# Homework 2.5 - More ANN Fundamentals

## Due Wednesday, September 26, 2018

## 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). Likewise, FFNNs can learn in many ways but the most
fundamental way is supervised learning. 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, supervised learning, and classification
— are the topics of this homework.

## 2. Goal

The goal of this assignment is to give you experience with supervised
learning using backpropagation of error using FFNNs for classification.

## 3. Assignment

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 with
sigmoidal activation functions with λ=1 and *η* = 0.5.
The target values for γ_{1} and γ_{2} are
0.9 and 0.1, respectively. There are three ANs in the hidden layer and
one in the output layer, with the weights
*v*_{1,1}=−0.1, *v*_{2,1}=0.2,
*v*_{3,1}=−0.9, *v*_{1,2}=0.6,
*v*_{2,2}=0.8, *v*_{3,2}=−0.1,
*v*_{1,3}=−0.2, *v*_{2,3}=−0.2,
*v*_{3,3}=−0.4, *w*_{1}=−0.2,
*w*_{2}=0.6, *w*_{3}=0.0, and
*w*_{4}=0.1. (Note that this is the same network structure
with the same weights as found in Homework 2.)

Complete the following exercises:

**Explain** how its weights would be updated, using the
backpropagation algorithm we covered in class, if presented with the
data item (−1.0, −1.0) γ_{1}. **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) γ_{1}, 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) γ_{1} increased or decreased due to
learning.

## 4. What to Turn In

Turn in a neatly handwritten copy of your answers to the exercises for
this assignment. You may also turn in a scanned electronic copy of this
assignment as a backup in case your paper copy is misplaced.