# SAGE: Shaping Anchors for Guided Exploration in RLVR of LLMs

arXiv: https://arxiv.org/abs/2605.18864 · HTML: https://mlsanity.com/paper/2605.18864/

## Key facts

- Cited by 3 papers in the map (data through 14 Oct 2026).
- Placed in Reinforcement learning for reasoning, one of 17 directions in LLM reasoning.
- Builds on 30 of its 40 references that are in the map.
- First posted on arXiv 15 May 2026.

## Placed in

- LLM reasoning / [Reinforcement learning for reasoning](https://mlsanity.com/lr/reinforcement-learning-for-reasoning/)

## Builds on (most cited first, 30 of 30)

- [LoRA: Low-Rank Adaptation of Large Language Models](https://mlsanity.com/paper/2106.09685/) (2021)
- [DeepSeekMath: Pushing the Limits of Mathematical Reasoning in Open Language Models](https://mlsanity.com/paper/2402.03300/) (2024)
- [Evaluating Large Language Models Trained on Code](https://mlsanity.com/paper/2107.03374/) (2021)
- [Efficient Memory Management for Large Language Model Serving with PagedAttention](https://mlsanity.com/paper/2309.06180/) (2023)
- [Measuring Mathematical Problem Solving With the MATH Dataset](https://mlsanity.com/paper/2103.03874/) (2021)
- [DAPO: An Open-Source LLM Reinforcement Learning System at Scale](https://mlsanity.com/paper/2503.14476/) (2025)
- [HybridFlow: A Flexible and Efficient RLHF Framework](https://mlsanity.com/paper/2409.19256/) (2024)
- [Understanding R1-Zero-Like Training: A Critical Perspective](https://mlsanity.com/paper/2503.20783/) (2025)
- [Qwen2.5-Math Technical Report: Toward Mathematical Expert Model via Self-Improvement](https://mlsanity.com/paper/2409.12122/) (2024)
- [Tulu 3: Pushing Frontiers in Open Language Model Post-Training](https://mlsanity.com/paper/2411.15124/) (2024)
- [Does Reinforcement Learning Really Incentivize Reasoning Capacity in LLMs Beyond the Base Model?](https://mlsanity.com/paper/2504.13837/) (2025)
- [Beyond the 80/20 Rule: High-Entropy Minority Tokens Drive Effective Reinforcement Learning for LLM Reasoning](https://mlsanity.com/paper/2506.01939/) (2025)
- [Reasoning with Exploration: An Entropy Perspective](https://mlsanity.com/paper/2506.14758/) (2025)
- [Pass@k Training for Adaptively Balancing Exploration and Exploitation of Large Reasoning Models](https://mlsanity.com/paper/2508.10751/) (2025)
- [On Memorization of Large Language Models in Logical Reasoning](https://mlsanity.com/paper/2410.23123/) (2024)
- [SEED-GRPO: Semantic Entropy Enhanced GRPO for Uncertainty-Aware Policy Optimization](https://mlsanity.com/paper/2505.12346/) (2025)
- [Pass@K Policy Optimization: Solving Harder Reinforcement Learning Problems](https://mlsanity.com/paper/2505.15201/) (2025)
- [TreeRL: LLM Reinforcement Learning with On-Policy Tree Search](https://mlsanity.com/paper/2506.11902/) (2025)
- [An Empirical Study on Eliciting and Improving R1-like Reasoning Models](https://mlsanity.com/paper/2503.04548/) (2025)
- [T1: Advancing Language Model Reasoning through Reinforcement Learning and Inference Scaling](https://mlsanity.com/paper/2501.11651/) (2025)
- [AceReason-Nemotron 1.1: Advancing Math and Code Reasoning through SFT and RL Synergy](https://mlsanity.com/paper/2506.13284/) (2025)
- [Depth-Breadth Synergy in RLVR: Unlocking LLM Reasoning Gains with Adaptive Exploration](https://mlsanity.com/paper/2508.13755/) (2025)
- [The Choice of Divergence: A Neglected Key to Mitigating Diversity Collapse in Reinforcement Learning with Verifiable Reward](https://mlsanity.com/paper/2509.07430/) (2025)
- [Unlocking Exploration in RLVR: Uncertainty-aware Advantage Shaping for Deeper Reasoning](https://mlsanity.com/paper/2510.10649/) (2025)
- [KL-Regularized Reinforcement Learning is Designed to Mode Collapse](https://mlsanity.com/paper/2510.20817/) (2025)
- [On the Design of KL-Regularized Policy Gradient Algorithms for LLM Reasoning](https://mlsanity.com/paper/2505.17508/) (2025)
- [BNPO: Beta Normalization Policy Optimization](https://mlsanity.com/paper/2506.02864/) (2025)
- [Rethinking Entropy Regularization in Large Reasoning Models](https://mlsanity.com/paper/2509.25133/) (2025)
- [Decomposing the Entropy-Performance Exchange: The Missing Keys to Unlocking Effective Reinforcement Learning](https://mlsanity.com/paper/2508.02260/) (2025)
- [Unlocking Reasoning Capabilities in LLMs via Reinforcement Learning Exploration](https://mlsanity.com/paper/2510.03865/) (2025)

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

- [Toward Plasticity-Preserving KL Regularization for Capability Retention in LLM Reinforcement Learning](https://mlsanity.com/paper/2608.01743/) (2026)
- [The Model Knows Another Way: Strategy Switching for Effective RLVR Exploration](https://mlsanity.com/paper/2609.33085/) (2026)
- [Mid-Harness: Scaling Actions Between Model and Harness for Terminal Agents](https://mlsanity.com/paper/2609.39982/) (2026)

## Abstract (arXiv)

Recent studies observe that reinforcement learning with verifiable rewards (RLVR) reliably improves pass@1 on reasoning tasks, yet often fails to yield comparable gains in pass@k, raising the question of whether RLVR genuinely enables large language models to acquire novel reasoning abilities or merely enhances the efficiency of sampling reasoning modes already present in the base model. Prior analyses largely support the latter view, attributing this limitation to structural properties of standard RLVR objectives that result in insufficient exploration pressure. In this work, we argue that a central structural constraint arises from reverse-KL regularization, which stabilizes training but inherently anchors the policy to the reference distribution, thereby suppressing the emergence of alternative reasoning modes. However, we show that neither removing the KL term nor replacing it with forward-KL provides a satisfactory solution, as both disrupt the efficiency-coverage trade-off by either inducing reward hacking or allocating probability mass to off-target regions. To resolve this tension, we propose SAGE, a principled framework that enables controllable empirical support expansion by reshaping the reverse-KL anchor distribution itself through a guide function q(x,y), achieving consistent improvements in both pass@1 and pass@k across challenging mathematical reasoning benchmarks. Our code is available at https://github.com/tally0818/SAGE.