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17.11.2020 by Łukasz Gebel & Piotr Czajka

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achine Learning: The Bare Math Behind Libraries 23.06.2021 by Łukasz Gebel & Piotr Czajka

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Super Heroes & Super Powers

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Agenda ● What is Machine Learning? ● Supervised learning ● Unsupervised learning ● Q&A

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Machine Learning „Field of study that gives computers the ability to learn without being explicitly programmed.” Arthur Samuel

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Machine Learning „I consider every method that needs training as intelligent or machine learning method.” Our Lecturer

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Supervised learning ● It is similar to be tought by a teacher.

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Supervised learning ● Build your model that performs particular task: – Prepare data set consisting of examples & expected outputs

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Supervised learning ● Build your model that performs particular task: – Prepare data set consisting of examples & expected outputs – Present examples to your model

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Supervised learning ● Build your model that performs particular task: – Prepare data set consisting of examples & expected outputs – Present examples to your model – Check how it responds (model’s output values)

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Supervised learning ● Build your model that performs particular task: – Prepare data set consisting of examples & expected outputs – Present examples to your model – Check how it responds (model’s output values) – Adjust model’s params by comparing output values with expected output values

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Neural Networks ● Inspired by biological brain mechanisms ● Many applications: – Computer vision – Speech recognition – Compression

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Biological Neuron http://www.marekrei.com/blog/wp-content/uploads/2014/01/neuron.png

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Artificial Neuron ● Inputs (x 1, … , x n ) are features of single example ● Multiply each input by weight, sum it and put the sum as an argument of activation function w 1 w 2 w n w 0 Σ x 1 x 2 x n ... s y

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Activation function ● Sigmoid – Maps sum of neurons signals to value from 0 to 1 – Continous, nonlinear – If input is positive it gives values > 0.5 f (x)= 1 1+e(−βx)

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Linear Regression ● Method for modelling relationship between variables ● Simplest form: how x relates to y ● Examples: – House size vs house price – Voltage vs electric current

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Real life problem

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What defines a superhero?

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Costume

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Costume

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Costume price vs number of issues ● For given amount of money predict in how many comic book issues you’ll appear. Costume price(x) Number of issues (y) 240 6370 480 8697 ... ... 26 2200

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Interesting fact ● Invisible woman costume costs $120 Invisible woman

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Linear regression ● Let's have a function: f (x ,Θ)=Θ1 x+Θ0 f (x,Θ)−number of comicbookissues x−costume price Θ−parameters

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Let's plot

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Let's plot

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Let's plot

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Warning!

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Objective function Q(Θ)= 1 2 N ∑ j=0 N (f ( xj ,Θ)−y j )2 Q(Θ)−objectivefunction N−numberof datasamples j−indexof particulardatasample

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Objective function Q(Θ)= 1 2 N ∑ j=0 N (f ( xj ,Θ)−y j )2 Q(Θ)−objectivefunction N−numberof datasamples j−indexof particulardatasample

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Objective function Q(Θ)= 1 2 N ∑ j=0 N (f ( xj ,Θ)−y j )2 Q(Θ)−objectivefunction N−numberof datasamples j−indexof particulardatasample

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Objective function Q(Θ)= 1 2 N ∑ j=0 N (f ( xj ,Θ)−y j )2 Q(Θ)−objectivefunction N−numberof datasamples j−indexof particulardatasample

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Objective function - intuition f (xj ,Θ)=4 y=2 (4−2)2 =22 =4 sum+=4

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Objective function - intuition f (xj ,Θ)=1 y=1 (1−1)2 =02 =0 sum+=0

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Gradient descent ● Find the minimum of the objective function ● Iteratively update function parameters: Θ0 (t +1)=Θ0 (t)−α 1 N ∑ j=0 N (f ( xj ,Θ)− y j ) Θ1 (t +1)=Θ1 (t )−α 1 N ∑ j=0 N (f (xj ,Θ)− yj ) x t−number of iteration α−learning step

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Gradient descent ● Find the minimum of the objective function ● Iteratively update function parameters: Θ0 (t +1)=Θ0 (t)−α 1 N ∑ j=0 N (f ( xj ,Θ)− y j ) Θ1 (t +1)=Θ1 (t )−α 1 N ∑ j=0 N (f (xj ,Θ)− yj ) x t−number of iteration α−learning step

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Gradient descent ● Find the minimum of the objective function ● Iteratively update function parameters: Θ0 (t +1)=Θ0 (t)−α 1 N ∑ j=0 N (f ( xj ,Θ)− y j ) Θ1 (t +1)=Θ1 (t )−α 1 N ∑ j=0 N (f (xj ,Θ)− yj ) x t−number of iteration α−learning step

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Gradient descent

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Gradient descent −α ∗ +derivative < 0

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Gradient descent −α ∗ +derivative < 0

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Gradient descent −α ∗ +derivative < 0

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Gradient descent −α ∗ +derivative < 0

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Gradient descent −α ∗ −derivative > 0

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Gradient descent −α ∗ −derivative > 0

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Gradient descent −α ∗ −derivative > 0

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Gradient descent −α ∗ −derivative > 0

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Demo

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Linearly separable problem

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Not linearly separable problem

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NN – random weights init +1 +1 x 1 x 2 y

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NN – feed forward +1 +1 x 1 x 2 y

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NN – feed forward +1 +1 x 1 x 2 y

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NN – feed forward +1 +1 x 1 x 2 y

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NN – feed forward +1 +1 x 1 x 2 y

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NN – compute error +1 +1 x 1 x 2 y ( y−expected output)2

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NN – backpropagation step ● Use gradient descent and computed error ● Update every weight of every neuron from hidden and output layer

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NN – backpropagation step +1 +1 x 1 x 2 y ( y−expected output)2

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NN – backpropagation step +1 +1 x 1 x 2 y

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Neural Network „Neural networks are nothing more than randomized optimization.” Another Lecturer

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Real life problem ● You said it can solve non linear problems, let's generate superhero logo using it.

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Unsupervised learning

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What? They can learn by themselves?

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What? They can learn by themselves?

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Why would we let them? ● Less complex mathematical apparatus than in supervised learning. ● It is similar to discovering world on your own.

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Why would we let them? Used mostly for sorting and grouping when: ● Sorting key can’t be easily figured out. ● Data is very complex and finding the key is not trivial.

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Idea behind – groups

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Idea behind – groups

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Idea behind – groups

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Hebbian learning

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Hebbian learning ● Works similar to the nature ● Great for beginners and biological simulations :) ● Simple Hebbian learning algorithm Δ w ij =η⋅x ij ⋅y i Δ w ij −change of j weight of ineuron η−learningcoefficient x ij − jinput of ineuron y i −output of i neuron

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Hebbian learning ● Works similar to the nature ● Great for beginners and biological simulations :) ● Generalised Hebbian learning algorithm Δ w ij =F(x ij , y j ) Δ w ij −change of j weight of ineuron η−learningcoefficient x ij − jinput of ineuron y i −output of i neuron

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Hebb’s neuron model w 1 w 2 w n w 0 Δ w ij =F(x ij , y j ) Σ x 1 x 2 x n ... 1 s y

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Hebb’s neuron model 0.230 0.010 0.900 0.110 Δ w ij =F(x ij , y j ) Σ 0.200 0.300 0.100 1

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Hebb’s neuron model 0.230 0.010 0.900 0.110 Δ w ij =F(x ij , y j ) Σ 1 0.046 0.003 0.090 0.110 0.200 0.300 0.100

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Hebb’s neuron model 0.230 0.010 0.900 0.110 Δ w ij =F(x ij , y j ) Σ 1 0.046 0.003 0.090 0.110 0.249 0.200 0.300 0.100

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Hebb’s neuron model 0.230 0.010 0.900 0.110 Δ w ij =F(x ij , y j ) Σ 1 0.046 0.003 0.090 0.110 0.249 0.562 0.562 0.200 0.300 0.100

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Hebb’s neuron model 0.230 0.010 0.900 0.110 Δ w ij =F(x ij , y j ) Σ 1 0.562 0.200 0.300 0.100

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Hebb’s neuron model 0.230 0.010 0.900 0.110 Δ w ij =F(x ij , y j ) Σ 1 0.562 +0.011 +0.016 +0.005 +0.056 0.300 0.100 0.200

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Hebb’s neuron model 0.241 0.026 0.905 0.166 Δ w ij =F(x ij , y j ) Σ x 1 x 2 x 3 1

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Demo

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Imagine superhero teams

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Marvel database to the rescue Intelligence Strength Speed Durability Energy projection Fighting skills Iron Man 6 6 5 6 6 4 Spiderman 4 4 3 3 1 4 Black Panter 5 3 2 3 3 5 Wolverine 2 4 2 4 1 7 Thor 2 7 7 6 6 4 Dr Strange 4 2 7 2 6 6 Hulk 2 7 3 7 5 4 Cpt. America 3 3 2 3 1 6 Mr Fantastic 6 2 2 5 1 3 Human Torch 2 2 5 2 5 3 Invisible Woman 3 2 3 6 5 3 The Thing 3 6 2 6 1 5 Luke Cage 3 4 2 5 1 4 She Hulk 3 7 3 6 1 4 Ms Marvel 2 6 2 6 1 4 Daredevil 3 3 2 2 4 5

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Hebbian learning weaknesses ● Unstable. ● Prone to rise the weights ad infinitum. ● Some groups can trigger no response. ● Some groups may trigger response from too many neurons.

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And now – self organising networks

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Self organising?

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Learning with concurrency ● You try to generalize input vector in weights vector. ● Instead of checking the reaction to input - you check distance between both vectors. ● Ideally – each neuron specializes in one class generalization. ● Two main strategies: – Winner Takes All (WTA) – Winner Takes Most (WTM)

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Idea behind Neuron weights 3.0 2.0 2.0 Example 1.0 2.0 3.0

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Example 1.0 2.0 3.0 Idea behind Distance 2.0 0.0 -1.0 Neuron weights 3.0 2.0 2.0 d i =w i −x i Euclidiandistance √∑ i=1 n d i 2 =√5

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Distance 2.0 0.0 -1.0 Idea behind Neuron weights 3.0 2.0 2.0 Learning coefficient η=0.100 Learning Step 0.2 0.0 -0.1 Δ w i =η⋅d i Example 1.0 2.0 3.0 d i =w i −x i Learning coefficient η=0.100

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Idea behind Distan ce 2.0 0.0 -1.0 Neuron weights 2.8 = 3.0 – 0.2 2.0 = 2.0 – 0.0 2.1 = 2.0 -(-0.1) Exam ple 1.0 2.0 3.0 d i =w i −x i Learning coefficient η=0.100 Learning Step 0.2 0.0 -0.1 w' i =w i −Δw i Δ w i =η⋅d i Example 1.0 2.0 3.0 Learning Step 0.2 0.0 -0.1 Δ w i =η⋅d i Distance 2.0 0.0 -1.0 d i =w i −x i

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Idea behind Neuron weights 2.8 2.0 2.1

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Idea behind – initial setup 1 2 3

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Idea behind – mid learning 1 2 3

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Idea behind – homeostasis 1 2 3

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Demo time

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Learning with concurrency ● Gives more diverse groups. ● Less prone to clustering (than Hebb’s). ● Searches wider spectrum of answers. ● First step to more complex networks.

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Learning with concurency - weaknesses ● WTA – works best if teaching examples are evenly distributed in solution space. ● WTM – works best if weights’ vectors are evenly distributed in solution space. ● Still can stick to local optimum.

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Teuvo Kohonen’s SOM Created by this nice guy here

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Kohonen’s self-organizing map ● The most popular self-organizing network with concurrency algorithm. ● It teaches groups of neurons with WTM alghoritm ● Special features: – Neurons are organised in a grid – Nevertheless – they are treated as a single layer

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Kohonen’s self-organizing map w ij (s+1)=w ij (s)+Θ(k best ,i , s)⋅η(s)⋅(I j (s)−w ij (s)) s−epochnumber k best −best neuron w ij (s)− j weight of ineuron Θ(k best ,i,s)−neighbourhood function η(s)−learning coefficient for sepoch I j (s)− jchunk of example for s epoch

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SOM model By Mcld - Own work, CC BY-SA 3.0, https://commons.wikimedia.org/w/index.php?curid=10373592

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Common weaknesses of artificial neuron systems ● We are still dependant on randomized weights. ● All algorithms can stick to local optimum.

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Bibliography ● Presentation + code: https://bitbucket.org/medwith/public/downloads/mluvr-MLCon.zip ● https://www.coursera.org/learn/machine-learning ● https://www.coursera.org/specializations/deep-learning ● Math for Machine Learning - Amazon Training and Certification ● Linear and Logistic Regression - Amazon Training and Certification ● Grus J., Data Science from Scratch: First Principles with Python ● Patterson J., Gibson A., Deep Learning: A Practitioner's Approach ● Trask A., Grokking Deep Learning ● Stroud K. A., Booth D. J, Engineering Mathematics ● https://github.com/massie/octave-nn- neural network Octave implementation ● https://www.desmos.com/calculator/dnzfajfpym - Nanananana … Batman equation ;) ● https://xkcd.com/605/ - extrapolating ;) ● http://dilbert.com/strip/2013-02-02 - Dilbert & Machine Learning

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Thank you