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This paper introduces a single-loop multi-objective optimization method that combines a norm-constrained linear minimization oracle with entropic task weight updates, achieving an $O(T^{-1/2})$ ergodic stationarity rate in the deterministic setting and $O(T^{-1/4})$ with stochastic gradients. The analysis establishes static regret for the inner weighting objective and iteration-wise inner-gap control under specific conditions, extending prior spectral geometry work to general norms. This approach offers a principled framework for multi-task optimization that accounts for non-Euclidean geometries and dynamic task importance.
A novel multi-objective optimizer achieves $O(T^{-1/4})$ convergence with stochastic gradients, independent of task count, by integrating norm-constrained oracles and entropic weight updates.
Multiple-gradient descent is typically formulated in Euclidean geometry, whereas matrix-aware optimizers use directions induced by other norms. We analyze a single-loop multi-objective method combining a norm-constrained linear minimization oracle with entropic updates of task weights. For any finite-dimensional norm pair, we establish an $O(T^{-1/2})$ ergodic stationarity rate in the deterministic setting, deriving the required trajectory gradient bound from smoothness and lower-boundedness in the unregularized case. With stochastic gradients and a bounded-gradient assumption, one persistent blended-gradient momentum buffer yields an $O(T^{-1/4})$ rate whose leading term has no explicit dependence on the number of tasks. We also establish static regret for the inner weighting objective on arbitrary bounded gradient sequences and, with entropy regularization and a weight floor, iteration-wise inner-gap control after burn-in while retaining ergodic stationarity, conditional on a trajectory gradient bound. The optimal blended gradient can be nonunique in spectral and sign geometries, motivating a selection-independent inner gap. Our analysis extends the spectral construction studied by concurrent MOON to general norms. Experiments compare the method with existing multi-task optimizers and examine its oracle, weighting, and momentum choices.