When is a better LLM worth paying for? Stochastic Optimization with Switching Mixed-Fidelity Oracles

Abstract: LLMs are increasingly used upstream of decision-making to clean data, detect label errors, and reconstruct missing values. Yet LLM-assisted data curation creates a new operational tradeoff: higher-fidelity models and richer prompts can be substantially more expensive, while the value of this additional fidelity depends on the downstream optimization accuracy. This raises the central question of this work: when should one pay for a better LLM, given a specific downstream task at hand? We study this question through stochastic optimization with mixed-fidelity first-order oracles, which can have systematic bias and state-dependent variance. We develop two cost-aware algorithmic frameworks that allow adaptive switching between stochastic oracles. With known bias, the first meta-algorithm applies a restarted, cost-aware policy that adaptively selects oracle fidelity across optimization stages. Its cumulative cost is no larger than that of the best fixed oracle in hindsight and can be substantially smaller. With unknown bias, we develop a budgeted certification scheme that uses occasional trusted-gradient evaluations to certify the optimization progress and eliminate unsuccessful oracle sources, and show an explicit tradeoff between certification cost and delayed detection of biased sources. We provide concrete realizations of these meta-algorithms by applying biased stochastic gradient extrapolation (SGE) and recursive regularization under state-dependent noise as subroutines. Synthetic and LLM-completion experiments illustrate the qualitative switching behavior predicted by the theory: the adaptive policy skips several expensive LLM oracles and substantially reduces cumulative cost relative to fixed high-fidelity alternatives while achieving comparable accuracy.