Impact of Data Distribution on Reward Model Training

A researcher is training a reward model using a small preference dataset, $\mathcal{D}_r$, which contains exactly two preference pairs:

1. For input $\mathbf{x}_1$, response $\mathbf{y}_{1a}$ is preferred over $\mathbf{y}_{1b}$.
2. For input $\mathbf{x}_2$, response $\mathbf{y}_{2a}$ is preferred over $\mathbf{y}_{2b}$.

Given the empirical loss formula $\mathcal{L}_r(\phi) = -\frac{1}{|\mathcal{D}_r|} \sum_{(\mathbf{x},\mathbf{y}_a,\mathbf{y}_b)\in\mathcal{D}_r} \log \text{Pr}_{\phi}(\mathbf{y}_a \succ \mathbf{y}_b|\mathbf{x})$, which of the following expressions correctly represents the loss for this specific dataset?

Comparing Reward Model Performance