When Do Surrogate Updates Improve Decisions? A Local Theory of Trajectory-Wise Transfer

Models trained or adapted from trajectories often use tractable surrogate losses but are evaluated by downstream decisions. A trajectory-induced surrogate reduction need not track its decision utility. At a fixed checkpoint and restricted update space, we define the corresponding reductions in population surrogate loss and decision risk as learnability and decision utility, respectively, and study their transfers. The one-step bound separates nonnegatively calibrated gradient misalignment from curvature, while its pathwise extension accumulates both terms. For a nonzero accessible surrogate gradient, universal first-order transfer holds exactly under positive collinearity with the accessible decision gradient. The calibration gap also bounds learnability-based selection regret; restricting it to candidate-gradient differences sharpens the guarantee by retaining only ranking-relevant directions. Across nested update spaces, we establish an approximation-calibration trade-off. For test-time continual learning, this provides a local diagnostic for whether an experience-induced update is aligned with the downstream decision objective before such updates accumulate along deployment. Controlled gridworld and LLM post-training experiments are consistent with theoretical predictions.