水如歌对《机器学习》的笔记(3)

水如歌(A muggle.)

• 书名: 机器学习
• 作者: (美)Tom Mitchell
• 页数: 282
• 出版社: 机械工业出版社
• 出版年: 2008-3
• 第7页
1
($\lim_{b \to \rm{end}} \hat{V}(\rm{Successor}(b)) = V_{\rm{train}} (b)$) Indeed, consider the case when ($\rm{Successor}(b) = b_{\rm{end}}$).2And, if ($E = 0$) (recall ($E$) is non-negative), then
($$V_{\rm{train}} = \hat{V} = \hat{V} \circ \rm{Successor} \implies \hat{V} = \rm{Const}$$)
While notice the boundary values of ($\hat{V}$), we get
($$\lim_{E \to 0^+} \hat{V} = V$$)
which is what we hope.
2017-01-26 12:43:32 回应
• 第23页
Another equivalent, but more intuitive, definitions of ($G$) and ($S$) are:
($$G := \{ g \in H | \rm{Consistent}(g, D) \text{ and: } \forall g' \in H, \rm{Consistent}(g', D), \text{ must have } g \geq_{g} g' \}$$)
($$S := \{ s \in H | \rm{Consistent}(s, D) \text{ and: } \forall s' \in H, \rm{Consistent}(s', D), \text{ must have } s' \geq_{g} s \}$$)Or say,
($$G := \sup_{\geq_{g}} (\rm{VS}_{H, D}) \;,\; S:= \inf_{\geq_{g}} (\rm{VS}_{H, D})$$)
2017-01-26 12:51:31 回应
• 第23页
Proof: (i) If ($g \geq_g h \geq_g s$), then ($\forall x \in X$),
($$s(x) = 1 \implies h(x) = 1 \implies g(x) = 1$$)
or
($$g(x) = 0 \implies g(x) = 0 \implies s(x) = 0$$)
And since ($\forall x \in D, g(x) = c(x), s(x) = c(x)$), we have
($$\forall x \in \{ x \in D | c(x) = 1 \}, s(x) = 1 \implies h(x) = 1$$)
and
($$\forall x \in \{ x \in D | c(x) = 0 \}, g(x) = 0 \implies h(x) = 0$$)
So, ($\forall h$) satisfies ($g \geq_g h \geq_g s$) is in ($\rm{VS}_{H, D}$). (ii) By definitions of ($G$) and ($S$) (the equivalent version), ($\forall h \in \rm{VS}_{H, D}$) has ($g \geq_g h \geq_g s$). Q.E.D.Bug: 显示有问题，但预览时是好的。
2017-01-26 12:58:25 回应