Distributionally Robust Multi-Objective Optimization

Many machine learning problems involve multiple inherent trade-offs that are best addressed by gradient-based multi-objective optimization (MOO) algorithms. Existing methods are often proposed with various motivations, analyzed case by case, and differ algorithmically in how the component gradients are aggregated at each step. In this work, we develop a unifying framework for gradient aggregation in MOO, establishing (optimal) rates of convergence to Pareto stationarity, the standard measure of performance in MOO. Central to our analysis is a sufficient alignment condition, from which we derive a theorem showing that non-conflicting directions, when chosen within the convex hull of gradients, form a fundamental sufficient condition for convergence. We further show that feasibility can be ensured through projection onto the dual cone, broadening the scope of methods that admit convergence guarantees. In parallel, we present a primal optimization perspective of gradient aggregation that encompasses established algorithms, clarifies their theoretical relationships, and enables the design of new variants. As an illustration, we introduce capped MGDA, derived from a CVaR-based formulation, and demonstrate its robustness in adversarial federated learning. Finally, we validate our theory through experiments on synthetic problems and practical benchmarks.

We study distributionally robust optimization (DRO) for robust inference when the worst-case distribution is continuous, leading to significant computational challenges due to the infinite-dimensional nature of the optimization problem. Unlike traditional discrete DRO approaches, which often suffer from scalability issues, limited generalization, and costly worst-case inference, our framework exploits Brenier's theorem to characterize the least favorable distribution as the pushforward of a transport map from a continuous reference measure. This characterization motivates our study of the minimax problem in Wasserstein space. We propose an iterative algorithmic framework with multiple variants and establish global convergence guarantees under mild assumptions, deriving complexity bounds in terms of subgradient evaluations and inexact Jordan-Kinderlehrer-Otto updates. Numerical results with neural network-based transport maps demonstrate that the proposed method enables both stable training of robust classifiers and effective worst-case inference for classification tasks.

Multi-task reinforcement learning (MTRL) aims to train a single agent to efficiently optimize performance across multiple tasks simultaneously. However, jointly optimizing all tasks often yields imbalanced learning: agents quickly solve easy tasks but learn slowly on harder ones. While prior work primarily attributes this imbalance to conflicting task gradients and proposes gradient manipulation or specialized architectures to address it, we instead focus on a distinct and under-explored challenge: imbalanced data allocation. Standard MTRL allocates an equal number of environment interactions to each task, which over-allocates data to easy tasks that require relatively few interactions to solve and under-allocates data to hard tasks that require substantially more experience to solve. To address this challenge, we introduce Distributionally Robust Adaptive Task Sampling (DRATS), an algorithm that adaptively prioritizes sampling tasks furthest from being solved. We derive DRATS by formalizing MTRL as a feasibility problem from which we derive a minimax objective for minimizing the worst-case return gap, the difference between a desired target return and the agent's return on a task. In benchmarks like MetaWorld-MT10 and MT50, DRATS improves data efficiency and increases worst-task performance compared to existing task sampling algorithms.