A Stochastic Multi-Gradient Algorithm for Wasserstein Distributionally Robust Multi-objective Optimization

Distributionally robust multi-objective optimization (DRMOO) has been proposed to improve the robustness of multi-task learning models under distribution shifts. However, existing DRMOO methods mainly construct ambiguity sets based on $f$-divergence, which may have limited capability in capturing the geometric structure of the sample space and changes in the support of distributions. To address this limitation, we propose a Wasserstein distributionally robust multi-objective optimization algorithm, which leverages the geometry of the sample space to characterize distributional uncertainty more flexibly. Specifically, by applying the dual reformulation of Wasserstein distributionally robust optimization, each robust objective is reformulated as an expectation involving an inner adversarial maximization problem. Based on this, we develop a double-loop optimization algorithm, where the inner loop employs $K$-step gradient ascent to approximately solve the inner maximization problem, and the outer loop adopts an MGDA-type method to construct a common descent direction among multiple objectives. Theoretically, under non-convex and generalized $\ell$-smooth assumptions, we establish the convergence guarantees of the proposed algorithm, proving that it achieves an $ϵ$-Pareto stationary point with a total sample complexity of $\mathcal{O}({ϵ}^{-4})$. Experimental results demonstrate the practical performance of the proposed algorithm.