Scalable Min-Max Optimization via Primal-Dual Exact Pareto Optimization
This addresses a scalability problem in multi-objective optimization for fairness applications, but it is incremental as it builds on primal-dual consensus techniques.
The paper tackled the slow convergence of min-max optimization for multi-objective fairness by proposing a smooth variant using an augmented Lagrangian, resulting in the EPO-AL algorithm that scales better with objectives and has lower per-iteration complexity than existing methods.
In multi-objective optimization, minimizing the worst objective can be preferable to minimizing the average objective, as this ensures improved fairness across objectives. Due to the non-smooth nature of the resultant min-max optimization problem, classical subgradient-based approaches typically exhibit slow convergence. Motivated by primal-dual consensus techniques in multi-agent optimization and learning, we formulate a smooth variant of the min-max problem based on the augmented Lagrangian. The resultant Exact Pareto Optimization via Augmented Lagrangian (EPO-AL) algorithm scales better with the number of objectives than subgradient-based strategies, while exhibiting lower per-iteration complexity than recent smoothing-based counterparts. We establish that every fixed-point of the proposed algorithm is both Pareto and min-max optimal under mild assumptions and demonstrate its effectiveness in numerical simulations.