RADAR: Reasoning-Ability and Difficulty-Aware Routing for Reasoning LLMs

TL;DR

RADAR tackles the practical challenge of deploying reasoning LLMs under cost constraints by routing each query to an optimal {RLM, reasoning budget} configuration. It discretizes configurations, formulates routing as a two-objective optimization over predicted performance and cost, and leverages an IRT-based calibration to model query difficulty and configuration ability, with adaptive testing to add new models efficiently. By employing linear and Chebyshev scalarization, RADAR constructs Pareto-optimal frontiers and demonstrates superior performance and generalization across eight challenging benchmarks, including robust out-of-distribution evaluation. The framework remains plug-and-play and low-latency, offering interpretable routing decisions and scalable integration of new models, making it practical for real-world heterogeneous RLM ecosystems.

Abstract

Reasoning language models have demonstrated remarkable performance on many challenging tasks in math, science, and coding. Choosing the right reasoning model for practical deployment involves a performance and cost tradeoff at two key levels: model size and reasoning budget, where larger models and higher reasoning budget lead to better performance but with increased cost and latency. In this work, we tackle this tradeoff from the angle of model configuration routing for different queries, and present RADAR (Reasoning-Ability and Difficulty-Aware Routing), a lightweight, interpretable, and scalable routing framework. Inspired by psychometrics, RADAR learns an item response model from model responses with different budgets to different queries, with interpretable parameters including query difficulties and model-budget abilities. RADAR then routes queries with higher difficulty to model-budget pairs with higher ability, and vice versa. We conduct extensive experiments on 8 widely used challenging reasoning benchmarks, demonstrating the superior performance of RADAR compared to state-of-the-art model routing methods. RADAR also exhibits query generalization capabilities, showing strong performance on out-of-distribution queries in all benchmarks. RADAR is also scalable and can efficiently integrate additional models by dynamically selecting a small set of evaluation queries to estimate their abilities.

Figures (6)

• Figure 1: Left: Our pilot study on MATH-500 hendrycks2021measuring shows a performance differential over (RLM, reasoning budget) configurations with the smallest RLM already solving over $50\%$ of the queries with minimal reasoning. Right: Radar exploits this performance differential by jointly estimating query difficulties and configuration abilities, and routing queries to sufficiently able configurations, thereby optimizing performance-cost tradeoffs towards the Pareto frontier. On out-of-domain queries from FRAMES krishna2024fact, Radar can match $90\%$ of the performance of OpenAI o4-mini with high reasoning effort at just $10\%$ of its cost, with the next best method song2025irt requiring $30\%$ of the cost.
• Figure 2: Illustration of our Radar framework. Left: Radar jointly estimates interpretable query difficulties and RLM configuration abilities using IRT (simplified for illustration purposes; full details in Section \ref{['sec:irt']}). New RLM configurations can be rapidly added by estimating their ability on a small subset of dynamically selected queries using adaptive testing (Section \ref{['sec:model-generalization']}). Right: Radar formulates routing as multi-objective optimization and routes queries to sufficiently capable configurations, optimizing performance-cost tradeoffs towards the Pareto frontier (Section \ref{['sec:moo']}).
• Figure 3: Radar estimates interpretable query difficulties and RLM configuration abilities. Left: Mean predicted correctness probability of configurations on questions with $5$ different ground-truth difficulty levels in MATH-500. As difficulties increase, configurations with higher abilities are predicted to perform better. Right: Fraction of routing calls on MATH-500 queries spread across RLM configurations when varying the performance-cost tradeoff weight. A lower (higher) weight leverages a higher fraction of Qwen3 (o4-mini) configurations, prioritizing cost (performance).
• Figure 4: Fraction of routing calls on OOD queries from FRAMES spread across RLM configurations when varying the performance-cost tradeoff weight before (left) and after (right) adding new RLM configuration from Qwen3-14B. Radar rapidly estimates the ability of Qwen3-14B at $16$K reasoning budget to leverage it for improved performance.
• Figure 5: We show the Pareto performance-cost tradeoff curves for all methods on ID queries across benchmarks. Radar outperforms baselines denoting better performance-cost tradeoffs towards the Pareto frontier.