# A Note on Hybrid Online Reinforcement and Imitation Learning for LLMs: Formulations and Algorithms

arXiv: https://arxiv.org/abs/2512.23097 · HTML: https://mlsanity.com/paper/2512.23097/

## Key facts

- Cited by 3 papers in the map (data through 14 Oct 2026).
- Placed in Language-conditioned policies and VLAs, one of 13 directions in Offline RL and imitation learning from data.
- Builds on 3 of its 10 references that are in the map.
- First posted on arXiv 28 Dec 2025.

## Placed in

- Offline RL and imitation learning from data / [Language-conditioned policies and VLAs](https://mlsanity.com/offlinerl/language-conditioned-policies-and-vlas/)

## Builds on (most cited first, 3 of 3)

- [Distilling the Knowledge in a Neural Network](https://mlsanity.com/paper/1503.02531/) (2015)
- [On-Policy Distillation of Language Models: Learning from Self-Generated Mistakes](https://mlsanity.com/paper/2306.13649/) (2023)
- [Scheduled Sampling for Sequence Prediction with Recurrent Neural Networks](https://mlsanity.com/paper/1506.03099/) (2015)

## Built on by (most cited first, 3 of 3)

- [A Survey of On-Policy Distillation for Large Language Models](https://mlsanity.com/paper/2604.00626/) (2026)
- [Multi-Rollout On-Policy Distillation via Peer Successes and Failures](https://mlsanity.com/paper/2605.12652/) (2026)
- [A First-Principles Derivation of LLM Policy Optimization: From Expected Reward to GRPO and Its Structural Extensions](https://mlsanity.com/paper/2606.16733/) (2026)

## Abstract (arXiv)

We present a unified framework for Large Language Model (LLM) fine-tuning that integrates Imitation Learning and Reinforcement Learning. By analyzing the gradient of a composite objective combining trajectory-level KL divergence with task rewards, we derive a natural decomposition into two components: (1) an analytically computable Dense Gradient for token-level imitation, and (2) a Monte Carlo estimated Sparse Gradient for long-horizon reward optimization. The Dense Gradient admits a closed-form logit-level formula, enabling efficient GPU implementation.