The goal of multi-objective optimization (MOO) is to learn under multiple, potentially conflicting, objectives. One widely used technique to tackle MOO is through linear scalarization, where one fixed preference vector is used to combine the objectives into a single scalar value for optimization. However, recent work (Hu et al., 2024) has shown linear scalarization often fails to capture the non-convex regions of the Pareto Front, failing to recover the complete set of Pareto optimal solutions. In light of the above limitations, this paper focuses on Tchebycheff scalarization that optimizes for the worst-case objective. In particular, we propose an online mirror descent algorithm for Tchebycheff scalarization, which we call OMD-TCH. We show that OMD-TCH enjoys a convergence rate of $O(\sqrt{\log m/T})$ where $m$ is the number of objectives and $T$ is the number of iteration rounds. We also propose a novel adaptive online-to-batch conversion scheme that significantly improves the practical performance of OMD-TCH while maintaining the same convergence guarantees. We demonstrate the effectiveness of OMD-TCH and the adaptive conversion scheme on both synthetic problems and federated learning tasks under fairness constraints, showing state-of-the-art performance.

翻译：多目标优化（MOO）的目标是在多个可能相互冲突的目标下进行学习。处理MOO的一种广泛使用技术是线性标量化，即通过一个固定的偏好向量将多个目标组合成单个标量值进行优化。然而，近期研究（Hu等人，2024）表明线性标量化往往无法捕捉帕累托前沿的非凸区域，从而无法恢复完整的帕累托最优解集。鉴于上述局限性，本文聚焦于针对最坏情况目标进行优化的切比雪夫标量化方法。具体而言，我们提出了一种用于切比雪夫标量化的在线镜像下降算法，称为OMD-TCH。我们证明OMD-TCH具有$O(\sqrt{\log m/T})$的收敛速率，其中$m$为目标数量，$T$为迭代轮数。同时，我们提出了一种新颖的自适应在线到批量转换方案，在保持相同收敛保证的前提下显著提升了OMD-TCH的实际性能。通过在公平约束下的合成问题与联邦学习任务上的实验，我们验证了OMD-TCH及自适应转换方案的有效性，并展示了其最先进的性能表现。