Distributionally Robust Multi-Objective Optimization

TL;DR

This paper introduces a distributionally robust approach to multi-objective optimization, developing algorithms with provable guarantees that effectively handle data shifts and improve sample efficiency.

Contribution

It proposes the first distributionally robust multi-objective optimization framework with novel Pareto solution concepts and efficient gradient-based algorithms.

Findings

Double-loop MGDA achieves $ ilde{O}( ext{sample complexity})$ for $ ext{epsilon}$-Pareto-stationary points.

Single-loop double-clip MGDA significantly improves sample complexity to $ ilde{O}( ext{epsilon}^{-4})$.

Algorithms are theoretically sound for nonconvex problems and perform well in experiments.

Abstract

Multi-objective optimization (MOO) has received growing attention in applications that require learning under multiple criteria. However, the existing MOO formulations do not explicitly account for distributional shifts in the data. We introduce distributionally robust multi-objective optimization (DR-MOO), which minimizes multiple objectives under their respective worst-case distributions. We propose Pareto-type solution concepts for DR-MOO and develop multi-gradient descent algorithms (MGDA) with provable guarantees. Leveraging a Lagrangian dual reformulation, we first design a double-loop MGDA that uses an inner loop to estimate dual variables and achieves a total sample complexity $\mathcal{O}(\epsilon^{-12})$ for reaching an $\epsilon$-Pareto-stationary point. To further improve efficiency, we incorporate gradient clipping to handle generalized-smooth and biased gradient estimates, removing the need for double sampling. This yields a single-loop double-clip MGDA with substantially improved sample complexity $\mathcal{O}(\epsilon^{-4})$. Our theory applies to the nonconvex setting and does not require bounded objectives or gradients. Experiments demonstrate that our methods are competitive with state-of-the-art MGDA baselines.