# LEGO-OPD: Factorized Teacher Composition for Multimodal On-Policy Distillation

arXiv: https://arxiv.org/abs/2610.00333 · HTML: https://mlsanity.com/paper/2610.00333/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Placed in Multimodal reasoning, one of 17 directions in LLM reasoning.
- Builds on 24 of its 27 references that are in the map.
- First posted on arXiv 29 Sep 2026.

## Placed in

- LLM reasoning / [Multimodal reasoning](https://mlsanity.com/lr/multimodal-reasoning/)
- Knowledge distillation / [Multimodal distillation](https://mlsanity.com/distill/multimodal-distillation/)

## Builds on (most cited first, 24 of 24)

- [Let's Verify Step by Step](https://mlsanity.com/paper/2305.20050/) (2023)
- [MMMU: A Massive Multi-discipline Multimodal Understanding and Reasoning Benchmark for Expert AGI](https://mlsanity.com/paper/2311.16502/) (2023)
- [MMLU-Pro: A More Robust and Challenging Multi-Task Language Understanding Benchmark](https://mlsanity.com/paper/2406.01574/) (2024)
- [MathVista: Evaluating Mathematical Reasoning of Foundation Models in Visual Contexts](https://mlsanity.com/paper/2310.02255/) (2023)
- [OlympiadBench: A Challenging Benchmark for Promoting AGI with Olympiad-Level Bilingual Multimodal Scientific Problems](https://mlsanity.com/paper/2402.14008/) (2024)
- [On-Policy Distillation of Language Models: Learning from Self-Generated Mistakes](https://mlsanity.com/paper/2306.13649/) (2023)
- [Mitigating Object Hallucinations in Large Vision-Language Models through Visual Contrastive Decoding](https://mlsanity.com/paper/2311.16922/) (2023)
- [MathVerse: Does Your Multi-modal LLM Truly See the Diagrams in Visual Math Problems?](https://mlsanity.com/paper/2403.14624/) (2024)
- [BLINK: Multimodal Large Language Models Can See but Not Perceive](https://mlsanity.com/paper/2404.12390/) (2024)
- [Measuring Multimodal Mathematical Reasoning with MATH-Vision Dataset](https://mlsanity.com/paper/2402.14804/) (2024)
- [MMMU-Pro: A More Robust Multi-discipline Multimodal Understanding Benchmark](https://mlsanity.com/paper/2409.02813/) (2024)
- [DExperts: Decoding-Time Controlled Text Generation with Experts and Anti-Experts](https://mlsanity.com/paper/2105.03023/) (2021)
- [Rethinking On-Policy Distillation of Large Language Models: Phenomenology, Mechanism, and Recipe](https://mlsanity.com/paper/2604.13016/) (2026)
- [DeepMath-103K: A Large-Scale, Challenging, Decontaminated, and Verifiable Mathematical Dataset for Advancing Reasoning](https://mlsanity.com/paper/2504.11456/) (2025)
- [Q-Bench: A Benchmark for General-Purpose Foundation Models on Low-level Vision](https://mlsanity.com/paper/2309.14181/) (2023)
- [A General Framework for Updating Belief Distributions](https://mlsanity.com/paper/1306.6430/) (2013)
- [Tuning Language Models by Proxy](https://mlsanity.com/paper/2401.08565/) (2024)
- [R-Bench: Graduate-level Multi-disciplinary Benchmarks for LLM & MLLM Complex Reasoning Evaluation](https://mlsanity.com/paper/2505.02018/) (2025)
- [MMFineReason: Closing the Multimodal Reasoning Gap via Open Data-Centric Methods](https://mlsanity.com/paper/2601.21821/) (2026)
- [Visual-Advantage On-Policy Distillation for Vision-Language Models](https://mlsanity.com/paper/2605.21924/) (2026)
- [Decomposed On-Policy Distillation for Vision-Language Reasoning: Steering Gradients for Visual Grounding](https://mlsanity.com/paper/2606.00564/) (2026)
- [H-OPD: Confidence Aware Heterogeneous Multi-Teacher Multimodal On-policy Distillation](https://mlsanity.com/paper/2607.02592/) (2026)
- [Seeing Before Reasoning: Decoupling Perception and Reasoning for Shortcut-Resilient Multimodal On-Policy Self-Distillation](https://mlsanity.com/paper/2606.19120/) (2026)
- [Universal Reasoner: A Single, Composable Plug-and-Play Reasoner for Frozen LLMs](https://mlsanity.com/paper/2505.19075/) (2025)

## Abstract (arXiv)

Multimodal on-policy distillation (OPD) aims to improve visual grounding while preserving the strong reasoning capabilities of language models. Recent multi-teacher approaches combine LLM and VLM teachers to provide complementary supervision. However, directly using a VLM's full predictive distribution entangles its visual grounding signal with its own language prior, preventing the grounding information from being transferred independently. Conversely, increasing the strength of visual supervision can improve perception but may overemphasize visual evidence and degrade language reasoning. To address this trade-off, we introduce LEGO-OPD, which selectively composes factors from a Language Expert and a Grounding expert into One teacher distribution for multimodal OPD. Under a generalized Bayesian formulation, the language expert provides a prior over candidate tokens, while the grounding expert contributes a visual likelihood that updates this prior, rather than transferring its complete predictive distribution. This factorized composition allows language reasoning and visual grounding to be controlled independently. We further introduce adaptive calibration to determine how strongly the visual likelihood should update the language prior at each decoding prefix. Specifically, LEGO-OPD uses the grounding expert's image-induced prediction shift as a prefix-dependent reference, preventing both insufficient and excessive visual supervision. Experiments with Qwen3 models show that LEGO-OPD consistently outperforms the evaluated single- and multi-teacher OPD baselines on both multimodal and text-only reasoning tasks. Moreover, it improves the initial student's visual perception while preserving text-only reasoning.