Homework 2 - More ANN Fundamentals
Due Monday, September 9, 2019
Due Friday, September 13, 2019
1. Motivation
Artificial neurons (ANs) can be combined in many ways to compute more
complex functions than could be computed by a single AN. The most
fundamental way is by combining ANs into a layered, feedforward neural
network (FFNN). Moreover, FFNNs may be used for many tasks but the two
most fundamental are classification and function approximation, of which
classification is the easier to visualize. These ANN fundamentals —
FFNNs and classification — are the topics of this homework.
2. Goal
The goal of this assignment is to give you experience with basic FFNNs
and classification.
3. Assignment
Complete the following exercises:
Part 1 — FFNN Representation
- Consider a two-layer FFNN — that is, one with two layers of
computational elements (ANs) — used for classification in a 2D
space with augmented vectors. The ANs in this FFNN are all SUs and their
activation functions are identical to fAN as given in
Homework 1, Subpart 1.1. There are three ANs in the hidden layer and one
in the output layer. Given the weights v1,1=−0.3,
v2,1=−0.3, v3,1=0.1,
v1,2=−0.1, v2,2=0.6,
v3,2=0.6, v1,3=0.1,
v2,3=0.3, v3,3=0.4,
w1=0.5, w2=0.0,
w3=0.1, and w4=0.2,
draw this FFNN.
- Draw the decision region encoded by this FFNN. Be sure to
indicate the γ1 side of the region. Be sure to indicate
which portion of the decision region is due to each hidden layer AN.
(Hint: If you're having difficulty figuring out exactly how the decision
region works out given the weights above, try plotting the points from
Exercise 3 immediately below and calculate their classes by plugging
their point values into the equations defined by the weights above in
Exercise 1.)
- Add the following points on the graph you just drew and
label the class of each according to the AN.
- (−0.7, 0.3)
- (−0.7, −0.5)
- (0.4, 0.8)
- (0.8, −1.0)
- (0.0, 0.0)
- (−0.4, 0.9)
- Explain the significance of the sign of
w4.
- Explain the significance of the value of
w2.
- Explain the significance of the relative values of
w1 and w4.
- Explain the significance of the relative values of
w3 and w4.
- Explain how the decision region for this FFNN would change if
the value of w2 were changed to −0.4 rather than 0.0
and explain which points, if any, from those above would be
classified differently and which would be classified the same. Be sure
to discuss the relative values of w1,
w2, w3, and w4.
- Explain how the decision region for this FFNN would change if
γ2 were changed to −1 rather than 0 and
explain which points, if any, from those above would be classified
differently and which would be classified the same. (For this
hypothetical, use a value of −0.4 for w2.)
- Give an example weight vector w for which a
γ2 value of −1 would give a different
classification for one of the points from exercise 3 than would a
γ2 value of 0. Explain your answer.
- Explain how the decision region for this FFNN would change if
fAN for the hidden layer ANs were changed to be a
linear activation function, in particular, if
fAN(net)=net for the hidden layer ANs.
(For the output AN, fAN remains unchanged in this
hypothetical.) Explain which points, if any, from those above
would be classified differently and which would be classified the same.
(For this hypothetical, use a value of −0.4 for w2.)
Part 2 — FFNN Learning
Consider the two-layer FFNN described above in Part 1, Exercise 1x,
except using sigmoidal logistic activation functions with λ=1. This
FFNN uses the backpropagation algorithm we covered in class with
η = 1.0. The target values for γ1 and
γ2 are 0.9 and 0.1, respectively.
Complete the following exercises:
- Explain how the weights of this FFNN would be updated if
presented with the data item (−1.0, −1.0)
γ1. Show your work. Keep track of four
significant digits.
- Calculate the output value of the FFNN above if, after
learning on (−1.0, −1.0) γ1, you were to
present this data item to the FFNN again. Show your work. Keep
track of four significant digits.
- Explain whether the error value for the input (−1.0,
−1.0) γ1 increased or decreased due to
learning.
4. What to Turn In
You may write and draw your responses to this assignment neatly by hand
or type your answers and use graphing software to complete the exercises
for this assignment. If drawn, the diagrams should be neatly drawn on
engineering or graph paper. In any case, you should turn in to the
appropriate Canvas dropbox an electronic (perhaps scanned) copy of your
assignment, so that I can grade it online.