Linear regression

$\underset{\alpha ,\beta \in {\mathbb{R}}^n}{\mathrm{argmin}}\{{\sum }_i\Vert \alpha +\beta x_i-y_i{\Vert }_2^2\}$. The formula corresponds to finding the best parameters $\alpha ,\beta \in {\mathbb{R}}^n$ that map an input $x_i\in {\mathbb{R}}^n$ to an output $y_i\in {\mathbb{R}}^m$ by minimizing the L2 loss function. Variations of this might consider other Metric spaces in which to minimize Distance. For example,

Polynomial Regression: $\underset{{\beta }_k\in {\mathbb{R}}^n}{\mathrm{argmin}}\{{\sum }_i\Vert y_i-{\sum }_k^n{\beta }_k{x}_i^k{\Vert }_2^2\}$ Ridge Regression: $\underset{{\beta }_k\in {\mathbb{R}}^n}{\mathrm{argmin}}\{{\sum }_i\Vert y_i-{\sum }_k^n{\beta }_k{x}_i^k{\Vert }_2^2+{\sum }_k^n{\beta }_k^2\}$ LASSO Regression: $\underset{{\beta }_k\in {\mathbb{R}}^n}{\mathrm{argmin}}\{{\sum }_i\Vert y_i-{\sum }_k^n{\beta }_k{x}_i^k{\Vert }_2^2+{\sum }_k^n\Vert {\beta }_k{\Vert }_1\}$ Logistic Regression: $\underset{{\beta }_k\in {\mathbb{R}}^n}{\mathrm{argmin}}\{{\sum }_i\Vert y_i-\sigma ({\sum }_k^n{\beta }_k{x}_i^k){\Vert }_2^2\}$ where $\sigma (x):=\frac{1}{1+e^{-x}}$