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optimization_set [2016/05/20 21:40] nikolaj |
optimization_set [2016/05/20 22:04] nikolaj |
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| @#55CCEE: context | @#55CCEE: $ B $ | | | @#55CCEE: context | @#55CCEE: $ B $ | | ||
| @#55CCEE: context | @#55CCEE: $ \langle Y, \le \rangle $ ... Non-strict partially ordered set | | | @#55CCEE: context | @#55CCEE: $ \langle Y, \le \rangle $ ... Non-strict partially ordered set | | ||
- | | @#55CCEE: context | @#55CCEE: $ s:B\to Y $ | | + | | @#55CCEE: context | @#55CCEE: $ r:B\to Y $ | |
- | | @#FF9944: definition | @#FF9944: $ O_s := \{\beta\in B\mid \forall(b\in X).\,s(\beta)\le{s(b)}\}$ | | + | | @#FF9944: definition | @#FF9944: $ O_r := \{\beta\in B\mid \forall(b\in X).\,r(\beta)\le{r(b)}\}$ | |
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- | If ${\mathrm{min}(s)}\subseteq B$ denote the minimum values of $s$, then | + | If ${\mathrm{min}(r)}\subseteq B$ denote the minimum values of $r$, then |
- | $O_s = s^{-1}({\mathrm{min}(s)})$ | + | $O_r = R^{-1}({\mathrm{inf}(r)})$ |
- | with $s^{-1}:{\mathcal P}Y\to{\mathcal P}B$. | + | with $r^{-1}:{\mathcal P}Y\to{\mathcal P}B$. |
Compare with [[Solution set]]. | Compare with [[Solution set]]. | ||
- | |||
- | == Example == | ||
- | For | ||
- | |||
- | $s:{\mathbb R}\to{\mathbb R}$ | ||
- | |||
- | $s(x):=(x-7)^2$ | ||
- | |||
- | we get | ||
- | |||
- | $O_s=\{7\}$ | ||
=== Parametrized regression === | === Parametrized regression === | ||
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and find from this set find the optimal fit (given by optimal $\beta\in B$) w.r.t. loss function $V:Y\to Y$ by optimizing | and find from this set find the optimal fit (given by optimal $\beta\in B$) w.r.t. loss function $V:Y\to Y$ by optimizing | ||
- | $s(\beta):=V(f(\beta,x),y)$ | + | $r(\beta):=V(f(\beta,x),y)$ |
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. | 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. |