## Optimization set

### Set

context | $ B $ |

context | $ \langle Y, \le \rangle $ … Non-strict partially ordered set |

context | $ r:B\to Y $ |

definition | $ O_r := \{\beta\in B\mid \forall(b\in B).\,r(\beta)\le{r(b)}\}$ |

todo

#tag

If p are parameters and c_p(x) curves with x_min(c_p)=f(p) known, try to find x_min(c') by fitting c_p to c'. Now what's p here. Is there a scheme so that we can extend the list p to have guaranteed that there are parameters so that eventually c_p=c'?

If ${\mathrm{min}(r)}\subseteq Y$ denote the minimum values of $r$, then

$O_r = r^{-1}({\mathrm{min}(r)})$

with $r^{-1}:{\mathcal P}Y\to{\mathcal P}B$.

Compare with Solution set.

#### Parametrized regression

Consider a test set

$\langle x_0,y_0\rangle \in X\times Y$,

where $x_0$ somehow depends on $y_0$.

Use $B$-family of fit functions

$f:B\to(X\to Y)$

(the indexed subspace of $X\to Y$ is called hypotheses space)

and find from this set find the optimal fit (given by optimal $\beta\in B$) w.r.t. loss function

$V:Y\times Y\to Y$

by optimizing

$r(\beta):=V(f(\beta,x),y)$

As a remark, given a function $f$ (resp. a $\beta$), the value $V(f(\beta,x_0),y_0)$ (or a multiple thereof) is called “empirical risk” in Statistical learning theory.

##### Linear regression w.r.t. least square

$f(\beta,x):=\beta_0+\sum_{i=1}^N\beta_ix_i$

with loss function

$V({\hat y},y)=({\hat y}-y)\cdot({\hat y}-y)$

In practice, $x_i$ may be vectors and then $V$ is taken to be an inner product.