ODAR: Principled Adaptive Routing for LLM Reasoning via Active Inference

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.

Figures (12)

• Figure 1: Multi-dimensional performance comparison across 23 benchmarks. The radar chart visualizes ODAR's average accuracy across 8 task categories relative to frontier models (GPT-5.1, Claude-4.5) and the strongest inference-time baseline (Self-Consistency). ODAR significantly expands the performance envelope, particularly in Mathematics (+20.2% on IMO 2025) and Advanced Cognition (+12.0% on HLE), while maintaining superior cost-efficiency.
• Figure 2: ODAR System Architecture (Overview). Given input $x$, a rule-based dispatch layer (ER/MR/SS; fixed heuristics and priority rules) extracts coarse task features and selects a base model identifier for dispatch. In parallel, the Difficulty Estimator predicts $d\in[0,1]$ and the Strategy Selector routes the query using fixed thresholds $\tau_1{=}0.3$ and $\tau_2{=}0.7$: Simple ($d{<}\tau_1$): Fast-only ($c{=}1$); Medium ($\tau_1{\leq}d{<}\tau_2$): Fast + Slow verification ($c{=}2$); Hard ($d{\geq}\tau_2$): Best-of-$N$ with $N{=}5$ Slow candidates plus FEP fusion ($c{=}6$). Our evaluation emphasizes difficulty-based routing and FEP fusion; ER/MR provide system-level dispatch/model selection.
• Figure 3: Multi-dimensional Ablation Matrix. Columns denote accuracy drop ($\Delta$ Acc %) relative to Full ODAR across 16 representative benchmarks. The red line indicates normalized inference cost. Results confirm that the DE prevents cost explosions while the Slow Agent ensures reasoning depth.
• Figure 4: Mapping from theoretical principles to computational implementation. Each theoretical framework contributes a distinct component: FEP enables principled fusion, Active Inference guides adaptive routing, and theta-gamma coupling motivates the dual-agent architecture.
• Figure 5: Theoretical framework overview. Left panel: neuroscientific principles including gamma oscillations, theta rhythms, and the Free Energy Principle. Middle panel: corresponding computational principles of fast pattern matching, slow verification, and uncertainty-driven resource allocation. Right panel: ODAR implementation comprising the Fast Agent, Slow Agent, and Difficulty Estimator.