ODAR: Principled Adaptive Routing for LLM Reasoning via Active Inference

TL;DR

ODAR-Expert introduces an adaptive routing framework for LLM reasoning that dynamically allocates resources between heuristic and deliberative agents, improving accuracy and efficiency through active inference and free-energy principles.

Contribution

This work presents ODAR-Expert, a novel adaptive routing method for LLMs that balances accuracy and compute cost using active inference and a free-energy-based decision mechanism.

Findings

Achieves 98.2% accuracy on MATH benchmark.

Reduces computational costs by 82% compared to homogeneous sampling.

Improves the compute-accuracy trade-off across 23 benchmarks.

Abstract

The paradigm of large language model (LLM) reasoning is shifting from parameter scaling to test-time compute scaling, yet many existing approaches still rely on uniform brute-force sampling (for example, fixed best-of-N or self-consistency) that is costly, hard to attribute, and can trigger overthinking with diminishing returns. We propose ODAR-Expert, an adaptive routing framework that optimizes the accuracy-efficiency trade-off via principled resource allocation. ODAR uses a difficulty estimator grounded in amortized active inference to dynamically route queries between a heuristic Fast Agent and a deliberative Slow Agent. We further introduce a free-energy-principled, risk-sensitive fusion mechanism that selects answers by minimizing a variational free energy objective, balancing log-likelihood with epistemic uncertainty (varentropy) as a principled alternative to ad hoc voting over heterogeneous candidates. Extensive evaluation across 23 benchmarks shows strong and consistent gains, including 98.2% accuracy on MATH and 54.8% on Humanity's Last Exam (HLE), while improving the compute-accuracy frontier under compute-matched settings. We also validate reproducibility on a fully open-source stack (Llama 4 + DeepSeek), where ODAR surpasses homogeneous sampling strategies while reducing computational costs by 82%. Overall, our results suggest that thinking-optimal scaling requires adaptive resource allocation with free-energy-based decision-making rather than simply increasing test-time compute.