Machine Learning: The Bare Math Behind Libraries

During this presentation, we will answer how much you’ll need to invest in a superhero costume to be as popular as Superman. We will generate a unique logo which will stand against the ever popular Batman and create new superhero teams. We shall achieve it using linear regression and neural networks.

Machine learning is one of the hottest buzzwords in technology today as well as one of the most innovative fields in computer science – yet people use libraries as black boxes without basic knowledge of the field. In this session, we will strip them to bare math, so next time you use a machine learning library, you’ll have a deeper understanding of what lies underneath.

During this session, we will first provide a short history of machine learning and an overview of two basic teaching techniques: supervised and unsupervised learning.

We will start by defining what machine learning is and equip you with an intuition of how it works. We will then explain the gradient descent algorithm with the use of simple linear regression to give you an even deeper understanding of this learning method. Then we will project it to supervised neural networks training.

Within unsupervised learning, you will become familiar with Hebb’s learning and learning with concurrency (winner takes all and winner takes most algorithms). We will use Octave for examples in this session; however, you can use your favourite technology to implement presented ideas.

Our aim is to show the mathematical basics of neural networks for those who want to start using machine learning in their day-to-day work or use it already but find it difficult to understand the underlying processes. After viewing our presentation, you should find it easier to select parameters for your networks and feel more confident in your selection of network type, as well as be encouraged to dive into more complex and powerful deep learning methods.

Transcript

1. achine Learning: The Bare Math Behind Libraries Machine Learning: The Bare Math Behind Libraries Machine Learning: The Bare Math Behind Libraries Piotr Czajka & Łukasz Gebel TomTom @medwith @rauluka7
2. Super Heroes & Super Powers
4. Agenda • What is Machine Learning? • Supervised learning • Unsupervised learning • Q&A
5. Machine Learning „Field of study that gives computers the ability to learn without being explicitly programmed.” Arthur Samuel
6. Machine Learning „I consider every method that needs training as intelligent or machine learning method.” Our Lecturer
7. Supervised learning • It is similar to be tought by a teacher.
8. Supervised learning • Build your model that performs particular task: – Prepare data set consisting of examples & expected outputs
9. Supervised learning • Build your model that performs particular task: – Prepare data set consisting of examples & expected outputs – Present examples to your model
10. 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)
11. 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
12. Neural Networks • Inspired by biological brain mechanisms • Many applications: – Computer vision – Speech recognition – Compression
13. Biological Neuron http://www.marekrei.com/blog/wp-content/uploads/2014/01/neuron.png
14. Artifcial 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
15. 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)
16. 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
17. Real life problem
18. What defnes a superhero?
19. Costume
20. 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
21. Interesting fact • Invisible woman costume costs $120 Invisible woman
22. Linear regression • Let's have a function: f (x ,Θ)=Θ1 x+Θ0 f (x,Θ)−number of comicbookissues x−costume price Θ−parameters
23. Let's plot
24. Let's plot
25. Let's plot
26. Warning!
27. 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
28. 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
29. 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
30. 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
31. Objective function - intuition f (xj ,Θ)=4 y=2 (4−2)2 =22 =4 sum+=4
32. Objective function - intuition f (xj ,Θ)=1 y=1 (1−1)2 =02 =0 sum+=0
33. 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
34. 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
35. 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
36. Gradient descent
37. Gradient descent −α ∗ +derivative < 0
38. Gradient descent −α ∗ +derivative < 0
39. Gradient descent −α ∗ +derivative < 0
40. Gradient descent −α ∗ +derivative < 0
41. Gradient descent −α ∗ −derivative > 0
42. Gradient descent −α ∗ −derivative > 0
43. Gradient descent −α ∗ −derivative > 0
44. Gradient descent −α ∗ −derivative > 0
45. Demo
46. Linearly separable problem
47. Not linearly separable problem
48. NN – random weights init +1 +1 x 1 x 2 y
49. NN – feed forward +1 +1 x 1 x 2 y
50. NN – feed forward +1 +1 x 1 x 2 y
51. NN – feed forward +1 +1 x 1 x 2 y
52. NN – feed forward +1 +1 x 1 x 2 y
53. NN – compute error +1 +1 x 1 x 2 y ( y−expected output)2
54. NN – backpropagation step • Use gradient descent and computed error • Update every weight of every neuron from hidden and output layer
55. NN – backpropagation step +1 +1 x 1 x 2 y ( y−expected output)2
56. NN – backpropagation step +1 +1 x 1 x 2 y
57. Neural Network „Neural networks are nothing more than randomized optimization.” Another Lecturer
58. Real life problem • You said it can solve non linear problems, let's generate superhero logo using it.
59. Unsupervised learning
60. What? They can learn by themselves?
61. What? They can learn by themselves?
62. Why would we let them? • Less complex mathematical apparatus than in supervised learning. • It is similar to discovering world on your own.
63. Idea behind – groups
64. Idea behind – groups
65. Idea behind – groups
66. Why would we let them? Used mostly for sorting and grouping when: • Sorting key can’t be easily fgured out. • Data is very complex and fnding the key is not trivial.
67. Hebbian learning
68. 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
69. 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
70. 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
71. 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
72. 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
73. 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
74. 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
75. 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
76. 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
77. 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
78. Demo
79. Imagine superhero teams
80. 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
81. Hebbian learning weaknesses • Unstable. • Prone to rise the weights ad infnitum. • Some groups can trigger no response. • Some groups may trigger response from too many neurons.
82. And now – self organising networks
83. Self organising?
84. 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)
85. Idea behind Neuron weights 3.0 2.0 2.0 Example 1.0 2.0 3.0
86. 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
87. 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
88. 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
89. Idea behind Neuron weights 2.8 2.0 2.1
90. Idea behind – initial setup 1 2 3
91. Idea behind – mid learning 1 2 3
92. Idea behind – homeostasis 1 2 3
93. Demo time
94. 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.
95. 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.
96. Teuvo Kohonen’s SOM Created by this nice guy here
97. 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
98. 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
99. SOM model By Mcld - Own work, CC BY-SA 3.0, https://commons.wikimedia.org/w/index.php?curid=10373592
100. Common weaknesses of artifcial neuron systems • We are still dependant on randomized weights. • All algorithms can stick to local optimum.
101. Bibliography • Presentation + code: https://bitbucket.org/medwith/public/downloads/ml-math-devoxPL19.zip • https://www.coursera.org/learn/machine-learning • https://www.coursera.org/specializations/deep-learning • Math for Machine Learning - Amazon Training and Certifcation • Linear and Logistic Regression - Amazon Training and Certifcation • 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
102. Thank you