# Policy-Guided Stepwise Model Routing for Cost-Effective Reasoning

arXiv: https://arxiv.org/abs/2605.06116 · HTML: https://mlsanity.com/paper/2605.06116/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Placed in Test-time compute and efficient thinking, one of 17 directions in LLM reasoning.
- Builds on 29 of its 30 references that are in the map.
- First posted on arXiv 7 May 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, 29 of 29)

- [Chain-of-Thought Prompting Elicits Reasoning in Large Language Models](https://mlsanity.com/paper/2201.11903/) (2022)
- [Training Verifiers to Solve Math Word Problems](https://mlsanity.com/paper/2110.14168/) (2021)
- [Scaling Laws for Neural Language Models](https://mlsanity.com/paper/2001.08361/) (2020)
- [Efficient Memory Management for Large Language Model Serving with PagedAttention](https://mlsanity.com/paper/2309.06180/) (2023)
- [Self-Consistency Improves Chain of Thought Reasoning in Language Models](https://mlsanity.com/paper/2203.11171/) (2022)
- [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)
- [Fast Inference from Transformers via Speculative Decoding](https://mlsanity.com/paper/2211.17192/) (2022)
- [Qwen2.5-Math Technical Report: Toward Mathematical Expert Model via Self-Improvement](https://mlsanity.com/paper/2409.12122/) (2024)
- [Accelerating Large Language Model Decoding with Speculative Sampling](https://mlsanity.com/paper/2302.01318/) (2023)
- [Large Language Monkeys: Scaling Inference Compute with Repeated Sampling](https://mlsanity.com/paper/2407.21787/) (2024)
- [Constrained Policy Optimization](https://mlsanity.com/paper/1705.10528/) (2017)
- [IMPALA: Scalable Distributed Deep-RL with Importance Weighted Actor-Learner Architectures](https://mlsanity.com/paper/1802.01561/) (2018)
- [FrugalGPT: How to Use Large Language Models While Reducing Cost and Improving Performance](https://mlsanity.com/paper/2305.05176/) (2023)
- [RouteLLM: Learning to Route LLMs with Preference Data](https://mlsanity.com/paper/2406.18665/) (2024)
- [Deep Learning Scaling is Predictable, Empirically](https://mlsanity.com/paper/1712.00409/) (2017)
- [Hybrid LLM: Cost-Efficient and Quality-Aware Query Routing](https://mlsanity.com/paper/2404.14618/) (2024)
- [Robots That Ask For Help: Uncertainty Alignment for Large Language Model Planners](https://mlsanity.com/paper/2307.01928/) (2023)
- [Omni-MATH: A Universal Olympiad Level Mathematic Benchmark For Large Language Models](https://mlsanity.com/paper/2410.07985/) (2024)
- [Beyond Chinchilla-Optimal: Accounting for Inference in Language Model Scaling Laws](https://mlsanity.com/paper/2401.00448/) (2023)
- [Reward-Guided Speculative Decoding for Efficient LLM Reasoning](https://mlsanity.com/paper/2501.19324/) (2025)
- [Fly-Swat or Cannon? Cost-Effective Language Model Choice via Meta-Modeling](https://mlsanity.com/paper/2308.06077/) (2023)
- [Reasoning with Sampling: Your Base Model is Smarter Than You Think](https://mlsanity.com/paper/2510.14901/) (2025)
- [KOR-Bench: Benchmarking Language Models on Knowledge-Orthogonal Reasoning Tasks](https://mlsanity.com/paper/2410.06526/) (2024)
- [SpecReason: Fast and Accurate Inference-Time Compute via Speculative Reasoning](https://mlsanity.com/paper/2504.07891/) (2025)
- [Online conformal prediction with decaying step sizes](https://mlsanity.com/paper/2402.01139/) (2024)
- [xVerify: Efficient Answer Verifier for Reasoning Model Evaluations](https://mlsanity.com/paper/2504.10481/) (2025)
- [Confidence-Guided Stepwise Model Routing for Cost-Efficient Reasoning](https://mlsanity.com/paper/2511.06190/) (2025)
- [Conformal Constrained Policy Optimization for Cost-Effective LLM Agents](https://mlsanity.com/paper/2511.11828/) (2025)

## Abstract (arXiv)

Inference-time computation has greatly enhanced the performance of large language models (LLMs) on challenging reasoning tasks, but this strategy can incur high inference costs. One solution is to route intermediate chain-of-thought (CoT) states to language models of different sizes; however, existing approaches rely on handcrafted routing strategies that limit performance, or on training large process reward models that may be infeasible in many applications. We formulate stepwise model routing as a constrained decision-making problem, which we solve by training a small control policy using reinforcement learning in conjunction with threshold calibration to tune the performance-efficiency tradeoff. We validate our method on three math benchmarks (GSM8K, MATH500, and OmniMath) on both open and closed models. Our method consistently improves the accuracy-cost tradeoff compared to handcrafted approaches, while achieving a comparable tradeoff to methods that require training large process reward models.