A Unified Framework for Gradient Aggregation in 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.

An Operator Splitting View of Federated Learning

Over the past few years, the federated learning ($\texttt{FL}$) community has witnessed a proliferation of new $\texttt{FL}$ algorithms. However, our understating of the theory of $\texttt{FL}$ is still fragmented, and a thorough, formal comparison of these algorithms remains elusive. Motivated by this gap, we show that many of the existing $\texttt{FL}$ algorithms can be understood from an operator splitting point of view. This unification allows us to compare different algorithms with ease, to refine previous convergence results and to uncover new algorithmic variants. In particular, our analysis reveals the vital role played by the step size in $\texttt{FL}$ algorithms. The unification also leads to a streamlined and economic way to accelerate $\texttt{FL}$ algorithms, without incurring any communication overhead. We perform numerical experiments on both convex and nonconvex models to validate our findings.

FedMGDA+: Federated Learning meets Multi-objective Optimization

Federated learning has emerged as a promising, massively distributed way to train a joint deep model over large amounts of edge devices while keeping private user data strictly on device. In this work, motivated from ensuring fairness among users and robustness against malicious adversaries, we formulate federated learning as multi-objective optimization and propose a new algorithm FedMGDA+ that is guaranteed to converge to Pareto stationary solutions. FedMGDA+ is simple to implement, has fewer hyperparameters to tune, and refrains from sacrificing the performance of any participating user. We establish the convergence properties of FedMGDA+ and point out its connections to existing approaches. Extensive experiments on a variety of datasets confirm that FedMGDA+ compares favorably against state-of-the-art.