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.

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.

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

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.

Non-strict partial order

Solution set