ENTROROUTER: Learning Efficient Model Routing via Entropy Regulation

Large Language Models (LLMs) present a fundamental trade-off: larger models offer superior reasoning but incur high costs, while smaller models are efficient but often lack the capability for complex tasks. While multi-round routing agents have been proposed to manage this trade-off, they often suffer from a structural failure mode where the router prematurely stops calling external experts. This paper investigates why this happens and proposes a more robust, single-round routing alternative.

The authors identify a phenomenon they call Trust Region Collapse, where the deep coupling of reasoning and routing, combined with strong pre-training priors, causes the router to ignore external expert responses and default to its own limited capabilities. To solve this, they introduce EntroRouter, a two-stage framework that treats entropy regulation as a primary objective:

1. Soft Supervision: Instead of training the router to mimic a single hard label, the model is trained on a distribution of viable experts. This establishes a high-entropy prior that preserves the policy's ability to handle problem ambiguity.
2. Soft Anchor: During reinforcement learning, the authors regularize the policy using a reference distribution derived from offline capability estimates. This creates a safe trust region that prevents the policy from degenerating while guiding it toward cost-efficient, high-confidence decisions.

EntroRouter effectively decouples the planning (routing) phase from the reasoning phase. By treating routing as a meta-analysis task rather than an interleaved reasoning step, the framework avoids the interference of the router's internal pre-training priors. Experiments across seven mathematical reasoning benchmarks demonstrate that EntroRouter achieves 98.3% of the accuracy of the strongest fixed-expert baseline while reducing computational costs by nearly half. The authors also show that the model generalizes well to out-of-distribution tasks and maintains robustness even when offline capability estimates contain noise.