In large-scale hypothesis testing, computing exact $p$-values or $e$-values is often resource-intensive, creating a need for budget-aware inferential methods. We propose a general framework for active hypothesis testing that leverages inexpensive auxiliary statistics to allocate a global computational budget. For each hypothesis, our data-adaptive procedure probabilistically decides whether to compute the exact test statistic or a transformed proxy, guaranteeing a valid $p$-value or $e$-value while satisfying the exact budget constraint. Theoretical guarantees are established for our constructions, showing that the procedure achieves optimality for $e$-values and for $p$-values under independence, and admissibility for $p$-values under general dependence. Empirical results from simulations and two real-world applications, including a large-scale genome-wide association study (GWAS) and a clinical prediction task leveraging large language models (LLM), demonstrate that our framework improves statistical efficiency under fixed resource limits.

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