Test 2 for the course

CS 4365/CS 5354, Spring 2015

Name: _____________________________________________________

1-2. A student is taking two important classes. There is only
time t left for studying for both classes, so if the student
spends time t_{1} studying for the first class, this student
will have time t_{2} = t − t_{1} left for the
second class.

- The grade for the first class depends on the time
t
_{1}as f_{1}(t_{1}) = (t − t_{1})^{2}. - The grade for the second class depends on the time
t
_{2}as f_{2}= (t − t_{2})^{2}, i.e., since t_{2}= t − t_{1}, as

f_{2}(t_{1})= (t_{1})^{2}.

- What is the solution to this multi-objective optimization
problem when we select the equal weights w
_{1}= w_{2}= 0.5? - What is the solution to in the general cases,
when we select an arbitrary value w
_{1}from the interval [0, 1], and take w_{2}= 1 − w_{1}? - Why multi-objective optimization is important for cloud computing?

3. Let us now assume that every day, the company uses
at least m = 100 computations, and the probabilities of different
numbers of computations x is described by the power law
p(x) = A * x^{−α}, with α = 3. Assume that
the cost of a unit in-house computation is c_{0} = 10 money
units per
computation unit and the cost of computing in the cloud is
c_{1} = 20 money units per computation unit.

- How
much computing power x
_{0}should be purchased for in-house computations? - What is the resulting expected cost?
- Compute the expected costs when we do all the computations exceeding m in the cloud, and show that the optimal arrangement indeed saves money.

In these computations, you can use the formulas that we derived in class for the power law case:

- the optimal amount of computing power to purchase is
x
_{0}= m * (c_{1}/ c_{0})^{1 / (α − 1)}; - the resulting expected cost is equal to
((α − 1) / (α − 2)) *
c
_{0}^{(α − 2) / (α − 1)}* c_{1}^{1 / (α − 1)}* m; - if we do all the computations exceeding m in the cloud, then
the resulting cost is

m * (c_{0}+ c_{1}/ (α − 2)).

4. Estimate the costs and decide whether it is beneficial to sign a contract with the cloud provider for T = 2 years:

- the cost of buying a unit of
computations on a year-by-year basis is c
_{0}= 1; - the contract
offer a discount price c
_{1}= 0.9, - the discount rate is q = 0.8, and
- the price of computing decreases yearly by a factor of v = 0.9.

5. Neural networks:

- Why neural networks are needed for cloud computing?
- Let us assume that we have are training a neural network with K = 2
neurons to
learn multiplication. This means that for inputs x
_{1}= 2.0 and x_{2}= 3.0, the desired output is Y = 6.0. On a certain iteration, we have w_{ki}= 0 for all k and i, so that the outputs of the non-linear neurons are equal to 0.5: y_{1}= y_{2}= 0.5. Suppose that at this iteration, W_{1}= W_{2}= 1.0 and W_{0}= 0.0. Assuming that α = 0.1, describe the values of all the weights on the next iteration.

*Reminder*:

- In the first
layer, we have K non-linear neurons. Each neuron k (k = 1, 2, ...,
K) transforms the input signals x
_{1}, ..., x_{n}into a signal

y_{k}= s_{0}(w_{k1}* x_{1}+ ... + w_{kn}* x_{n}− w_{k0}),

where s_{0}(z) = 1 / (1 + exp(−z)). - The last linear neuron then transform these
signals y
_{k}into a single output

y = W_{1}* y_{1}+ ... + W_{K}* y_{K}− W_{0}.

- ΔW
_{0}= α * Δ y, for some small number α > 0; -
ΔW
_{k}= − y_{k}* ΔW_{0}; - Δw
_{k0}= − ΔW_{k}* W_{k}* (1 − y_{k}); - Δw
_{ki}= − x_{i}* Δw_{k0}.

6. Fuzzy techniques:

- Why fuzzy techniques are needed for cloud computing?
- Briefly describe the main steps of fuzzy techniques.
- Which "and"-operations are typically used in fuzzy techniques? If the expert's degree of belief in a statement A is 0.6 and the expert's degree of belief in a statement B is 0.8, what is the expert's degree of belief in the statement "A and B"?

7. What is green computing? Explain the need for saving energy in cloud computing.

8-9. Describe, in detail, the paper that you reviewed as a project for this class:

- what problem is addressed in his paper,
- what solution is proposed for this problem, and
- (if applicable) what are the remaining open problems.

10. Briefly describe someone else's project for this class.