A Regime Theory of Controller Class Selection for LLM Action Decisions

Language and vision-language models must decide whether to answer directly, retrieve evidence, defer to a stronger model, or abstain. Existing routing methods refine routers of a preselected type but offer little guidance on choosing the type itself. We address this gap through controller class selection, where a controller maps inputs to actions. We organize controllers into four classes: fixed actions, partition routers, instance-level learned controllers, and prior-gated controllers. With finite data, greater flexibility is not always better: different statistical regimes favor different classes. Our framework guides class selection using three measurable properties: how much improvement is possible beyond the best fixed action, whether there are enough samples to reliably detect gains from instance-level signals, and how much improvement a coarse partition can recover. We derive a finite-sample threshold for certifying selective prediction gains, with a matching-order lower bound, alongside bounds on residual improvement and partition gains. These results motivate a selection procedure that computes its diagnostics from training data alone. Across ten benchmark settings evaluated under strict nested cross-validation, the predicted class matches the empirical winner in nine. Replacing the judge on the same dataset shifts the diagnostics and reverses the winning class, as predicted. On TextVQA, an OCR experiment further separates the benefit of additional information from that of a deterministic gate, which outperforms the tested learners given the same information. Code is available at https://github.com/Anonymous-Awesome-Submissions/Regime-Theory.