Necessity of retaining path supervision during endpoint fine-tuning

Determine whether retaining the teacher-forced path objective during Stage II autonomous endpoint fine-tuning is necessary for task-level performance gains after path pretraining has established a suitable initialization for the constrained Kuramoto system.

Background

The training procedure uses teacher-forced path supervision in Stage I to initialize the constrained Kuramoto system and autonomous endpoint supervision in Stage II to improve finite-time inference. The paper compares PRETRAIN-END-ONLY, which optimizes only the endpoint loss after Stage-I pretraining, with TWO-STAGE-END, which retains both the path and endpoint losses during continuation.

In the reported single-run ablation, the two variants achieve very similar performance on MNIST and Fashion-MNIST. Consequently, the experiments do not resolve whether continued path supervision provides an independent benefit once path pretraining has already produced a usable dynamical initialization. Establishing this would require broader experiments across datasets, initialization schemes, optimization settings, and repeated runs.

References

Within this single-run ablation, there is therefore no clear evidence that retaining $\mathcal L_{\rm path}$ during Stage II is necessary for the task-level gain once path pretraining has already established a suitable initialization.

— A Constrained Kuramoto Gradient-Flow System Can Perform High-Accuracy Finite-Time Inference  (2609.01539 - Cheng et al., 1 Sep 2026) in Appendix D, Section "Training-Protocol Ablations," subsection "Roles of Path and Endpoint Supervision" (Appendix D, subsection app:full_protocol_ablation)