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(Machine Learning Foundations)—Mathematical Foundations
,副教授 (Associate Professor),资讯工程学系 (Computer Science and Information Engineering)
The VC Dimension
Recap
Definition of VC Dimension
VC Dimension
the formal name of maximum non-break point
the Four VC Dimensions
VC Dimension and Learning
- finite dVC⟹g d V C ⟹ g can generalize Eout(g)≈Ein(g) E o u t ( g ) ≈ E i n ( g )
- regardless of learning algorithm A A 、input distribution P P 、target function f
VC Dimension of Perceptrons
2D PLA Revisited
d-D perceptrons: dVC=d+1 ? d V C = d + 1 ?
dVC≥d+1 d V C ≥ d + 1
- There are some
d+1 d + 1 inputs we can shatter.
- 每一行代表一个点
- 灰色部分(第一列)视作第 0 维,是常数,代表 threshhold
-
X X 可逆
- 任意的
Y Y 都可以表示出来
dVC≤d+1 d V C ≤ d + 1
- We cannot shatter any set of d+2 d + 2 inputs.
- linear dependence restricts dichotomy
- 任意一个可以 shatter 的 d+1 d + 1 向量组再加一维
- Xd+2 X d + 2 能被前 d+1 d + 1 个向量线性表出
Physical Intuition of VC Dimension
Degrees of Freedom 自由度
- dVC≈freeparameters d V C ≈ f r e e p a r a m e t e r s
Penalty for Model Complexity
with a high probability, Eout≤Ein+Ω(N,H,δ) E o u t ≤ E i n + Ω ( N , H , δ )
The VC Message
Sample Complexity