Class imbalance occurs when the class distribution is not equal. Namely, one class is under-represented (minority class), and the other class has significantly more samples in the data (majority class). The class imbalance problem is prevalent in many real world applications. Generally, the under-represented minority class is the class of interest. The synthetic minority over-sampling technique (SMOTE) method is considered the most prominent method for handling unbalanced data. The SMOTE method generates new synthetic data patterns by performing linear interpolation between minority class samples and their K nearest neighbors. However, the SMOTE generated patterns do not necessarily conform to the original minority class distribution. This paper develops a novel theoretical analysis of the SMOTE method by deriving the probability distribution of the SMOTE generated samples. To the best of our knowledge, this is the first work deriving a mathematical formulation for the SMOTE patterns’ probability distribution. This allows us to compare the density of the generated samples with the true underlying class-conditional density, in order to assess how representative the generated samples are. The derived formula is verified by computing it on a number of densities versus densities computed and estimated empirically.

Recently, active learning is considered a promising approach for data acquisition due to the significant cost of the data labeling process in many real world applications, such as natural language processing and image processing. Most active learning methods are merely designed to enhance the learning model accuracy. However, the model accuracy may not be the primary goal and there could be other domain-specific objectives to be optimized. In this work, we develop a novel active learning framework that aims to solve a general class of optimization problems. The proposed framework mainly targets the optimization problems exposed to the exploration-exploitation trade-off. The active learning framework is comprehensive, it includes exploration-based, exploitation-based and balancing strategies that seek to achieve the balance between exploration and exploitation. The paper mainly considers regression tasks, as they are under-researched in the active learning field compared to classification tasks. Furthermore, in this work, we investigate the different active querying approaches—pool-based and the query synthesis—and compare them. We apply the proposed framework to the problem of learning the price-demand function, an application that is important in optimal product pricing and dynamic (or time-varying) pricing. In our experiments, we provide a comparative study including the proposed framework strategies and some other baselines. The accomplished results demonstrate a significant performance for the proposed methods.

Dynamic pricing is a beneficial strategy for firms seeking to achieve high revenues. It has been widely applied to various domains such as the airline industry, the hotel industry, and e-services. Dynamic pricing is basically the problem of setting time-varying prices for a certain product or service for the purpose of optimizing revenue. However, a major challenge encountered when applying dynamic pricing is the lack of knowledge of the demand-price curve. The demand-price curve defines the customer’s response towards changing the price. In this work, we address the dynamic pricing problem in case of unknown demand-price relation. This work introduces a less myopic pricing approach based on looking ahead for one or several future steps in our quest to optimize revenue. Specifically, the proposed formulation maximizes the summation of the immediate revenue and the expected future revenues of one or multiple look-ahead steps. A key benefit of the proposed approach is that it automatically strikes a balance between the conflicting goals of revenue maximization and demand learning, by producing less myopic and profitable prices. We provide a formulation for the presented look-ahead pricing approach, and we implement two variants of it: one-step and two-step look-ahead methods. Experiments are conducted on synthetic and real datasets to compare the proposed pricing methods to other pricing strategies in literature. The experimental results indicate that the proposed look-ahead methods outperform their counterparts in terms of the achieved revenue gain.