## 1 Introduction  
In practical optimization problems, uncertainty is a pervasive feature--encountered in diverse scenarios such as uncertain travel durations in routing problems and unpredictable future demand in revenue management [1]. Two distinct methodologies exist for tackling optimization under uncertainty, primarily differentiated by their reliance on data [2]. The first category, which includes stochastic optimization [3] and robust optimization [4, 5], does not treat data as a primary component. These methods typically predefine distributions for uncertain parameters without using empirical data, which can lead to inaccuracies in distribution specifications and subsequently, suboptimal decisions [6]. On the other hand, the rapid advancement of internet technologies, facilitating the collection and storage of vast quantities of data, has given birth to a second category--using data to characterize uncertainty. This second category further branches out based on the nature of the data utilized: historical data of uncertain parameters or auxiliary data (contextual covariates) that assist in predicting these uncertain parameters. One subset of this category models scenarios or distributions of uncertain parameters using historical data, without taking into account any useful auxiliary data. Techniques such as the sample average approximation (SAA) method [7] and the data-driven distributionally robust optimization framework exemplify this subcategory [8, 9]. The other subset capitalizes on machine learning (ML) techniques to predict uncertain parameters, integrating both historical data and relavant auxiliary data. This integration of ML techniques with mathematical optimization models is termed prescriptive analytics. The most commonly used method in this domain is the two-stage predict-then-optimize(PTO) framework. Here, ML models are used to obtain point predictions using available auxiliary data, effectively transforming uncertain problems into simpler and deterministic problems. However, the PTO framework may not be apt for optimization problems in which the objective functions are not linear in their uncertain parameters, as stochastic problems may involve complex integrals with respect to the conditional distribution model [1, 6, 10, 11]. In such instances, the estimate-then-optimize (ETO) methods often yield superior decisions compared to PTO methods. ETO methods first estimate the distributions of uncertain parameters and then solve the resulting contextual stochastic optimization (CSO) problems. A prevalent non-parametric ETO method is the weighted SAA (wSAA) method, which estimates the conditional distribution by assigning weights to historical data, with the weights determined by the covariates [6, 10]. Within the context of the wSAA framework, this paper scrutinizes its decision-making performance based on the minimum relative regret metric, defined as the difference between the expected optimal profit (benefits) of an oracle possessing full information (the actual distribution) and the maximum expected out-of-sample profit realized by the data-driven policy, normalized by the former profit. Related Literature and ContributionsOur work contributes to two key areas of literature. Firstly, it relates to prescriptive analytics frameworks, in which decision-makers leverage contextual data about uncertainty to make informed decisions [1, 2, 11, 12, 13, 14]. Several advanced frameworks have been developed, including the smart predict-then-optimize framework [13, 15], the wSAA framework [10, 14], the empirical risk minimization framework [12, 14], and the kernel optimization framework [2, 12, 14, 16]. Among these frameworks, our focus lies on the wSAA framework, which predicts the distributions of uncertain parameters by assigning weights to empirical data [10, 12]. One of its strengths lies in its ability to reveal all necessary information for solving the stochastic problem when a perfect distributional prediction of the underlying conditional distribution can be achieved [6].