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Used in 121 equations

x^=Wdf(x)+bd,f(x)=JumpReLUθ(Wex+be),L(x)=∥x−x^∥22+λ∥f(x)∥0.\hat x = W_d f(x) + b_d, \qquad f(x) = \mathrm{JumpReLU}_\theta\big(W_e x + b_e\big), \qquad \mathcal L(x) = \lVert x - \hat x \rVert_2^2 + \lambda \lVert f(x) \rVert_0.

Probing, Sparse Autoencoders, Patching, and Steering: The Main Interpretability Methods, Compared · Equation 9

This equation states an equality: the expressions on both sides have the same value under the article’s assumptions.

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LPM(θ)=− E(x, yw, yl) ∼ DH∪DAI[log⁡σ(rθ(x,yw)−rθ(x,yl))].\mathcal{L}_{PM}(\theta) = -\,\mathbb{E}_{(x,\,y_w,\,y_l)\,\sim\, D_H \cup D_{AI}}\Big[\log \sigma\big(r_\theta(x,y_w) - r_\theta(x,y_l)\big)\Big].

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

This equation states an equality: the expressions on both sides have the same value under the article’s assumptions.

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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],

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

This equation states an equality: the expressions on both sides have the same value under the article’s assumptions.

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J(θ)=Ex∼D, y∼πθ(⋅∣x)[r(x,y)]−βDKL(πθ(⋅∣x) ∥ πref(⋅∣x))J(\theta) = \mathbb{E}_{x \sim \mathcal{D},\ y \sim \pi_\theta(\cdot \mid x)}\left[r(x,y)\right] - \beta D_{\mathrm{KL}}\left(\pi_\theta(\cdot \mid x) \,\|\, \pi_{\mathrm{ref}}(\cdot \mid x)\right)

Claude, From First Principles: Training, Constitutional Methods, and What Actually Shapes a Response · Equation 6

This equation states an equality: the expressions on both sides have the same value under the article’s assumptions.

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p(y∣x)≈∑z∈top-k(pη(⋅∣x))pη(z∣x) pθ(y∣x,z),p(y \mid x) \approx \sum_{z \in \mathrm{top}\text{-}k\left(p_\eta(\cdot \mid x)\right)} p_\eta(z \mid x) \, p_\theta(y \mid x, z),

From BM25 to Agentic Retrieval: A History of Retrieval-Augmented Generation · Equation 21

This equation gives an approximation: it relates the quantities while allowing an approximation.

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