Linear models and loss
Pick a parametric family $f_\theta(x)$, then choose $\theta$ by minimizing a loss on the data.
$$\hat y = f(x,\theta),\quad L=L(\hat y,y),\quad \theta=\arg\min_\theta L$$
Often we use gradients of $L$ w.r.t. $\theta$ to walk downhill (see Optimization).
Cannon range
Physics says range $R$ depends on angle $\phi$ and muzzle speed $v_0$. You measure noisy ranges at a few angles and fit $v_0$ by minimizing mean absolute error between predicted and measured $R$. That is supervised learning as parameter fitting.
Key takeaways
- Learning as optimization over $\theta$.
- Loss defines what “fit” means.
- Gradients power many trainers.