Stochastic Aligned Multi-Objective Optimization

Aligned Multi-Objective Optimization (AMOO) studies how to minimize multiple objectives that share a common optimum. Existing deterministic guarantees do not directly cover noisy minibatch updates. Unrestricted weight scaling can amplify noise, while choosing weights from the update batch can bias the combined gradient. We propose delayed update normalized AMOO (DunAMOO), which selects nonnegative weights that sum to one and applies them one iteration later. This controls weight scaling and separates the applied weights from fresh gradient noise, using one minibatch per iteration. We provide, to our knowledge, the first convergence analysis of stochastic AMOO that handles noisy Polyak weight selection and delayed updates. Under the stated curvature and noise conditions, we prove finite expected entry into a near region. After entry, the expected squared error contributed by paths with no earlier exit has a geometrically decaying bound down to a noise-dependent level. Experiments on tasks with loss-scale imbalance show favorable convergence trends for DunAMOO.