Learning problems in which multiple conflicting objectives must be considered simultaneously often arise in various fields, including engineering, drug design, and environmental management. Traditional methods for dealing with multiple black-box objective functions, such as scalarization and identification of the Pareto set under the componentwise order, have limitations in incorporating objective preferences and exploring the solution space accordingly. While vector optimization offers improved flexibility and adaptability via specifying partial orders based on ordering cones, current techniques designed for sequential experiments either suffer from high sample complexity or lack theoretical guarantees. To address these issues, we propose Vector Optimization with Gaussian Process (VOGP), a probably approximately correct adaptive elimination algorithm that performs black-box vector optimization using Gaussian process bandits. VOGP allows users to convey objective preferences through ordering cones while performing efficient sampling by exploiting the smoothness of the objective function, resulting in a more effective optimization process that requires fewer evaluations. We establish theoretical guarantees for VOGP and derive information gain-based and kernel-specific sample complexity bounds. We also conduct experiments on both real-world and synthetic datasets to compare VOGP with the state-of-the-art methods.

翻译：在工程、药物设计和环境管理等多个领域中，同时考虑多个相互冲突目标的学习问题频繁出现。处理多个黑箱目标函数的传统方法，如标量化以及在分量序下识别帕累托集，在纳入目标偏好并据此探索解空间方面存在局限性。虽然向量优化通过基于序锥指定偏序提供了更高的灵活性和适应性，但当前为序贯实验设计的技术要么样本复杂度高，要么缺乏理论保证。为解决这些问题，我们提出了基于高斯过程的向量优化（VOGP），这是一种概率近似正确的自适应消除算法，它利用高斯过程赌博机执行黑箱向量优化。VOGP允许用户通过序锥传达目标偏好，同时通过利用目标函数的平滑性进行高效采样，从而实现更有效的优化过程，并减少所需的评估次数。我们为VOGP建立了理论保证，并推导了基于信息增益以及核函数特定的样本复杂度界。我们还在真实世界和合成数据集上进行了实验，将VOGP与最先进的方法进行了比较。