# Depth-Breadth Synergy in RLVR: Unlocking LLM Reasoning Gains with Adaptive Exploration

arXiv: https://arxiv.org/abs/2508.13755 · HTML: https://mlsanity.com/paper/2508.13755/

## Key facts

- Cited by 43 papers in the map (data through 14 Oct 2026).
- Placed in Reinforcement learning for reasoning, one of 17 directions in LLM reasoning.
- Builds on 25 of its 42 references that are in the map.
- First posted on arXiv 19 Aug 2025.

## Placed in

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

## Builds on (most cited first, 25 of 25)

- [GPT-4 Technical Report](https://mlsanity.com/paper/2303.08774/) (2023)
- [Training language models to follow instructions with human feedback](https://mlsanity.com/paper/2203.02155/) (2022)
- [Proximal Policy Optimization Algorithms](https://mlsanity.com/paper/1707.06347/) (2017)
- [The Llama 3 Herd of Models](https://mlsanity.com/paper/2407.21783/) (2024)
- [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)
- [DeepSeek-R1: Incentivizing Reasoning Capability in LLMs via Reinforcement Learning](https://mlsanity.com/paper/2501.12948/) (2025)
- [Let's Verify Step by Step](https://mlsanity.com/paper/2305.20050/) (2023)
- [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)
- [OpenAI o1 System Card](https://mlsanity.com/paper/2412.16720/) (2024)
- [Kimi k1.5: Scaling Reinforcement Learning with LLMs](https://mlsanity.com/paper/2501.12599/) (2025)
- [Qwen2.5-Math Technical Report: Toward Mathematical Expert Model via Self-Improvement](https://mlsanity.com/paper/2409.12122/) (2024)
- [Does Reinforcement Learning Really Incentivize Reasoning Capacity in LLMs Beyond the Base Model?](https://mlsanity.com/paper/2504.13837/) (2025)
- [SimpleRL-Zoo: Investigating and Taming Zero Reinforcement Learning for Open Base Models in the Wild](https://mlsanity.com/paper/2503.18892/) (2025)
- [Cognitive Behaviors that Enable Self-Improving Reasoners, or, Four Habits of Highly Effective STaRs](https://mlsanity.com/paper/2503.01307/) (2025)
- [VAPO: Efficient and Reliable Reinforcement Learning for Advanced Reasoning Tasks](https://mlsanity.com/paper/2504.05118/) (2025)
- [Learning to Reason under Off-Policy Guidance](https://mlsanity.com/paper/2504.14945/) (2025)
- [Echo Chamber: RL Post-training Amplifies Behaviors Learned in Pretraining](https://mlsanity.com/paper/2504.07912/) (2025)
- [OctoThinker: Mid-training Incentivizes Reinforcement Learning Scaling](https://mlsanity.com/paper/2506.20512/) (2025)
- [SRFT: A Single-Stage Method with Supervised and Reinforcement Fine-Tuning for Reasoning](https://mlsanity.com/paper/2506.19767/) (2025)
- [Pangu Embedded: An Efficient Dual-system LLM Reasoner with Metacognition](https://mlsanity.com/paper/2505.22375/) (2025)
- [Rethinking Reflection in Pre-Training](https://mlsanity.com/paper/2504.04022/) (2025)
- [Decomposing Elements of Problem Solving: What "Math" Does RL Teach?](https://mlsanity.com/paper/2505.22756/) (2025)

## Built on by (most cited first, 30 of 43)

- [The Landscape of Agentic Reinforcement Learning for LLMs: A Survey](https://mlsanity.com/paper/2509.02547/) (2025)
- [A Survey of Reinforcement Learning for Large Reasoning Models](https://mlsanity.com/paper/2509.08827/) (2025)
- [Jointly Reinforcing Diversity and Quality in Language Model Generations](https://mlsanity.com/paper/2509.02534/) (2025)
- [Improving Sampling Efficiency in RLVR through Adaptive Rollout and Response Reuse](https://mlsanity.com/paper/2509.25808/) (2025)
- [Prune-OPD: Efficient and Reliable On-Policy Distillation for Long-Horizon Reasoning](https://mlsanity.com/paper/2605.07804/) (2026)
- [Learning to Self-Verify Makes Language Models Better Reasoners](https://mlsanity.com/paper/2602.07594/) (2026)
- [Beyond the Sampled Token: Preserving Candidate Support in RLVR](https://mlsanity.com/paper/2510.14807/) (2025)
- [Adaptive Rollout Allocation for Online Reinforcement Learning with Verifiable Rewards](https://mlsanity.com/paper/2602.01601/) (2026)
- [Representation-Based Exploration for Language Models: From Test-Time to Post-Training](https://mlsanity.com/paper/2510.11686/) (2025)
- [EnvFactory: Scaling Tool-Use Agents via Executable Environments Synthesis and Robust RL](https://mlsanity.com/paper/2605.18703/) (2026)
- [Learn Hard Problems During RL with Reference Guided Fine-tuning](https://mlsanity.com/paper/2603.01223/) (2026)
- [Random Policy Valuation is Enough for LLM Reasoning with Verifiable Rewards](https://mlsanity.com/paper/2509.24981/) (2025)
- [V0: A Generalist Value Model for Any Policy at State Zero](https://mlsanity.com/paper/2602.03584/) (2026)
- [When Silence Is Golden: Can LLMs Learn to Abstain in Temporal QA and Beyond?](https://mlsanity.com/paper/2602.04755/) (2026)
- [Off-Context GRPO: Learning to Reason on Hard Problems using Privileged Information](https://mlsanity.com/paper/2607.19313/) (2026)
- [Video-STR: Reinforcing MLLMs in Video Spatio-Temporal Reasoning with Relation Graph](https://mlsanity.com/paper/2510.10976/) (2025)
- [Accordion-Thinking: Self-Regulated Step Summaries for Efficient and Readable LLM Reasoning](https://mlsanity.com/paper/2602.03249/) (2026)
- [Composition-RL: Compose Your Verifiable Prompts for Reinforcement Learning of Large Language Models](https://mlsanity.com/paper/2602.12036/) (2026)
- [Efficient RLVR Training via Weighted Mutual Information Data Selection](https://mlsanity.com/paper/2603.01907/) (2026)
- [ResRL: Boosting LLM Reasoning via Negative Sample Projection Residual Reinforcement Learning](https://mlsanity.com/paper/2605.00380/) (2026)
- [CARE What Fails: Contrastive Anchored-REflection for Verifiable Multimodal Reasoning](https://mlsanity.com/paper/2512.19554/) (2025)
- [Low-rank Optimization Trajectories Modeling for LLM RLVR Acceleration](https://mlsanity.com/paper/2604.11446/) (2026)
- [Optimizing What Policies Learn From: Recoverability-aware Rollout Intervention Learning](https://mlsanity.com/paper/2608.05080/) (2026)
- [SAGE: Shaping Anchors for Guided Exploration in RLVR of LLMs](https://mlsanity.com/paper/2605.18864/) (2026)
- [Tail-Likelihood Reinforcement Learning](https://mlsanity.com/paper/2609.02987/) (2026)
- [Minerva: Reinforcement Learning with Verifiable Rewards for Cyber Threat Intelligence LLMs](https://mlsanity.com/paper/2602.00513/) (2026)
- [WebGen-R1: Incentivizing Large Language Models to Generate Functional and Aesthetic Websites with Reinforcement Learning](https://mlsanity.com/paper/2604.20398/) (2026)
- [Modularized Reinforcement Learning on LLMs: From MDP Creation to Exploration and Learning](https://mlsanity.com/paper/2606.21943/) (2026)
- [Critique to Verify: Accurate and Honest Test-Time Scaling with RL-Trained Verifiers](https://mlsanity.com/paper/2509.23152/) (2025)
- [Group Pattern Selection Optimization: Let LRMs Pick the Right Pattern for Reasoning](https://mlsanity.com/paper/2601.07238/) (2026)

## Abstract (arXiv)

Reinforcement Learning with Verifiable Reward (RLVR) is a powerful method for enhancing the reasoning abilities of Large Language Models, but its full potential is limited by a lack of exploration in two key areas: Depth (the difficulty of problems) and Breadth (the number of training instances). Our analysis of the popular GRPO algorithm reveals a bias that down-weights difficult, low-accuracy problems, which are crucial for improving reasoning skills. To address this, we introduce Difficulty Adaptive Rollout Sampling (DARS), a method that re-weights difficult problems by using targeted, multi-stage rollouts. DARS increases the number of rollout outcomes for these harder problems according to our proposed re-balancing schedules and leads to consistent gains in Pass@K. We discovered that increasing rollout size alone does not improve performance and may actually impair it. In contrast, scaling the batch size to increase breadth via full-batch updates significantly boosted Pass@1 metrics. This improvement stems from higher token-level entropy, ensuring robust exploration and minimized gradient noise. We further present DARS-Breadth, a combined approach that uses DARS with a large breadth of training data. This method demonstrates simultaneous gains in both Pass@K and Pass@1, confirming that depth (adaptive exploration) and breadth (scaling iteration instances) are orthogonal and complementary dimensions for unlocking the full power of RLVR.