Homework 1 - ANN Fundamentals

1. Motivation

The foundational computing element for an artificial neural network (ANN) is the artificial neuron (AN). ANs can learn in many ways but the most fundamental way is supervised learning. Moreover, ANs may be used for many tasks but the most basic is classification. These ANN basics — ANs, supervised learning, and classification — are the topics of this homework.

2. Goal

The goal of this assignment is to give you experience with basic ANs, the basic steps involved in updating weights of an AN, and using ANs for classification problems.

3. Assignment

Complete the following exercises:

Part 1 — AN Representation

1. Consider a single AN used for classification in a 2D space with an augmented vector as discussed in the Engelbrecht text. This AN is a summation unit (SU) and its activation function f_AN is a step function with outputs γ₁=1 for f_AN(net) ≥ 0 and γ₂=0 for f_AN(net) < 0. Given the weights v₁=1.0, v₂=0.9, and v₃=−0.7, draw this AN.
2. Draw (on graph or engineering paper or by using software) the decision boundary encoded by this AN. Be sure to indicate the γ₁ side of the boundary.
3. Add the following points on the graph you just drew and label the class of each according to the AN.
   1. (−0.1, 1.0)
   2. (−0.8, 0.9)
   3. (0, 0)
   4. (0.1, 0.9)
   5. (−1.0, −0.1)
4. Assume that the AN’s classification of each of the points above is correct. List a new labeled data item which, if added to the data set above, would necessarily cause the AN to misclassify at least one data item. (That is, given this new data item, there would be no set of possible weights that would allow this AN to correctly classify all data items.) Explain why the AN would necessarily misclassify at least one data item.
5. Explain how the decision boundary for this AN would change if γ₂ were changed to −1, rather than 0.
   [Note for explanatory questions, like this one: To explain the answer, describe what the answer is — in this case, what changes (if any) there are to the decision boundary — and also why that is the correct answer — in this case that answer would be based on how the value of γ affects the decision boundary.]
6. Explain how the decision boundary for this AN would change if f_AN were changed to be a sigmoidal function.
7. Explain how the classification of each of the points listed above would change if f_AN were a sigmoidal function.

Part 2 — AN Learning

1. Consider the same AN given above in 1.1 together with the following learning rule (and ignore the points in 1.3 and 1.4 and possible modifications to the AN listed in 1.5 – 1.7):

   v_i(t) = v_i(t−1) + (t_p − o_p) z_i,p

   Explain how the AN weights would be updated given each of the following labeled data points, presented in the following order:
   1. (0, 0) γ₂
   2. (0.4, 0.0) γ₂
   3. (1.0, −1.0) γ₁
   4. (0.7, −0.9) γ₁
2. Draw the graph of the data and the decision boundary after each weight update for the data given in 2.1.
3. Explain which of the points in 2.1 are correctly classified by this AN after all of the weight updates from one learning pass through this data and explain what this tells us about the use of this learning rule.
4. Explain how many more passes through the data are needed before all of the points from 2.1 are correctly classified by this AN. (Note that you may answer this question on a scale of "one, two, many." That is, if all of the points are not correctly classified after two more passes through the data, you may stop at a total of three passes and report that it still had not correctly classified all of the points after three passes.) Show your work.
5. Explain the likely effect of introducing a learning rate parameter (η) into the equation given.

4. What to Turn In

Turn in a neatly handwritten or typed copy of your answers to the exercises for this assignment. The diagrams should be neatly drawn on engineering or graph paper or by using software. You should turn in both a paper and electronic (perhaps scanned) copy of this assignment.