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

Abstract

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 systems and six kinds of real verifiers, all from public releases, SPLIT lands within 0.02-0.8 percentage points of the best allocation in hindsight, while every fixed rule we test loses at least 2.5 points in some setting. Regret depends only on how well SPLIT ranks configurations, and it stays near-optimal under compute, price and latency costs and online. The formula also explains when asking more LLM judges lowers precision. As a forecaster far beyond the pilot, SPLIT beats power-law fitting when the verifier caps success, and we report where it loses.