多目标分布优化新算法，能同时优化多个概率分布目标。
Multiple Wasserstein Gradient Descent Algorithm for Multi-Objective Distributional Optimization
- 基于粒子的迭代算法，通过动态加权聚合梯度更新分布。
- 在合成与真实数据集上均有效降低多个目标函数值。
- 适合多目标生成建模、多任务学习等场景使用。
我们研究同时最小化多个目标泛函在概率分布族上的优化问题，该问题在机器学习与统计中常见，应用于多目标采样、多任务学习及多目标生成建模等领域。为此，我们提出一种基于粒子的迭代算法——多重瓦瑟斯坦梯度下降（MWGraD），构建一系列中间经验分布，每个分布由一组粒子表示，并逐步同时最小化多个目标泛函。具体而言，每轮迭代包含两步：首先基于当前粒子估计各目标泛函的瓦瑟斯坦梯度；随后利用动态调整权重将这些梯度聚合为单一瓦瑟斯坦梯度，并据此更新粒子。我们提供了理论分析，并在合成与真实数据集上展示了MWGraD的有效性。
原文摘要 · Abstract (English)
We address the optimization problem of simultaneously minimizing multiple objective functionals over a family of probability distributions. This type of Multi-Objective Distributional Optimization commonly arises in machine learning and statistics, with applications in areas such as multiple target sampling, multi-task learning, and multi-objective generative modeling. To solve this problem, we propose an iterative particle-based algorithm, which we call Muliple Wasserstein Gradient Descent (MWGraD), which constructs a flow of intermediate empirical distributions, each being represented by a set of particles, which gradually minimize the multiple objective functionals simultaneously. Specifically, MWGraD consists of two key steps at each iteration. First, it estimates the Wasserstein gradient for each objective functional based on the current particles. Then, it aggregates these gradients into a single Wasserstein gradient using dynamically adjusted weights and updates the particles accordingly. In addition, we provide theoretical analysis and present experimental results on both synthetic and real-world datasets, demonstrating the effectiveness of MWGraD.
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