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TABLE OF CONTENTS
EQUATION LIST
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BLEU Bilingual Evaluation Understudy
#nlp #BLEU #evaluation
$$ \text{BLEU}_{w}(\hat{S},S)=BP(\hat{S};S) \times \exp{\sum^{\infty}_{n=1}w_{n} \ln p_{n}(\hat{S};S)}, p_{n}(\hat{S};S)=\frac{\sum^{M}_{i=1}\sum_{s \in G_{n}(\hat{y})} \min(C(s,\hat{y}),\max_{y \in S_{i}} C(s,y))}{\sum^{M}_{i=1}\sum_{s \in G_{n}(\hat{y})}C(s, \hat{y})}, p_{n}(\hat{y};y)=\frac{\sum_{s \in G_{n}(\hat{y})} \min(C(s,\hat{y}), C(s,y))}{\sum_{s \in G_{n}(\hat{y})}C(s, \hat{y})}, BP(\hat{S};S) = e^{-(r/c-1)^{+}}$$
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Direct Policy Optimization DPO
#nlp #llm #RLHF
$$\pi_{r} (y|x) = \frac{1}{Z(x)} \pi_{ref} (y|x) \exp(\frac{1}{\beta} r(x,y) ) , r(x,y) = \beta \log \frac{\pi_{r} (y|x)}{\pi_{ref} (y|x)} + \beta \log Z(x) , p^{*}(y_{1} > y_{2} |x) = \frac{1}{1+\exp{(\beta \frac{\pi^{*} (y_{2}|x)}{\pi_{ref} (y_{2}|x)} - \beta \frac{\pi^{*} (y_{1}|x)}{\pi_{ref} (y_{1}|x)} )}} , \mathcal{L}_{DPO}(\pi_{\theta};\pi_{ref}) = -\mathbb{E}_{(x, y_{w},y_{l}) \sim D } [\log \sigma (\beta \frac{\pi_{\theta} (y_{w}|x)}{\pi_{ref} (y_{w}|x)} - \beta \frac{\pi_{\theta} (y_{l}|x)}{\pi_{ref} (y_{l}|x)} )] , \nabla \mathcal{L}_{DPO}(\pi_{\theta};\pi_{ref}) = - \beta \mathbb{E}_{(x, y_{w},y_{l}) \sim D } [ \sigma ( \hat{r}_{\theta} (x, y_{l}) - \hat{r}_{\theta} (x, y_{w})) [\nabla_{\theta} \log \pi (y_{w}|x) - \nabla_{\theta} \log \pi (y_{l}|x) ] ] , \hat{r}_{\theta} (x, y) = \beta \log (\frac{\pi_{\theta} (y|x)}{\pi_{ref} (y|x)})$$
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Hidden Markov Model
#machine learning #nlp
$$Q=\{q_{1},q_{2},...,q_{N}\}, V=\{v_{1},v_{2},...,v_{M}\} \\ I=\{i_{1},i_{2},...,i_{T}\},O=\{o_{1},o_{2},...,o_{T}\} \\ A=[a_{ij}]_{N \times N}, a_{ij}=P(i_{t+1}=q_{j}|i_{t}=q_{i}) \\ B=[b_{j}(k)]_{N \times M},b_{j}(k)=P(o_{t}=v_{k}|i_{t}=q_{j})$$
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Diffusion Model Forward Process Reparameterization
#machine learning #diffusion
$$x_{t}=\sqrt{\alpha_{t}}x_{t-1}+\sqrt{1-\alpha_{t}} \epsilon_{t-1}\\=\sqrt{\alpha_{t}\alpha_{t-1}}x_{t-2} + \sqrt{1-\alpha_{t}\alpha_{t-1}} \bar{\epsilon}_{t-2}\\=\text{...}\\=\sqrt{\bar{\alpha}_{t}}x_{0}+\sqrt{1-\bar{\alpha}_{t}}\epsilon \\\alpha_{t}=1-\beta_{t}, \bar{\alpha}_{t}=\prod_{t=1}^{T}\alpha_{t}$$
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Diffusion Model Reverse Process
#machine learning #diffusion
$$p_\theta(\mathbf{x}_{0:T}) = p(\mathbf{x}_T) \prod^T_{t=1} p_\theta(\mathbf{x}_{t-1} \vert \mathbf{x}_t) \\ p_\theta(\mathbf{x}_{t-1} \vert \mathbf{x}_t) = \mathcal{N}(\mathbf{x}_{t-1}; \boldsymbol{\mu}_\theta(\mathbf{x}_t, t), \boldsymbol{\Sigma}_\theta(\mathbf{x}_t, t))$$
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Diffusion Model Variational Lower Bound
#machine learning #diffusion
$$\begin{aligned} - \log p_\theta(\mathbf{x}_0) &\leq - \log p_\theta(\mathbf{x}_0) + D_\text{KL}(q(\mathbf{x}_{1:T}\vert\mathbf{x}_0) \| p_\theta(\mathbf{x}_{1:T}\vert\mathbf{x}_0) ) \\ &= -\log p_\theta(\mathbf{x}_0) + \mathbb{E}_{\mathbf{x}_{1:T}\sim q(\mathbf{x}_{1:T} \vert \mathbf{x}_0)} \Big[ \log\frac{q(\mathbf{x}_{1:T}\vert\mathbf{x}_0)}{p_\theta(\mathbf{x}_{0:T}) / p_\theta(\mathbf{x}_0)} \Big] \\ &= -\log p_\theta(\mathbf{x}_0) + \mathbb{E}_q \Big[ \log\frac{q(\mathbf{x}_{1:T}\vert\mathbf{x}_0)}{p_\theta(\mathbf{x}_{0:T})} + \log p_\theta(\mathbf{x}_0) \Big] \\ &= \mathbb{E}_q \Big[ \log \frac{q(\mathbf{x}_{1:T}\vert\mathbf{x}_0)}{p_\theta(\mathbf{x}_{0:T})} \Big] \\ \text{Let }L_\text{VLB} &= \mathbb{E}_{q(\mathbf{x}_{0:T})} \Big[ \log \frac{q(\mathbf{x}_{1:T}\vert\mathbf{x}_0)}{p_\theta(\mathbf{x}_{0:T})} \Big] \geq - \mathbb{E}_{q(\mathbf{x}_0)} \log p_\theta(\mathbf{x}_0) \end{aligned}$$
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Diffusion Model Variational Lower Bound Loss
#machine learning #diffusion
$$\begin{aligned} L_\text{VLB} &= L_T + L_{T-1} + \dots + L_0 \\ \text{where } L_T &= D_\text{KL}(q(\mathbf{x}_T \vert \mathbf{x}_0) \parallel p_\theta(\mathbf{x}_T)) \\ L_t &= D_\text{KL}(q(\mathbf{x}_t \vert \mathbf{x}_{t+1}, \mathbf{x}_0) \parallel p_\theta(\mathbf{x}_t \vert\mathbf{x}_{t+1})) \text{ for }1 \leq t \leq T-1 \\ L_0 &= - \log p_\theta(\mathbf{x}_0 \vert \mathbf{x}_1) \end{aligned}$$
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Hidden Markov Model
#machine learning #nlp
$$Q=\{q_{1},q_{2},...,q_{N}\}, V=\{v_{1},v_{2},...,v_{M}\} \\ I=\{i_{1},i_{2},...,i_{T}\},O=\{o_{1},o_{2},...,o_{T}\} \\ A=[a_{ij}]_{N \times N}, a_{ij}=P(i_{t+1}=q_{j}|i_{t}=q_{i}) \\ B=[b_{j}(k)]_{N \times M},b_{j}(k)=P(o_{t}=v_{k}|i_{t}=q_{j})$$
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