# SPIRAL: Learning to Search and Aggregate

arXiv: https://arxiv.org/abs/2606.23595 · HTML: https://mlsanity.com/paper/2606.23595/

## Key facts

- Cited by 3 papers in the map (data through 14 Oct 2026).
- Placed in Test-time compute and efficient thinking, one of 17 directions in LLM reasoning.
- Builds on 47 of its 72 references that are in the map.
- First posted on arXiv 22 Jun 2026.

## Placed in

- LLM reasoning / [Test-time compute and efficient thinking](https://mlsanity.com/lr/test-time-compute-and-efficient-thinking/)

## Builds on (most cited first, 30 of 47)

- [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)
- [Self-Consistency Improves Chain of Thought Reasoning in Language Models](https://mlsanity.com/paper/2203.11171/) (2022)
- [Reflexion: Language Agents with Verbal Reinforcement Learning](https://mlsanity.com/paper/2303.11366/) (2023)
- [Gemini 2.5: Pushing the Frontier with Advanced Reasoning, Multimodality, Long Context, and Next Generation Agentic Capabilities](https://mlsanity.com/paper/2507.06261/) (2025)
- [Continuous control with deep reinforcement learning](https://mlsanity.com/paper/1509.02971/) (2015)
- [Tree of Thoughts: Deliberate Problem Solving with Large Language Models](https://mlsanity.com/paper/2305.10601/) (2023)
- [Scaling LLM Test-Time Compute Optimally can be More Effective than Scaling Model Parameters](https://mlsanity.com/paper/2408.03314/) (2024)
- [Improving Factuality and Reasoning in Language Models through Multiagent Debate](https://mlsanity.com/paper/2305.14325/) (2023)
- [Understanding R1-Zero-Like Training: A Critical Perspective](https://mlsanity.com/paper/2503.20783/) (2025)
- [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)
- [STaR: Bootstrapping Reasoning With Reasoning](https://mlsanity.com/paper/2203.14465/) (2022)
- [Math-Shepherd: Verify and Reinforce LLMs Step-by-step without Human Annotations](https://mlsanity.com/paper/2312.08935/) (2023)
- [AlphaEvolve: A coding agent for scientific and algorithmic discovery](https://mlsanity.com/paper/2506.13131/) (2025)
- [Mastering Chess and Shogi by Self-Play with a General Reinforcement Learning Algorithm](https://mlsanity.com/paper/1712.01815/) (2017)
- [Automatic Chain of Thought Prompting in Large Language Models](https://mlsanity.com/paper/2210.03493/) (2022)
- [The Entropy Mechanism of Reinforcement Learning for Reasoning Language Models](https://mlsanity.com/paper/2505.22617/) (2025)
- [Mixture-of-Agents Enhances Large Language Model Capabilities](https://mlsanity.com/paper/2406.04692/) (2024)
- [Cognitive Behaviors that Enable Self-Improving Reasoners, or, Four Habits of Highly Effective STaRs](https://mlsanity.com/paper/2503.01307/) (2025)
- [Large Language Models Sensitivity to The Order of Options in Multiple-Choice Questions](https://mlsanity.com/paper/2308.11483/) (2023)
- [MiniMax-M1: Scaling Test-Time Compute Efficiently with Lightning Attention](https://mlsanity.com/paper/2506.13585/) (2025)
- [Chain of Agents: Large Language Models Collaborating on Long-Context Tasks](https://mlsanity.com/paper/2406.02818/) (2024)
- [Sample Efficient Actor-Critic with Experience Replay](https://mlsanity.com/paper/1611.01224/) (2016)
- [Learning to Discover at Test Time](https://mlsanity.com/paper/2601.16175/) (2026)
- [Skeleton-of-Thought: Prompting LLMs for Efficient Parallel Generation](https://mlsanity.com/paper/2307.15337/) (2023)
- [Winning Gold at IMO 2025 with a Model-Agnostic Verification-and-Refinement Pipeline](https://mlsanity.com/paper/2507.15855/) (2025)
- [Jointly Reinforcing Diversity and Quality in Language Model Generations](https://mlsanity.com/paper/2509.02534/) (2025)
- [Learning Adaptive Parallel Reasoning with Language Models](https://mlsanity.com/paper/2504.15466/) (2025)

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

- [QuasiMoTTo: Quasi-Monte Carlo Test-Time Scaling](https://mlsanity.com/paper/2607.01179/) (2026)
- [Stage-Replay Divergence Follows the KV Cache: Fixed-Prefix Precision Controls and Bidirectional Cache Transplantation](https://mlsanity.com/paper/2607.28495/) (2026)
- [Hermes: Learning Contextual Reasoning Unlocks Test-Time Scaling](https://mlsanity.com/paper/2609.38332/) (2026)

## Abstract (arXiv)

Language model reasoning can be substantially improved at test time via scaffolds that scale inference compute across different primitives -- sequential reasoning within a trace, independently sampled parallel traces, and aggregation of multiple reasoning traces into a final response. During post-training, however, language models are optimized only for sequential reasoning within a single trace. We introduce Sequential-Parallel-Aggregative Reinforcement Learning (SPIRAL), a framework in which a language model is trained to use all three primitives, as part of a unified inference compute pipeline. Concretely, the language model first samples a set of independent traces in parallel, each produced through sequential chain-of-thought reasoning, and then generates a final aggregation trace conditioned on those traces; all components are optimized end-to-end against the reward of the final aggregated response. To train this system, SPIRAL uses set reinforcement learning to teach models to produce a set of traces that are collectively useful for an aggregator and standard reinforcement learning to teach models to aggregate the set into improved final responses. Our experiments on reasoning tasks show that SPIRAL effectively scales with inference compute, outperforming GRPO by up to 11× scaling efficiency and 15% higher performance when all three compute primitives are scaled.