A Unified Framework for Gradient Aggregation in Multi-Objective Optimization
This work provides a theoretical unification for gradient aggregation in multi-objective optimization, which is significant for researchers and practitioners working on MOO algorithms by clarifying existing relationships and enabling new designs.
This paper develops a unified framework for gradient aggregation in multi-objective optimization (MOO), establishing optimal rates of convergence to Pareto stationarity. It introduces capped MGDA, derived from a CVaR-based formulation, and demonstrates its robustness in adversarial federated learning.
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.