Accelerated Multiple Wasserstein Gradient Flows for Multi-objective Distributional Optimization

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.

💡 Research Summary

The paper addresses multi‑objective optimization over probability distributions in the 2‑Wasserstein space, a problem the authors term Multi‑Objective Distributional Optimization (MODO). Building on the recently proposed Multiple Wasserstein Gradient Descent (MWGraD) algorithm (Nguyen et al., 2025), which jointly minimizes several objective functionals by aggregating their Wasserstein gradients into a single descent direction, the authors introduce an accelerated variant called A‑MWGraD. The acceleration is inspired by Nesterov’s momentum and is formulated as a damped Hamiltonian system in the probability space.

The continuous‑time dynamics of the original MWGraD flow are first revisited: \