A Unified Framework for Gradient Aggregation in Multi-Objective Optimization
Authors: Zeou Hu, Kelvin Ho, Yaoliang Yu
Organizations: 1Cheriton School of Computer Science, University of Waterloo, Waterloo, ON, Canada · 3Vector Institute · 2Computer Science, The Chinese University of Hong Kong
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.
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 O(ε−12) for reaching an ε-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 O(ε−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.
Last-iterate convergence and generalization guarantees in first-order convex learning hinge on the monotonicity of the update operator. While linear averaging preserves the monotonicity of gradient updates, this property is often violated when gradients are aggregated non-affinely, as in modern pipelines enforcing constraints like adaptivity, privacy, robustness or fairness. Whether it is possible to design non-affine aggregation rules that maintain monotonicity has remained an open question. We answer this question negatively: we prove that the monotonicity of aggregated gradients is preserved if and only if the aggregation rule is positively affine. Consequently, non-affine aggregation prevents steady convergence and substantially degrade algorithmic stability. We quantify these drawbacks and propose a path forward by identifying sufficient conditions under which monotonicity can be restored. Our results provide a unified theoretical framework explaining the disparate failure modes observed in modern learning systems.
A unified framework for first-order optimization algorithms fornonconvex unconstrained optimization is proposed that uses adaptivelypreconditioned gradients and includes popular methods such as full anddiagonal AdaGrad, AdaNorm, as well as adpative variants of Shampoo andMuon. This framework also allows combining heterogeneous geometriesacross different groups of variables while preserving a unifiedconvergence analysis. A fully stochastic global rate-of-convergenceanalysis is conducted for all methods in the framework, with andwithout two types of momentum, using reasonable assumptions on thevariance of the gradient oracle and without assuming boundedstochastic gradients or small enough stepsize.