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Taylor Expansion
Taylor Expansion
6 selected
Difficulty 2-7
6 unseen
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Foundation
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1 foundation
3 intermediate
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Question 1 of 6
120s
foundation (2/10)
state theorem
For a
C
2
function
f
near
x
, the second-order Taylor expansion is
f
(
x
+
h
)
≈
f
(
x
)
+
g
⊤
h
+
2
1
h
⊤
H
h
for small
h
. What are
g
and
H
?
Hide and think first
A.
g
=
∇
f
(
x
)
(gradient),
H
=
∇
2
f
(
x
)
(Hessian). Standard quadratic Taylor model.
B.
g
=
∇
2
f
(
x
)
(Hessian),
H
=
∇
f
(
x
)
(
∇
f
(
x
)
)
⊤
(rank-one outer product).
C.
g
=
∇
f
(
x
)
as a row,
H
=
identity matrix; second-order term is
2
1
∥
h
∥
2
.
D.
g
=
h
(the perturbation itself),
H
=
∇
2
f
(
x
)
. The linear term is the step, not the gradient.
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