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Equation 11 · Part 6 · How Constitutional AI Actually Constrains a Model's Behavior

Symbol r_θ

LPMAI(θ)=− E(x, yA, yB, p) ∼ DAI[p log⁡σ(rθ(x,yA)−rθ(x,yB))+(1−p) log⁡σ(rθ(x,yB)−rθ(x,yA))],\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],
rθr_\theta

What this part means

r_θ appears inside an expected value, so its contribution is averaged under the distribution or condition shown by that operator.

Its job in the formula

r_θ appears inside an expected value, so its contribution is averaged under the distribution or condition shown by that operator.

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, ywy_w, yly_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 yAy_A over yBy_B rather than a hard choice, the corresponding cross-entropy term is LPMAI(θ)=− E(x, yA, yB, p) ∼ DAI[p log⁡σ(rθ(x,yA)−rθ(x,yB))+(1−p) log⁡σ(rθ(x,yB)−rθ(x,yA))]\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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