Formula

Objective Function for Fine-Tuning a Strong LLM with Weak Supervision

The process of fine-tuning a strong Large Language Model using synthetic data generated by a weak model can be mathematically formalized. Given a collection of inputs XX, where each input x∈X\mathbf{x} \in X includes an instruction and any necessary user input, a weak LLM denoted by Pr⁡w(⋅)\Pr^{w}(\cdot) generates a prediction y^=arg⁡max⁡yPr⁡w(y∣x)\hat{\mathbf{y}} = \arg\max_{\mathbf{y}} \Pr^{w}(\mathbf{y}|\mathbf{x}). The strong LLM, denoted by Prθs(⋅)\mathrm{Pr}^{s}_{\theta}(\cdot), is then trained on these predictions. The objective is to find the optimal model parameters θ~\tilde{\theta} that maximize the log-probability of the weak model's generated predictions: θ~=arg⁡max⁡θ∑x∈Xlog⁡Prθs(y^∣x)\tilde{\theta} = \arg\max_{\theta} \sum_{\mathbf{x} \in X} \log \mathrm{Pr}_{\theta}^{s}(\hat{\mathbf{y}}|\mathbf{x}).

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Updated 2026-05-01

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Ch.4 Alignment - Foundations of Large Language Models

Foundations of Large Language Models

Foundations of Large Language Models Course

Computing Sciences

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