Vitamin 10 | Data 102
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University of California, Berkeley *
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Course
102
Subject
Computer Science
Date
May 8, 2024
Type
Pages
11
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10/10
Questions Answered
Saved on Apr 17 at 1:22 PM
Vitamin 10
Q1
1 Point
Calculate the Hoe
ff
ding bound for a variable which is the sample mean of 10 random variables which are each bounded between 3 and 8. In other words, compute a bound for the following probability where is the expected value of .
Explanation
Hoe
ff
ding’s inequality states . Remember that is the sample mean of 10 variables which are bounded on the interval . Therefore, . Then, we can rewrite the probability as which is a direct expression of Hoe
ff
ding's inequality where . This is Z
P
(
Z
−
μ
≥
t
)
μ
Z
10
10
E
[
X
]
i
25
2
1
t
>=0
min
e
10
t
E
[
e
]
t
(
Z
−
μ
)
exp
−
(
5
4
t
2
)
P
(
X
−
E
[
X
])
≥
ϵ
≤
(
n
1
∑
i
=1
n
i
i
)
exp
−
(
(
b
−
a
)
2
2
n
ϵ
2
)
Z
X
i
[
a
,
b
] = [3, 8]
Z
=
X
10
1
∑
i
=1
10
i
P
(
Z
−
μ
≥
t
)
P
(
X
−
E
[
X
])
≥
t
(
n
1
∑
i
=1
10
i
i
)
n
= 10
2
⋅
10
t
2
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then upper bounded by . Save Answer
Last saved on Apr 17 at 9:16 AM
Q2 Bandits
2 Points
Let be the set of arms in the setting of the bandits problem. For each arm , let the random variable be the random reward received for pulling arm at time and let be the expected reward from pulling arm . Finally, let the random variable be the arm pulled at time (so is the reward received at time ).
All expectations (e.g., , ) are taken over the randomness in the payouts as well as the arm pulled at each timestep.
Q2.1 Regret
1 Point
Select all of the definitions which are true.
exp
−
=
(
(8
−
3)
2
2
⋅
10
t
2
)
exp(
−
=
(
5
2
20
t
2
)
exp
−
(
5
4
t
2
)
A
a
∈
A
X
i
a
a
i
μ
=
a
E[
X
]
i
a
a
A
i
i
X
i
A
i
i
E[
X
]
i
a
E[
X
]
i
A
i
X
i
a
A
i
The regret is .
R
(
t
)
X
−
X
i
=1
∑
t
(
a
∈
A
max
i
a
i
A
i
)
The regret is .
R
(
t
)
μ
−
X
i
=1
∑
t
(
a
∈
A
max
a
i
A
i
)
The regret is .
R
(
t
)
X
−
E[
X
]
i
=1
∑
t
(
a
∈
A
max
i
a
i
A
i
)
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Explanation
This is by the definition of regret. Simplifying gets the second correct option.
Save Answer
Last saved on Apr 17 at 9:43 AM
Q2.2 Expected Regret and Pseudo-Regret
1 Point
Select all of the definitions which are true.
i
=1
The regret is .
R
(
t
)
μ
−
E[
X
]
i
=1
∑
t
(
a
∈
A
max
a
i
A
i
)
The regret is R
(
t
)
t
μ
−
a
∈
A
max
a
X
i
=1
∑
t
i
A
i
The regret is R
(
t
)
t
μ
−
a
∈
A
max
a
E[
X
]
i
=1
∑
t
i
A
i
μ
=
i
=1
∑
t
a
∈
A
max
a
t
μ
a
∈
A
max
a
The expected regret is the expectation of the regret
over the randomness in the payout, conditioned on the
sequence of choices made.
The expected regret is the expectation of the regret
over the randomness in the payout and the sequence of
choices made.
X
−
X
∑
t
(
a
A
)
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Explanation
By the definitions of expected regret and pseudo-regret. Pseudo-regret is conditioned on the sequence of choices made, so it is , where is the arm pulled at timestep .
The expected regret is .
X
−
X
i
=1
∑
(
a
∈
A
max
i
a
i
A
i
)
The expected regret is .
X
−
E[
X
]
i
=1
∑
t
(
a
∈
A
max
i
a
i
A
i
)
The expected regret is μ
−
E[
X
]
i
=1
∑
t
(
a
∈
A
max
a
i
A
i
)
The pseudo-regret is the expectation of the regret over
the randomness in the payout, conditioned on the
sequence of choices made.
The pseudo-regret is the expectation of the regret over
the randomness in the payout and the sequence of
choices made.
The pseudo-regret is .
X
−
X
i
=1
∑
t
(
a
∈
A
max
i
a
i
A
i
)
The pseudo-regret is .
X
−
E[
X
]
i
=1
∑
t
(
a
∈
A
max
i
a
i
A
i
)
The pseudo-regret is μ
−
E[
X
]
i
=1
∑
t
(
a
∈
A
max
a
i
A
i
)
μ
−
μ
i
=1
∑
t
(
a
∈
A
max
a
a
i
)
a
i
i
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0.1
3
4
0.1
0.1
6.
7
0.1
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0.2
1
0.1
0.1
0.1
0.1
0.1
0.1
0.2
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0.00000
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0.00000
0.00000
0.00000
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estimate the area under a curve. Fit the interpolating
polynomial to the following set of points. These points are
the actual values of f(x) = sin (e* – 2)
0.4
0.8
1.2
1.6
y -0.8415 |-0.4866 0.2236 0.9687 0.1874
a) Plot the function f(x) and the interpolating polynomial, using different colors. Use polyfit and
polyval. Also include the data points using discrete point plotting.
b) We wish to estimate the area under the curve, but this function is difficult to integrate. Hence, instead
1.6
of finding ° sin(e* – 2) dx (which is the same as finding the area under the curve sin (e* – 2) ),
we will compute the area under the interpolating polynomial over the domain 0
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Question 1:
a)Population Growth
The world population data spans from 1960 to 2017. We'd like to build a predictive model that can give us the best guess at what the population growth rate in a given year might be. We will calculate the population growth rate as follows:-
Growth_rate=current_year_population−previous_year_populationprevious_year_population
As such, we can only calculate the growth rate for the year 1961 onwards.
Write a function that takes the population_df and a country_code as input and computes the population growth rate for a given country starting from the year 1961. This function must return a return a 2-d numpy array that contains the year and corresponding growth rate for the country.
Function Specifications:
Should take a population_df and country_code string as input and return a numpy array as output.
The array should only have two columns containing the year and the population growth rate, in other words, it should have a shape (?, 2) where ? is…
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Recommended textbooks for you
- Operations Research : Applications and AlgorithmsComputer ScienceISBN:9780534380588Author:Wayne L. WinstonPublisher:Brooks Cole
Operations Research : Applications and Algorithms
Computer Science
ISBN:9780534380588
Author:Wayne L. Winston
Publisher:Brooks Cole