Label Adaptivity in Best-of-n Budget Selection

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Choosing a deployment budget for best-of-n search requires correctness information about candidates selected at different score scales. We study a fixed pool of M independent scored candidates with all scores visible and a budget of B correctness labels. For a finite menu D, let N=maxD, let J count multiplicatively separated scales, and set d=1-min(D)/N. We prove matching minimax expected-regret bounds of order dmin{1,sqrt(N/M+J/B)} for adaptive label queries and dmin{1,sqrt(N/M+J log(eJ)/B)} when the query set is chosen without labels. The logarithmic adaptivity cost enters only the label term; the diameter factor covers arbitrarily narrow menus. Lower bounds permit arbitrary score--label distributions and expected label budgets, while the upper constructions satisfy integer hard budgets. As a companion result, we obtain the largest pointwise mean squared error rate min{1,N/M+J/B} in this fixed-pool, general-menu experiment, building on the previously established full-menu two-resource estimation law. The analysis combines classical rank-based estimation, importance sampling, tournaments, and information inequalities. Exact checks and synthetic estimation diagnostics illustrate finite-budget limitations; they do not test the adaptive selection construction or establish real-task gains.