提出加速版水石流优化算法，提升多目标分布优化效率
Accelerated Multiple Wasserstein Gradient Flows for Multi-objective Distributional Optimization
- 基于Nesterov加速思想改进水石梯度下降法
- 理论证明收敛速度达O(1/t²)且实测更快采样
- 适合需要高效多目标分布生成的场景
我们研究在水石空间中对概率分布进行多目标优化。近期，Nguyen等人（2025）提出了多重水石梯度下降（MWGraD）算法，利用水石空间的几何结构实现多目标联合优化。本文在此基础上提出一种加速版本A-MWGraD，受Nesterov加速启发。我们分析了连续时间动力学，证明其在概率空间中收敛至弱帕累托最优点。理论结果表明，A-MWGraD在测地凸目标下达到O(1/t²)的收敛速率，在β-强测地凸目标下为O(e^{-√βt})，优于MWGraD在测地凸情形下的O(1/t)速率。我们进一步设计了一种实用的核基离散化方法，并通过数值实验验证其在多目标采样任务中显著优于MWGraD，表现出更快的收敛速度和更高的采样效率。
原文摘要 · Abstract (English)
We study multi-objective optimization over probability distributions in Wasserstein space. Recently, Nguyen et al. (2025) introduced Multiple Wasserstein Gradient Descent (MWGraD) algorithm, which exploits the geometric structure of Wasserstein space to jointly optimize multiple objectives. Building on this approach, we propose an accelerated variant, A-MWGraD, inspired by Nesterov's acceleration. We analyze the continuous-time dynamics and establish convergence to weakly Pareto optimal points in probability space. Our theoretical results show that A-MWGraD achieves a convergence rate of O(1/t^2) for geodesically convex objectives and O(e^{-\sqrtβt}) for $β$-strongly geodesically convex objectives, improving upon the O(1/t) rate of MWGraD in the geodesically convex setting. We further introduce a practical kernel-based discretization for A-MWGraD and demonstrate through numerical experiments that it consistently outperforms MWGraD in convergence speed and sampling efficiency on multi-target sampling tasks.
Thank you to arXiv for use of its open access interoperability. PaperDance 不是 arXiv 官方产品；中文卡片由大模型生成，请以原文为准。