data. There are several algorithms for determining which data should be labeled, i.e., query strategies, such as uncertainty sampling, query by committee, expected model change, expected error reduction, among others [13]. The selected unlabeled data is then labeled and added in the next training data set. Finally, the current model is retrained using the new training data. The steps within the for-loop are repeated until the allowable budget is reached. The abovementioned active learning process is applied to traditional IDEM. The graphical procedure of the proposed method is shown in Figure 3. First, the design spaces at each level are defined, and they generate a large pool of unlabeled data. Among the unlabeled data, the initial training points to be evaluated are randomly chosen. Step 1 involves the same preparation procedure for active learning contexts. In Step 2, the initial points are evaluated using the employed mapping models to label them as feasible or infeasible. The feasibility check is performed in the same manner as traditional IDEM (same as Step 2 and 3 in Section 2.1) by calculating HD-EMIs. The next step is to train a model with previously evaluated feasible/infeasible points and iteratively update the model in an active learning manner. Here, our model (\(M\)) is a classification model to represent the feasible/infeasible region in design spaces. In traditional IDEM, feasible and infeasible regions are only a group of feasible and infeasible points, respectively, and this representation is directly influenced by the initially sampled grid points at each design level. That is, a large sample size is required for the high resolution of feasible/infeasible regions, which in turn increases computational expense. Therefore, the classification algorithm is employed to avoid excessive grid points evaluations. In this study, a support vector machine (SVM) is used to classify regions as feasible or infeasible for the given feasible/infeasible samples. SVMs are one of the most common and powerful classifiers and were observed to be highly suitable for the active learning setup [14; 15; 16; 17]. When the first SVM is used to classify feasible/infeasible regions, the initial feasible regions identified by the SVM are not expected to be accurate enough, as the number of initial training points is relatively small. Therefore, the current classifier requires more samples to update and additional sampling can be implemented through query strategies.