Formula

Optimal Parameters Formula in Fine-Tuning

The optimal parameters, denoted as θ~\tilde{\theta}, obtained through fine-tuning are found by maximizing an objective function over the tuning dataset Dtune\mathcal{D}_{\mathrm{tune}}. This relationship is formally expressed as:

θ~=arg maxθ^+sampleDtuneLθ^+(sample)\tilde{\theta} = \argmax_{\hat{\theta}^+} \sum_{\mathrm{sample} \in \mathcal{D}_{\mathrm{tune}}} \mathcal{L}_{\hat{\theta}^+}(\mathrm{sample})

In this equation, θ^+\hat{\theta}^+ represents the parameters being actively optimized, which are initialized from the pre-trained parameters θ^\hat{\theta}, and Lθ^+(sample)\mathcal{L}_{\hat{\theta}^+}(\mathrm{sample}) calculates the objective value for a given sample.

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Updated 2026-04-19

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Foundations of Large Language Models

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