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Past exam of the mathematics course of the University of Cambridge
/
2022
/
ii
/
Paper 2
/
31J
/
c
/
Solution
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Past exam of the mathematics course of the University of Cambridge
2022
ii
Paper 2
31J
c
Created
2026-09-24
Updated
2026-09-25
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Let
Z
=
(
X
,
Y
)
be an independent
test
example and write
Z
i
=
(
X
i
,
Y
i
)
. Exchangeability of
Z
and
Z
i
gives
E
ℓ
(
H
D
,
Z
)
=
n
1
∑
i
=
1
n
E
ℓ
(
H
D
i
(
Z
)
,
Z
i
)
.
(1)
The datasets
D
i
(
Z
)
and
D
differ in one coordinate, so stability gives
E
ℓ
(
H
D
i
(
Z
)
,
Z
i
)
≤
E
ℓ
(
H
D
,
Z
i
)
+
β
.
(2)
Averaging over
i
and subtracting the empirical loss proves
E
F
(
D
)
≤
β
.
(3)
Ancestors
(11)
c
31J
Paper 2
ii
2022
Past exam of the mathematics course of the University of Cambridge
Mathematics course of the University of Cambridge
Course of the University of Cambridge
University of Cambridge
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