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The passage around this formula

Nothing in that loss distinguishes where a pair came from; a human-labeled winner and an AI-labeled winner are interchangeable once written down as (x, $y_w$, $y_l$) . That is the precise, narrow sense in which Constitutional AI “differs mechanically from plain RLHF”: it changes the labeling function for one half of one dataset, not the loss, not the optimizer, not the use of a KL penalty against the supervised policy. For the soft-labeled AI comparisons specifically, where the feedback model outputs a probability p of preferring response $y_A$ over $y_B$ rather than a hard choice, the corresponding cross-entropy term is $\mathcal{L}^{AI}_{PM}(\theta) = -\,\mathbb{E}_{(x,\,y_A,\,y_B,\,p)\,\sim\, D_{AI}}\Big[p\,\log \sigma\big(r_\theta(x,y_A)-r_\theta(x,y_B)\big) + (1-p)\,\log \sigma\big(r_\theta(x,y_B)-r_\theta(x,y_A)\big)\Big]$. which exposes the other genuine mechanical difference:…

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