Large language models (LLMs) are now available from cloud API providers in various sizes and configurations. While this diversity offers a broad spectrum of choices, effectively leveraging the options to optimize computational cost and performance remains challenging. In this work, we present AutoMix, an approach that strategically routes queries to larger LMs, based on the approximate correctness of outputs from a smaller LM. Central to AutoMix is a few-shot self-verification mechanism, which estimates the reliability of its own outputs without requiring training. Given that verifications can be noisy, we employ a meta-verifier in AutoMix to refine the accuracy of these assessments. Our experiments using LLAMA2-13B and GPT-4, on five context-grounded reasoning datasets demonstrate that AutoMix surpasses established baselines, improving the incremental benefit per cost by up to 86%. Our code and data are available at https://github.com/automix-llm/automix.

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