1
$$
2
\hat{y} = \mathbf{w}^\top \mathbf{x} + b = \sum_{j=1}^{d} w_j x_j + b
3
$$

1
$$
2
\hat{\mathbf{y}} = \mathbf{X} \mathbf{w} + b, \quad \mathbf{X} \in \mathbb{R}^{N \times d}
3
$$

1
$$
2
\mathbf{w}^* = (\mathbf{X}^\top \mathbf{X})^{-1} \mathbf{X}^\top \mathbf{y}
3
$$

1
$$
2
P(y = 1 \mid \mathbf{x}) = \sigma(\mathbf{w}^\top \mathbf{x} + b) = \frac{1}{1 + e^{-(\mathbf{w}^\top \mathbf{x} + b)}}
3
$$

1
$$
2
\mathcal{L}_{\text{MSE}} = \frac{1}{N} \sum_{i=1}^{N} \lVert \mathbf{y}_i - \hat{\mathbf{y}}_i \rVert_2^2
3
$$

1
$$
2
\mathcal{L}_{\text{BCE}} = -\frac{1}{N} \sum_{i=1}^{N} \left[ y_i \log \hat{y}_i + (1 - y_i) \log (1 - \hat{y}_i) \right]
3
$$

1
$$
2
\mathcal{L}_{\text{CE}} = -\frac{1}{N} \sum_{i=1}^{N} \sum_{c=1}^{C} y_{ic} \log \hat{y}_{ic}
3
$$

1
$$
2
\mathcal{L}_{\text{hinge}} = \max(0, \, 1 - y_i f(\mathbf{x}_i))
3
$$

1
$$
2
\mathcal{L}_{\text{reg}} = \mathcal{L}(\mathbf{w}) + \lambda R(\mathbf{w}), \quad
3
R(\mathbf{w}) = \begin{cases}
4
\lVert \mathbf{w} \rVert_2^2 = \sum_j w_j^2, & \text{L2 (Ridge)} \\
5
\lVert \mathbf{w} \rVert_1 = \sum_j \lvert w_j \rvert, & \text{L1 (Lasso)}
6
\end{cases}
7
$$

1
$$
2
\theta_{t+1} = \theta_t - \eta \nabla_\theta \mathcal{L}(\theta_t)
3
= \theta_t - \eta \cdot \frac{1}{N} \sum_{i=1}^{N} \nabla_\theta \ell(f(\mathbf{x}_i; \theta), y_i)
4
$$

1
$$
2
\theta_{t+1} = \theta_t - \eta \nabla_\theta \ell(f(\mathbf{x}_i; \theta_t), y_i), \quad i \sim \mathcal{U}\{1, \dots, N\}
3
$$

1
$$
2
P(\theta \mid \mathcal{D}) = \frac{P(\mathcal{D} \mid \theta) \, P(\theta)}{P(\mathcal{D})}
3
= \frac{P(\mathcal{D} \mid \theta) \, P(\theta)}{\int_\Theta P(\mathcal{D} \mid \theta') P(\theta') \, d\theta'}
4
$$

1
$$
2
\theta_{\text{MAP}} = \arg\max_\theta \log P(\mathcal{D} \mid \theta) + \log P(\theta)
3
$$

1
$$
2
\theta_{\text{MLE}} = \arg\max_\theta \prod_{i=1}^{N} p(\mathbf{x}_i \mid \theta)
3
= \arg\max_\theta \sum_{i=1}^{N} \log p(\mathbf{x}_i \mid \theta)
4
$$

1
$$
2
p(\mathbf{x}) = \frac{1}{(2\pi)^{d/2} \lvert \boldsymbol{\Sigma} \rvert^{1/2}}
3
\exp\left( -\frac{1}{2} (\mathbf{x} - \boldsymbol{\mu})^\top \boldsymbol{\Sigma}^{-1} (\mathbf{x} - \boldsymbol{\mu}) \right)
4
$$

1
$$
2
\text{KL}(p \parallel q) = \int p(\mathbf{x}) \log \frac{p(\mathbf{x})}{q(\mathbf{x})} \, d\mathbf{x}
3
= \mathbb{E}_{x \sim p}\left[ \log \frac{p(x)}{q(x)} \right]
4
$$

1
$$
2
\mathcal{H}(X) = -\sum_{x} p(x) \log p(x), \quad
3
\mathcal{I}(X; Y) = \sum_{x, y} p(x, y) \log \frac{p(x, y)}{p(x) p(y)}
4
$$

1
$$
2
\text{Accuracy} = \frac{TP + TN}{TP + TN + FP + FN}, \quad
3
\text{Precision} = \frac{TP}{TP + FP}, \quad
4
\text{Recall} = \frac{TP}{TP + FN}
5
$$

1
$$
2
F_1 = 2 \cdot \frac{\text{Precision} \cdot \text{Recall}}{\text{Precision} + \text{Recall}}, \quad
3
\text{AUC} = P(s_{\text{pos}} > s_{\text{neg}})
4
$$

1
$$
2
d_{\text{Euclidean}}(\mathbf{x}, \mathbf{y}) = \lVert \mathbf{x} - \mathbf{y} \rVert_2, \quad
3
d_{\cos}(\mathbf{x}, \mathbf{y}) = 1 - \frac{\mathbf{x}^\top \mathbf{y}}{\lVert \mathbf{x} \rVert_2 \lVert \mathbf{y} \rVert_2}
4
$$

1
$$
2
\min_{\mathbf{w}, b} \frac{1}{2} \lVert \mathbf{w} \rVert^2 + C \sum_{i=1}^{N} \max(0, \, 1 - y_i(\mathbf{w}^\top \mathbf{x}_i + b))
3
$$

1
$$
2
\mathbf{z}^{(l)} = \mathbf{W}^{(l)} \mathbf{a}^{(l-1)} + \mathbf{b}^{(l)}, \quad
3
\mathbf{a}^{(l)} = f(\mathbf{z}^{(l)})
4
$$

1
$$
2
\frac{\partial \mathcal{L}}{\partial \mathbf{W}^{(l)}} = \frac{\partial \mathcal{L}}{\partial \mathbf{z}^{(l)}} \frac{\partial \mathbf{z}^{(l)}}{\partial \mathbf{W}^{(l)}}
3
= \boldsymbol{\delta}^{(l)} (\mathbf{a}^{(l-1)})^\top, \quad
4
\boldsymbol{\delta}^{(l)} = (\mathbf{W}^{(l+1)})^\top \boldsymbol{\delta}^{(l+1)} \odot f'(\mathbf{z}^{(l)})
5
$$

1
$$
2
\sigma(x) = \frac{1}{1 + e^{-x}}, \qquad
3
\tanh(x) = \frac{e^{x} - e^{-x}}{e^{x} + e^{-x}}, \qquad
4
\text{ReLU}(x) = \max(0, x)
5
$$

1
$$
2
\text{LeakyReLU}(x) = \begin{cases} x, & x \geq 0 \\ \alpha x, & x < 0 \end{cases}, \quad
3
\text{GELU}(x) = x \cdot \Phi(x) \approx x \sigma(1.702 x)
4
$$

1
$$
2
\text{softmax}(\mathbf{z})_i = \frac{e^{z_i}}{\sum_{j=1}^{C} e^{z_j}}
3
= \frac{e^{z_i - \max_k z_k}}{\sum_{j=1}^{C} e^{z_j - \max_k z_k}} \quad \text{(数值稳定版)}
4
$$

1
$$
2
(f * g)(i, j) = \sum_{m} \sum_{n} f(m, n) \, g(i - m, \, j - n)
3
$$

1
$$
2
\mathbf{Y}_{c_{\text{out}}} = \sum_{c_{\text{in}}=1}^{C_{\text{in}}} \mathbf{X}_{c_{\text{in}}} * \mathbf{K}_{c_{\text{out}}, c_{\text{in}}} + b_{c_{\text{out}}}
3
$$

1
$$
2
H_{\text{out}} = \left\lfloor \frac{H_{\text{in}} + 2p - k}{s} \right\rfloor + 1, \quad
3
W_{\text{out}} = \left\lfloor \frac{W_{\text{in}} + 2p - k}{s} \right\rfloor + 1
4
$$

1
$$
2
\mathbf{h}_t = \tanh(\mathbf{W}_{xh} \mathbf{x}_t + \mathbf{W}_{hh} \mathbf{h}_{t-1} + \mathbf{b}_h), \quad
3
\mathbf{y}_t = \mathbf{W}_{hy} \mathbf{h}_t + \mathbf{b}_y
4
$$

1
$$
2
\begin{aligned}
3
\mathbf{f}_t &= \sigma(\mathbf{W}_f [\mathbf{h}_{t-1}, \mathbf{x}_t] + \mathbf{b}_f) &&\text{遗忘门} \\
4
\mathbf{i}_t &= \sigma(\mathbf{W}_i [\mathbf{h}_{t-1}, \mathbf{x}_t] + \mathbf{b}_i) &&\text{输入门} \\
5
\mathbf{o}_t &= \sigma(\mathbf{W}_o [\mathbf{h}_{t-1}, \mathbf{x}_t] + \mathbf{b}_o) &&\text{输出门} \\
6
\tilde{\mathbf{c}}_t &= \tanh(\mathbf{W}_c [\mathbf{h}_{t-1}, \mathbf{x}_t] + \mathbf{b}_c) &&\text{候选状态} \\
7
\mathbf{c}_t &= \mathbf{f}_t \odot \mathbf{c}_{t-1} + \mathbf{i}_t \odot \tilde{\mathbf{c}}_t &&\text{细胞状态} \\
8
\mathbf{h}_t &= \mathbf{o}_t \odot \tanh(\mathbf{c}_t) &&\text{隐藏状态}
9
\end{aligned}
10
$$