# When Sparse Reward Meets Dense Distillation: Training Dynamics of On-Policy Distillation

arXiv: https://arxiv.org/abs/2609.34849 · HTML: https://mlsanity.com/paper/2609.34849/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Placed in On-policy and self-distillation for LLMs, one of 12 directions in Knowledge distillation.
- Builds on 21 of its 23 references that are in the map.
- First posted on arXiv 28 Sep 2026.

## Placed in

- Knowledge distillation / [On-policy and self-distillation for LLMs](https://mlsanity.com/distill/on-policy-and-self-distillation-for-llms/)

## Builds on (most cited first, 21 of 21)

- [Proximal Policy Optimization Algorithms](https://mlsanity.com/paper/1707.06347/) (2017)
- [DeepSeekMath: Pushing the Limits of Mathematical Reasoning in Open Language Models](https://mlsanity.com/paper/2402.03300/) (2024)
- [DAPO: An Open-Source LLM Reinforcement Learning System at Scale](https://mlsanity.com/paper/2503.14476/) (2025)
- [Neural Tangent Kernel: Convergence and Generalization in Neural Networks](https://mlsanity.com/paper/1806.07572/) (2018)
- [Multi-Task Learning Using Uncertainty to Weigh Losses for Scene Geometry and Semantics](https://mlsanity.com/paper/1705.07115/) (2017)
- [Understanding R1-Zero-Like Training: A Critical Perspective](https://mlsanity.com/paper/2503.20783/) (2025)
- [Gradient Surgery for Multi-Task Learning](https://mlsanity.com/paper/2001.06782/) (2020)
- [On-Policy Distillation of Language Models: Learning from Self-Generated Mistakes](https://mlsanity.com/paper/2306.13649/) (2023)
- [GradNorm: Gradient Normalization for Adaptive Loss Balancing in Deep Multitask Networks](https://mlsanity.com/paper/1711.02257/) (2017)
- [Multi-Task Learning as Multi-Objective Optimization](https://mlsanity.com/paper/1810.04650/) (2018)
- [End-to-End Multi-Task Learning with Attention](https://mlsanity.com/paper/1803.10704/) (2018)
- [Conflict-Averse Gradient Descent for Multi-task Learning](https://mlsanity.com/paper/2110.14048/) (2021)
- [Self-Distilled Reasoner: On-Policy Self-Distillation for Large Language Models](https://mlsanity.com/paper/2601.18734/) (2026)
- [Reinforcement Learning via Self-Distillation](https://mlsanity.com/paper/2601.20802/) (2026)
- [Multi-Task Learning as a Bargaining Game](https://mlsanity.com/paper/2202.01017/) (2022)
- [Defining and Characterizing Reward Hacking](https://mlsanity.com/paper/2209.13085/) (2022)
- [Self-Distilled RLVR](https://mlsanity.com/paper/2604.03128/) (2026)
- [Learning Dynamics of LLM Finetuning](https://mlsanity.com/paper/2407.10490/) (2024)
- [HDPO: Hybrid Distillation Policy Optimization via Privileged Self-Distillation](https://mlsanity.com/paper/2603.23871/) (2026)
- [Direct Preference Knowledge Distillation for Large Language Models](https://mlsanity.com/paper/2406.19774/) (2024)
- [NTKMTL: Mitigating Task Imbalance in Multi-Task Learning from Neural Tangent Kernel Perspective](https://mlsanity.com/paper/2510.18258/) (2025)

## Abstract (arXiv)

Reinforcement learning with verifiable rewards provides a sparse post-training signal: a single binary outcome evaluates the entire rollout, and every token receives the same sequence-level advantage regardless of its individual contribution. To complement this sparse supervision, a growing family of methods adds a scalar-weighted teacher KL term to the policy-gradient objective, providing dense token-level guidance that may be unreliable at some positions. Despite the benefits of combining these signals, their interaction during optimization can destabilize joint training. To understand how this instability develops, we study the learning dynamics of hybrid reward--distillation training through a neural tangent kernel (NTK) analysis. We introduce the cross-signal NTK KDR(n), a token-level statistic that measures the alignment between reward and distillation gradients at position n. Through this analysis, we identify two failure modes: 1 Magnitude drowning, where the reward gradient exceeds the distillation gradient by orders of magnitude, so that even weak directional conflict can cause the distillation loss to rise despite its explicit inclusion in the training objective; and 2 Localized directional conflict, where the sequence-level advantage and the teacher's position-specific distribution induce opposing updates at the same token (KDR(n)!<!0). The severity of these effects depends on the optimization regime: the gradient-norm ratio kappa!=!|nablaLR|/|nablaLD| varies by roughly an order of magnitude across tasks, and our experiments reveal an empirical threshold beyond which naive mixing can lead to persistent training collapse. Motivated by these findings, we introduce the M3 family, which combines magnitude normalization with three strategies...