# From Base Rollouts to RL Reasoning: A Budgeted Search Perspective

arXiv: https://arxiv.org/abs/2609.01274 · HTML: https://mlsanity.com/paper/2609.01274/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 19 of its 26 references that are in the map.
- First posted on arXiv 1 Sep 2026.

## Builds on (most cited first, 19 of 19)

- [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)
- [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)
- [Direct Preference Optimization: Your Language Model is Secretly a Reward Model](https://mlsanity.com/paper/2305.18290/) (2023)
- [Measuring Mathematical Problem Solving With the MATH Dataset](https://mlsanity.com/paper/2103.03874/) (2021)
- [Let's Verify Step by Step](https://mlsanity.com/paper/2305.20050/) (2023)
- [GPQA: A Graduate-Level Google-Proof Q&A Benchmark](https://mlsanity.com/paper/2311.12022/) (2023)
- [Scaling LLM Test-Time Compute Optimally can be More Effective than Scaling Model Parameters](https://mlsanity.com/paper/2408.03314/) (2024)
- [Language Models Don't Always Say What They Think: Unfaithful Explanations in Chain-of-Thought Prompting](https://mlsanity.com/paper/2305.04388/) (2023)
- [Large Language Monkeys: Scaling Inference Compute with Repeated Sampling](https://mlsanity.com/paper/2407.21787/) (2024)
- [Does Reinforcement Learning Really Incentivize Reasoning Capacity in LLMs Beyond the Base Model?](https://mlsanity.com/paper/2504.13837/) (2025)
- [Reasoning with Language Model is Planning with World Model](https://mlsanity.com/paper/2305.14992/) (2023)
- [SFT Memorizes, RL Generalizes: A Comparative Study of Foundation Model Post-training](https://mlsanity.com/paper/2501.17161/) (2025)
- [Towards Faithfully Interpretable NLP Systems: How should we define and evaluate faithfulness?](https://mlsanity.com/paper/2004.03685/) (2020)
- [Language Agent Tree Search Unifies Reasoning Acting and Planning in Language Models](https://mlsanity.com/paper/2310.04406/) (2023)
- [SimpleRL-Zoo: Investigating and Taming Zero Reinforcement Learning for Open Base Models in the Wild](https://mlsanity.com/paper/2503.18892/) (2025)
- [Echo Chamber: RL Post-training Amplifies Behaviors Learned in Pretraining](https://mlsanity.com/paper/2504.07912/) (2025)
- [Inference-Aware Fine-Tuning for Best-of-N Sampling in Large Language Models](https://mlsanity.com/paper/2412.15287/) (2024)

## Abstract (arXiv)

Reinforcement learning with verifiable rewards (RLVR) improves language-model reasoning, but how these gains relate to inference-time decoding and search remains unclear. Does RL create reasoning the base model lacks, or shift the rollout distribution toward trajectories it can already reach but rarely samples? We study this behaviorally with a Unified Decoding Framework (UDF), which expresses token-level sampling, beam-like search, tree search, and sequence-level resampling as executable policies over a shared budgeted operating space, scored post hoc with pass@k, self-consistency, best-of-N, and first-finish success. Using paired Base/RL checkpoints from SimpleRL-Zoo, we ask whether an RL default-policy curve can be approximated by a structured path of Base operating points. On Math500, AIME, GPQA, and IFEval, the pass@k recovery path follows a Budgeted Operating-Point Transition Rule (BOPTR), NBase ≈ α NRLβ, with benchmark-conditioned exponents. On Qwen2.5-7B, BOPTR gives the lowest transfer error among the non-oracle rules we test, 3.41 pp (95% CI [2.32, 5.53]); a three-seed replication gives 3.07 ± 0.39 pp. The rule extends to ten models across four families (3.28 to 4.87 pp on checkpoints added after fitting), to four benchmarks it was never fitted on (5.03 pp vs. 4.44 pp in fit), and holds without an RL checkpoint for the target model (4.19 pp) or without RL supervision of any kind (5.08 pp). These results support a qualified internalized-search reading: under the recipe we test, much of the measured RL gain corresponds to a change in sampling efficiency toward operating points the base model can already reach under search. We treat the scaling patterns as descriptive of this recipe and cohort, report where they break down, and use UDF and BOPTR as behavioral diagnostics rather than evidence of parameter-level equivalence.