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The efficacy of such sampling algorithms is limited, however, by the relationship between the LLM and the particular sampling task at hand, which has motivated the framework of test-time training (TTT). TTT works by updating a model's weights in response to partial generations and reward feedback received at inference time, thus adapting to the particular problem. In this work, we propose a formalization for TTT as the problem of producing a sample from a given probability measure $\\mu^\\star$ belonging to a known class ${F}$ of distributions, given an oracle $\\hat \\mu$ which yields approximate density estimates for $\\mu^\\star$. 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