On the Geometry of On-Policy Distillation
On-policy distillation updates occupy a sparse, low-dimensional parameter subspace that is functionally sufficient and geometrically distinct from supervised fine-tuning and reinforcement learning.
Published Jun 5, 2026▲ 75 on Hugging FacearXiv ↗
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OPD reveals a unique off-principal, subspace-locked geometry with a practical freeze shortcut that SFT cannot match, yet the analysis lacks reasoning benchmarks, data budgets, and baseline accuracy curves to prove its functional merit.
Abstract
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.