# PAST: Privileged Adaptation from Complete Student Trajectories for On-Policy Self-Distillation

arXiv: https://arxiv.org/abs/2608.08726 · HTML: https://mlsanity.com/paper/2608.08726/

## Key facts

- Cited by 6 papers in the map (data through 14 Oct 2026).
- Placed in Prompting and chain-of-thought reasoning, one of 17 directions in LLM reasoning.
- Builds on 14 of its 17 references that are in the map.
- First posted on arXiv 9 Aug 2026.

## Placed in

- LLM reasoning / [Prompting and chain-of-thought reasoning](https://mlsanity.com/lr/prompting-and-chain-of-thought-reasoning/)
- 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, 14 of 14)

- [DeepSeekMath: Pushing the Limits of Mathematical Reasoning in Open Language Models](https://mlsanity.com/paper/2402.03300/) (2024)
- [On-Policy Distillation of Language Models: Learning from Self-Generated Mistakes](https://mlsanity.com/paper/2306.13649/) (2023)
- [Self-Distilled Reasoner: On-Policy Self-Distillation for Large Language Models](https://mlsanity.com/paper/2601.18734/) (2026)
- [Unifying distillation and privileged information](https://mlsanity.com/paper/1511.03643/) (2015)
- [DistiLLM: Towards Streamlined Distillation for Large Language Models](https://mlsanity.com/paper/2402.03898/) (2024)
- [MiniLLM: On-Policy Distillation of Large Language Models](https://mlsanity.com/paper/2306.08543/) (2023)
- [Self-Distillation Zero: Self-Revision Turns Binary Rewards into Dense Supervision](https://mlsanity.com/paper/2604.12002/) (2026)
- [Not All Disagreement Is Learnable: Token Teachability in On-Policy Distillation](https://mlsanity.com/paper/2605.26844/) (2026)
- [When Are Teacher Tokens Reliable? Position-Weighted On-Policy Self-Distillation for Reasoning](https://mlsanity.com/paper/2605.21606/) (2026)
- [Distill Where You Fail: Recovering Learning Signals of Negative RL-Groups from Adaptive Teacher Guidance](https://mlsanity.com/paper/2608.00782/) (2026)
- [Is More Privileged Information Better? From Solution Traces to Problem-Solving Structure in Self-Distilled Reasoning](https://mlsanity.com/paper/2608.01589/) (2026)
- [PAINT: Partial-Solution Adaptive Interpolated Training for Self-Distilled Reasoners](https://mlsanity.com/paper/2604.26573/) (2026)
- [H2SD: Hybrid Hindsight Self-Distillation](https://mlsanity.com/paper/2607.18955/) (2026)
- [Exploring and Enhancing the Transfer of Distribution in Knowledge Distillation for Autoregressive Language Models](https://mlsanity.com/paper/2409.12512/) (2024)

## Built on by (most cited first, 6 of 6)

- [What Does Privileged Information Add to On-Policy Self-Distillation?](https://mlsanity.com/paper/2609.20612/) (2026)
- [SCOPE-OPSD: Fisher-Conditioned Privileged Subspaces for On-Policy Self-Distillation](https://mlsanity.com/paper/2609.12579/) (2026)
- [When Clipping Reverses Correction: Failure Dynamics of Pointwise Forward-KL On-Policy Self-Distillation](https://mlsanity.com/paper/2609.38995/) (2026)
- [Graph-Conditioned On-Policy Agent Distillation from Off-the-Shelf Teachers](https://mlsanity.com/paper/2609.37522/) (2026)
- [How Should Teachers Be Prepared? RL on Student-Induced States for On-Policy Distillation](https://mlsanity.com/paper/2610.04950/) (2026)
- [Learning from Self-Generated Responses: Reliability--Uncertainty Self-Distillation for Reasoning](https://mlsanity.com/paper/2610.19176/) (2026)

## Abstract (arXiv)

On-policy self-distillation (OPSD) uses a privileged teacher to supervise a reasoning model on prefixes sampled from its own rollouts. Yet each rollout also reveals how the student's response unfolds and whether it succeeds, student-specific hindsight that standard OPSD does not use to form the teacher. We introduce Privileged Adaptation from Student Trajectories (PAST), which treats each completed student trajectory as additional privileged information for the OPSD teacher while leaving the student's distillation prefixes unchanged. PAST preserves the student's next-token distribution on correct trajectories and uses failed trajectories to adapt the teacher toward verified success under student-proximity regularization. We characterize what such a trajectory-conditioned teacher can transfer to a prefix-only student. Forward-KL distillation projects the teacher distributions to their conditional arithmetic mean given the prefix. This projection separates trajectory-specific variation that remains privileged from the mean policy shift available to the student. For correct trajectories, the unclipped population objective also has the frozen student as an ideal distributional fixed point. Across three mathematical reasoning benchmarks, PAST improves the Avg@12 macro average over Vanilla OPSD by 5.6 percentage points. A 2×2 factorial study shows gains from both complete-trajectory access and teacher adaptation, while trajectory removal and shuffling confirm that the adapted teacher uses the matching hindsight context.