Learning to Route Supervision Along the Path: Route Optimization for Gradual Domain Adaptation

Gradual domain adaptation progressively updates a source-trained classifier through unlabeled intermediate domains before reaching the target. However, existing methods often prescribe the update rule or weight schedule in advance, while the reliability of different supervision sources can change along the adaptation path. Source predictions can anchor early updates but become less informative than the adapting model as the domain shifts. Therefore, choosing appropriate supervision at each stage without target labels remains challenging. Based on the above analysis, we introduce PrismBridge, which selects bridge rules that combine frozen source predictions, class-prototype probabilities, and current-model predictions. Specifically, we formulate this choice as route-relaxed path optimization, where fixed-route policies are treated as constrained special cases. Candidates starting from the same state are evaluated using bridge losses, source supervision, and view agreement. A tolerance-aware selector then chooses among near-best candidates according to a predetermined preference order, and source-supervised smoothing stabilizes the selected update. Furthermore, we derive cumulative risk and selection-regret bounds along each policy's route-dependent model trajectory. Experiments across geometric, appearance, temporal, tabular, and corruption shifts demonstrate the effectiveness of PrismBridge, achieving severity-5 accuracy improvements of 8.1 and 15.9 percentage points over the strongest compared baseline on CIFAR-10-C and CIFAR-100-C.