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θ\theta

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Used in 64 equations

max⁡θ  Ex, y∼πθ[rϕ(x,y)]  −  β DKL(πθ ∥ πref).\max_\theta \; \mathbb{E}_{x,\, y\sim\pi_\theta}\big[r_\phi(x,y)\big] \;-\; \beta \, D_{\mathrm{KL}}\big(\pi_\theta \,\Vert\, \pi_{\mathrm{ref}}\big).

The Main Technical Approaches to AI Alignment, Compared · Equation 2

This mathematical expression combines the displayed quantities; its precise role follows from the surrounding article text.

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θ(t+1)=θ(t)−η⋅1 ⁣[θ∈θbridge]⊙∇θ L(θ(t))\theta^{(t+1)} = \theta^{(t)} - \eta \cdot \mathbb{1}\!\left[\theta \in \theta_{\text{bridge}}\right] \odot \nabla_\theta \, \mathcal{L}(\theta^{(t)})

Adapters, Native Pretraining, and Unified Tokens: The Main Multimodal Architectures, Compared · Equation 1

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

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Ljoint(θ)=E(xtext, ximg, xaud)∼D[ Ltext(θ)+Limg(θ)+Laud(θ) ]\mathcal{L}_{\text{joint}}(\theta) = \mathbb{E}_{(x_{\text{text}},\, x_{\text{img}},\, x_{\text{aud}}) \sim \mathcal{D}}\Big[\, \mathcal{L}_{\text{text}}(\theta) + \mathcal{L}_{\text{img}}(\theta) + \mathcal{L}_{\text{aud}}(\theta) \,\Big]

Adapters, Native Pretraining, and Unified Tokens: The Main Multimodal Architectures, Compared · 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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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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∇θJ(θ)=Eπ ⁣[∑t=0T∇θlog⁡πθ(at∣st)(∑t′≥trt′)].\nabla_\theta J(\theta) = \mathbb{E}_\pi\!\left[\sum_{t=0}^{T} \nabla_\theta \log \pi_\theta(a_t \mid s_t) \left(\sum_{t' \ge t} r_{t'}\right)\right].

The Hardest Unsolved Problems in AI Agent Architecture · Equation 1

This equation states a bound: one expression must stay on the indicated side of the other 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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