On the Geometry of On-Policy Distillation
For researchers studying training dynamics of LLMs, this work provides a geometric understanding of OPD, distinguishing it from SFT and RLVR, though the findings are primarily diagnostic and incremental.
The paper characterizes the training dynamics of on-policy distillation (OPD) for large language models, showing that OPD updates are more localized and avoid principal directions compared to supervised fine-tuning (SFT) and reinforcement learning with verifiable rewards (RLVR). OPD exhibits subspace locking, where cumulative updates rapidly enter a narrow low-dimensional channel that is functionally sufficient for performance.
On-policy distillation (OPD) is increasingly used to improve large language model reasoning, but its training dynamics remain poorly understood. We characterize the trajectory of OPD updates in parameter space and compare it with supervised fine-tuning (SFT) and reinforcement learning with verifiable rewards (RLVR). A suite of parameter-space diagnostics consistently places OPD in a relaxed off-principal regime: compared with SFT, its updates affect fewer weights and avoid principal directions more strongly, while compared with RLVR, they remain less tightly constrained. Beyond this static localization, OPD exhibits subspace locking: its cumulative updates rapidly enter a narrow low-dimensional channel. Constraining training to the update subspace formed early in training preserves OPD performance but substantially degrades SFT, indicating that the locked subspace is functionally sufficient for OPD. Control experiments further show that sparsifying the update tokens and shifting rollout generation off-policy preserve the rank dynamics, whereas mixing the OPD objective with RLVR changes them. Overall, these results suggest that OPD is not merely an intermediate point between SFT and RLVR, but induces its own update geometry in parameter space.