Sample, Mix, or Verify? Allocating Test-Time Compute under Imperfect Verifiers

Srinivasan Iyengar

A common way to make language models and agents more accurate is to spend more compute at inference time: generate many candidate solutions and submit the one a verifier prefers. Success then requires that some candidate is correct and that the verifier picks it. The first grows predictably with the number of samples; the second does not, because real verifiers (unit tests, majority votes, LLM judges, reward models) accept wrong answers, reject right ones, and make correlated mistakes. With a fixed budget, a practitioner must decide how many samples to draw, which models to draw them from, and how much to spend on verification, and today these choices are made by rules of thumb. We introduce SPLIT, which makes them from a small labelled pilot. It rests on an exact formula for the probability of submitting a correct answer under accept/reject and top-score selection that accounts for varying task difficulty, blind spots shared across models, and correlated repeated verifier calls. SPLIT fits the formula by empirical Bayes and uses it to plan budgets on unseen tasks. Across seven benchmarks spanning math, science QA, code and software-engineering agents, 26 models, 175 agent…