Data imbalance is pervasive in machine learning, from rare words and anomalies to underrepresented patterns in heterogeneous or cross-tabulated data. We study idealized optimizers geometries in continuous time to model small-step training in deep learning. We assume that the source of imbalance is unobserved: the optimizer has only access to the aggregate training loss ignoring the exact contributions of the majority and minority groups. In this setting, we characterize a region where majority losses are optimized regardless of the admissible minority structure. We derive explicit equations of this zone and bounds on the time needed to leave it. These bounds exhibit a milder dependence on minority amplitude for sign, spectral, and Newton descent than for Euclidean gradient descent. Experiments with AdamW and Muon suggest similar advantages over SGD across language, tabular, and image tasks.

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