Equivalent Predictors, Different Updates: Canonical-Coordinate Strategy Fusion for Long-Tailed Recognition

Long-Tailed Recognition (LTR) increasingly relies on heterogeneous learning strategies with complementary strengths across class-frequency regions. Gradient-based strategy fusion coordinates these strategies through their full-parameter gradients. We identify a non-invariance: functionally equivalent parameterizations of an expert-private classifier can preserve the predictor, loss, and shared-parameter gradient while producing different coordinated updates. The chosen private coordinates therefore become an unintended factor in how LTR strategies are coordinated, a phenomenon we call coordinate leakage. We propose Canonical-Coordinate Strategy Fusion (CCSF), which transports the affected gradients to a common reference coordinate, applies the original fusion rule and optimizer there, and maps the resulting update back to parameter storage. Controlled interventions across multiple full-gradient coordinators reveal the predicted coordinate sensitivity, trace its effect on expert complementarity across frequency regions, and show that CCSF numerically recovers the corresponding reference trajectories. Experiments evaluate the complete configuration on CIFAR-100-LT, ImageNet-LT, and iNaturalist 2018. Code is provided in the supplementary material.