# The Weakest Link Tells It All: Outcome-Supervised Process Reward Modeling via Learnable Credit Assignment

arXiv: https://arxiv.org/abs/2606.27739 · HTML: https://mlsanity.com/paper/2606.27739/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Placed in Reinforcement learning for reasoning, one of 17 directions in LLM reasoning.
- Builds on 17 of its 55 references that are in the map.
- First posted on arXiv 26 Jun 2026.

## Placed in

- LLM reasoning / [Reinforcement learning for reasoning](https://mlsanity.com/lr/reinforcement-learning-for-reasoning/)
- LLM alignment / RLHF / preference optimisation / [Process reward models for reasoning](https://mlsanity.com/alignment/process-reward-models-for-reasoning/)

## Builds on (most cited first, 17 of 17)

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- [Training Verifiers to Solve Math Word Problems](https://mlsanity.com/paper/2110.14168/) (2021)
- [DeepSeekMath: Pushing the Limits of Mathematical Reasoning in Open Language Models](https://mlsanity.com/paper/2402.03300/) (2024)
- [Qwen3 Technical Report](https://mlsanity.com/paper/2505.09388/) (2025)
- [Qwen2.5 Technical Report](https://mlsanity.com/paper/2412.15115/) (2024)
- [Mistral 7B](https://mlsanity.com/paper/2310.06825/) (2023)
- [Qwen2.5-Coder Technical Report](https://mlsanity.com/paper/2409.12186/) (2024)
- [Qwen2.5-Math Technical Report: Toward Mathematical Expert Model via Self-Improvement](https://mlsanity.com/paper/2409.12122/) (2024)
- [Solving math word problems with process- and outcome-based feedback](https://mlsanity.com/paper/2211.14275/) (2022)
- [Process Reinforcement through Implicit Rewards](https://mlsanity.com/paper/2502.01456/) (2025)
- [Improve Mathematical Reasoning in Language Models by Automated Process Supervision](https://mlsanity.com/paper/2406.06592/) (2024)
- [OpenR: An Open Source Framework for Advanced Reasoning with Large Language Models](https://mlsanity.com/paper/2410.09671/) (2024)
- [A Survey of Process Reward Models: From Outcome Signals to Process Supervisions for Large Language Models](https://mlsanity.com/paper/2510.08049/) (2025)
- [Dynamic Policy-Driven Adaptive Multi-Instance Learning for Whole Slide Image Classification](https://mlsanity.com/paper/2403.07939/) (2024)
- [Learning and Interpreting Multi-Multi-Instance Learning Networks](https://mlsanity.com/paper/1810.11514/) (2018)
- [From Correlation to Causation: Max-Pooling-Based Multi-Instance Learning Leads to More Robust Whole Slide Image Classification](https://mlsanity.com/paper/2408.09449/) (2024)
- [DreamPRM-Code: Function-as-Step Process Reward Model with Label Correction for LLM Coding](https://mlsanity.com/paper/2512.15000/) (2025)

## Abstract (arXiv)

Process reward models (PRMs) enhance the reasoning capabilities of large language models (LLMs) by providing fine-grained feedback, yet training PRMs typically requires expensive stepwise annotations. Outcome-supervised PRMs offer a scalable alternative by learning from final-answer correctness alone, but this introduces a fundamental credit assignment challenge, i.e., attributing outcomes to responsible reasoning steps. Existing approaches rely on either uniform or causal assignment, both of which fail to anchor credit in step correctness and thus hinder process error identification. In this work, we propose Outcome-Supervised Process Reward Modeling via Learnable Credit Assignment (LCA), an outcome-supervised PRM framework that jointly learns credit assignment and reward modeling under the principle of Weakest Link Assignment: a reasoning chain is as strong as its weakest link. To address mutual dependence between credit assignment and reward modeling, we formalize outcome-supervised PRM as a Multiple Instance Learning (MIL) problem and introduce Softmax-Weighted-Sum (SWS) pooling, an MIL pooling technique tailored for strong dependence and redundancy among reasoning states. We establish instance-level Bayes consistency of the resulting learning objective under mild assumptions. Extensive experiments demonstrate that LCA outperforms state-of-the-art outcome-supervised PRMs across multiple tasks and backbones. Code is available at https://github.com/JiaTianyu20031016/MIL.git.