Introduction

In this notebook you'll visualize small neural networks using pen and paper. The idea is to gain an intuition about how the components of a neural network interact with each other, and how the input flows through the network to produce an output. In succeeding notebooks, you'll implement neural networks as code.

Requirements

Knowledge

Chapter 1 of Nerual networks and deep learning by Michael Nielsen (#NIE15)
You should have completed the previous course Introduction to Machine Learning

Exercises

Logistic Regression as a Neuron

Consider a basic neural network which consists of just one single neuron. As it turns out this is equivalent to the familiar model of logistic regression.

Given

the input vector$ \vec x $
the weights$ \vec \theta = \left[3, -1, 0.5, 2\right] $
the first element of$ \vec \theta $ (which is$ \theta_0 $) is the bias.

Illustrate a neuron which produces the logistic hypothesis as its ouput:

$ o = \sigma( \theta_0 + \theta_1 \cdot x_1 + \theta_2 \cdot x_2 + \theta_3 \cdot x_3 ) $

or written in vector form:

$ o = \sigma( \theta_0 + \vec \theta_{1:} \cdot \vec x) $

Your illustration should include the following components

Input layer
Output layer
Connections, weights, bias
Sigmoid function$ \sigma $

Draw the Network

Given is the following information for a neural network:

\begin{equation} W_{HIDDEN} = \begin{pmatrix} 10 & -20 & 20 & -40 \ 20 & -40 & 0 & 0 \end{pmatrix} \end{equation}

\begin{equation} W_{OUTPUT} = \begin{pmatrix} 20 & 40 & -40 \end{pmatrix} \end{equation}

Each row of$ W_{HIDDEN} $ and$ W_{OUTPUT} $ represents the weights of one neuron in layer HIDDEN, resp. the OUTPUT layer. The first column equals the bias(es).

Further, activation function$ g(z) $, which applies to all neurons in the network:

\begin{equation} g(z)=\left{\begin{array}{cc} 0 & z\le-10\ 1 & z\ge10 \ 0.5 & else\end{array} \right. \end{equation}

Task:

Draw a graph of the network$ N_{SIMPLE} $ including all neurons and their connections. Note all weight and bias values on the corresponding nodes and edges of the graph.

Calculate the forward pass

Use the given vectors$ x_1, x_2, x_3 $ to create a mini-batch matrix as input for the network and calculate its output. Only use matrix operations for the calculation and note all intermediate results.

\begin{equation} \vec{x}{(1)} = [0,1,1] ,\; \vec{x}{(2)} =[1,1,0] ,\; \vec{x}_{(3)} = [1,0,1] \end{equation}

Literature

Licenses

Notebook License (CC-BY-SA 4.0)

The following license applies to the complete notebook, including code cells. It does however not apply to any referenced external media (e.g., images).

Exercise - Neural Network - Pen & Paper
by Benjamin Voigt
is licensed under a Creative Commons Attribution-ShareAlike 4.0 International License.
Based on a work at https://gitlab.com/deep.TEACHING.

Code License (MIT)

The following license only applies to code cells of the notebook.

Permission is hereby granted, free of charge, to any person obtaining a copy of this software and associated documentation files (the "Software"), to deal in the Software without restriction, including without limitation the rights to use, copy, modify, merge, publish, distribute, sublicense, and/or sell copies of the Software, and to permit persons to whom the Software is furnished to do so, subject to the following conditions:

The above copyright notice and this permission notice shall be included in all copies or substantial portions of the Software.

THE SOFTWARE IS PROVIDED "AS IS", WITHOUT WARRANTY OF ANY KIND, EXPRESS OR IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY, FITNESS FOR A PARTICULAR PURPOSE AND NONINFRINGEMENT. IN NO EVENT SHALL THE AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR OTHER LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING FROM, OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR OTHER DEALINGS IN THE SOFTWARE.

Exercise - Neural Network - Pen & Paper

Table of Contents