November 14, 2018





November 14, 2018





September 13, 2018

January 17, 2018
Historically, Windows users have had the most trouble installing deep learning, machine learning, data science, and AI libraries.
Just a few years ago this process would have taken you hours, if not days, to complete, if you completed it at all (as opposed to just giving up).
Nowadays, things are MUCH easier.
In this lecture, I walk you through how to install the following libraries on Windows:
Go to comments
December 19, 2017
Earlier this week, I mentioned Udemy was doing a promotion on Tech courses only (if you were signed up for my newsletter you would have gotten the announcement). I’ve just heard news that they’ve opened up the $10 sale to ALL courses for the next 3 days only!
What this means: All my courses will continue to be on sale for $10 (just click the below links). But in addition, you can find other courses (including calculus and probability prerequisites) for $10 too!
With some time off, now is the PERFECT time to catch up on your deep learning / machine learning / data science skills. It’s almost 2018 and AI is rising faster than ever.
What better way than to grab all the deep learning courses you’ll ever want to take, for just $10?
Don’t forget, this is the LOWEST possible price on Udemy – get these courses NOW. We really don’t know when the next big sale is going to be.
If you want to type in the coupon code manually, it’s: WINTER2017 (remember, this is only for my courses). However, the coupon codes are included automatically in the links below.
This sale lasts until Dec. 21 (3 days). Don’t wait!
https://www.udemy.com/datasciencelinearregressioninpython/?couponCode=WINTER2017
https://www.udemy.com/datasciencelogisticregressioninpython/?couponCode=WINTER2017
https://www.udemy.com/datasciencedeeplearninginpython/?couponCode=WINTER2017
https://www.udemy.com/datasciencenaturallanguageprocessinginpython/?couponCode=WINTER2017
https://www.udemy.com/datasciencedeeplearningintheanotensorflow/?couponCode=WINTER2017
https://www.udemy.com/sqlformarketersdataanalyticsdatasciencebigdata/?couponCode=WINTER2017
https://www.udemy.com/clusteranalysisunsupervisedmachinelearningpython/?couponCode=WINTER2017
https://www.udemy.com/unsuperviseddeeplearninginpython/?couponCode=WINTER2017
https://www.udemy.com/deeplearningrecurrentneuralnetworksinpython/?couponCode=WINTER2017
https://www.udemy.com/datasciencesupervisedmachinelearninginpython/?couponCode=WINTER2017
https://www.udemy.com/bayesianmachinelearninginpythonabtesting/?couponCode=WINTER2017
https://www.udemy.com/machinelearninginpythonrandomforestadaboost/?couponCode=WINTER2017
https://www.udemy.com/deepreinforcementlearninginpython/?couponCode=WINTER2017
https://www.udemy.com/deeplearninggansandvariationalautoencoders/?couponCode=WINTER2017
PREREQUISITE COURSE COUPONS
And just as important, $10 coupons for some helpful prerequisite courses. You NEED to know this stuff before you study machine learning:
General (sitewide): http://bit.ly/2oCY14Z
Python http://bit.ly/2pbXxXz
Calc 1 http://bit.ly/2okPUib
Calc 2 http://bit.ly/2oXnhpX
Calc 3 http://bit.ly/2pVU0gQ
Linalg 1 http://bit.ly/2oBBir1
Linalg 2 http://bit.ly/2q5SGEE
Probability (option 1) http://bit.ly/2prFQ7o
Probability (option 2) http://bit.ly/2p8kcC0
Probability (option 3) http://bit.ly/2oXa2pb
Probability (option 4) http://bit.ly/2oXbZSK
OTHER UDEMY COURSE COUPONS
As you know, I’m the “Lazy Programmer”, not just the “Lazy Data Scientist” – I love all kinds of programming!
iOS courses:
https://lazyprogrammer.me/ios
Android courses:
https://lazyprogrammer.me/android
Ruby on Rails courses:
https://lazyprogrammer.me/rubyonrails
Python courses:
https://lazyprogrammer.me/python
Big Data (Spark + Hadoop) courses:
https://lazyprogrammer.me/bigdatahadoopsparksql
Javascript, ReactJS, AngularJS courses:
https://lazyprogrammer.me/js
EVEN MORE COOL STUFF
Into Yoga in your spare time? Photography? Painting? There are courses, and I’ve got coupons! If you find a course on Udemy that you’d like a coupon for, just let me know and I’ll hook you up!
Remember, these links will selfdestruct on December 21 (3 days). Act NOW!
July 24, 2017
Since I am still busy hacking away at my next course, we are going to do another HUGE sale. ALL courses on Udemy are now $10. Take this opportunity to grab as many courses as you can because you never know when the next sale is going to be!
As usual, I’m providing $10 coupons for all my courses in the links below. Please use these links and share them with your friends!
You can also just type in the coupon code “JUL456”.
The $10 promo doesn’t come around often, so make sure you pick up everything you are interested in, or could become interested in later this year. The promo goes until July 31. Don’t wait!
At the end of this post, I’m going to provide you with some additional links to get machine learning prerequisites (calculus, linear algebra, Python, etc…) for $10 too!
If you don’t know what order to take the courses in, please check here: https://deeplearningcourses.com/course_order
Here are the links for my courses:
Deep Learning Prerequisites: Linear Regression in Python
https://www.udemy.com/datasciencelinearregressioninpython/?couponCode=JUL456
Deep Learning Prerequisites: Logistic Regression in Python
https://www.udemy.com/datasciencelogisticregressioninpython/?couponCode=JUL456
Deep Learning in Python
https://www.udemy.com/datasciencedeeplearninginpython/?couponCode=JUL456
Practical Deep Learning in Theano and TensorFlow
https://www.udemy.com/datasciencedeeplearningintheanotensorflow/?couponCode=JUL456
Deep Learning: Convolutional Neural Networks in Python
https://www.udemy.com/deeplearningconvolutionalneuralnetworkstheanotensorflow/?couponCode=JUL456
Unsupervised Deep Learning in Python
https://www.udemy.com/unsuperviseddeeplearninginpython/?couponCode=JUL456
Deep Learning: Recurrent Neural Networks in Python
https://www.udemy.com/deeplearningrecurrentneuralnetworksinpython/?couponCode=JUL456
Advanced Natural Language Processing: Deep Learning in Python
https://www.udemy.com/naturallanguageprocessingwithdeeplearninginpython/?couponCode=JUL456
Advanced AI: Deep Reinforcement Learning in Python
https://www.udemy.com/deepreinforcementlearninginpython/?couponCode=JUL456
Easy Natural Language Processing in Python
https://www.udemy.com/datasciencenaturallanguageprocessinginpython/?couponCode=JUL456
Cluster Analysis and Unsupervised Machine Learning in Python
https://www.udemy.com/clusteranalysisunsupervisedmachinelearningpython/?couponCode=JUL456
Unsupervised Machine Learning: Hidden Markov Models in Python
https://www.udemy.com/unsupervisedmachinelearninghiddenmarkovmodelsinpython/?couponCode=JUL456
Data Science: Supervised Machine Learning in Python
https://www.udemy.com/datasciencesupervisedmachinelearninginpython/?couponCode=JUL456
Bayesian Machine Learning in Python: A/B Testing
https://www.udemy.com/bayesianmachinelearninginpythonabtesting/?couponCode=JUL456
Ensemble Machine Learning in Python: Random Forest and AdaBoost
https://www.udemy.com/machinelearninginpythonrandomforestadaboost/?couponCode=JUL456
Artificial Intelligence: Reinforcement Learning in Python
https://www.udemy.com/artificialintelligencereinforcementlearninginpython/?couponCode=JUL456
SQL for Newbs and Marketers
https://www.udemy.com/sqlformarketersdataanalyticsdatasciencebigdata/?couponCode=JUL456
PREREQUISITE COURSE COUPONS
And last but not least, $10 coupons for some helpful prerequisite courses. You NEED to know this stuff before you study machine learning:
General (sitewide): http://bit.ly/2oCY14Z
Python http://bit.ly/2pbXxXz
Calc 1 http://bit.ly/2okPUib
Calc 2 http://bit.ly/2oXnhpX
Calc 3 http://bit.ly/2pVU0gQ
Linalg 1 http://bit.ly/2oBBir1
Linalg 2 http://bit.ly/2q5SGEE
Probability (option 1) http://bit.ly/2prFQ7o
Probability (option 2) http://bit.ly/2p8kcC0
Probability (option 3) http://bit.ly/2oXa2pb
Probability (option 4) http://bit.ly/2oXbZSK
OTHER UDEMY COURSE COUPONS
As you know, I’m the “Lazy Programmer”, not just the “Lazy Data Scientist” – I love all kinds of programming!
And I’ve got sales for everything:
iOS
Android
Ruby on Rails
Python
Big Data (Spark, Hadoop)
Javascript, ReactJS, AngularJS
Remember, these links will selfdestruct on July 31 (7 days). Act NOW!
Go to commentsJuly 11, 2017
Since I am still busy hacking away at my next course, we are going to do another HUGE sale. ALL courses on Udemy are now $10. Take this opportunity to grab as many courses as you can because you never know when the next sale is going to be!
As usual, I’m providing $10 coupons for all my courses in the links below. Please use these links and share them with your friends!
You can also just type in the coupon code “JUL123”.
The $10 promo doesn’t come around often, so make sure you pick up everything you are interested in, or could become interested in later this year. The promo goes until July 20. Don’t wait!
At the end of this post, I’m going to provide you with some additional links to get machine learning prerequisites (calculus, linear algebra, Python, etc…) for $10 too!
If you don’t know what order to take the courses in, please check here: https://deeplearningcourses.com/course_order
Here are the links for my courses:
Deep Learning Prerequisites: Linear Regression in Python
https://www.udemy.com/datasciencelinearregressioninpython/?couponCode=JUL123
Deep Learning Prerequisites: Logistic Regression in Python
https://www.udemy.com/datasciencelogisticregressioninpython/?couponCode=JUL123
Deep Learning in Python
https://www.udemy.com/datasciencedeeplearninginpython/?couponCode=JUL123
Practical Deep Learning in Theano and TensorFlow
https://www.udemy.com/datasciencedeeplearningintheanotensorflow/?couponCode=JUL123
Deep Learning: Convolutional Neural Networks in Python
https://www.udemy.com/deeplearningconvolutionalneuralnetworkstheanotensorflow/?couponCode=JUL123
Unsupervised Deep Learning in Python
https://www.udemy.com/unsuperviseddeeplearninginpython/?couponCode=JUL123
Deep Learning: Recurrent Neural Networks in Python
https://www.udemy.com/deeplearningrecurrentneuralnetworksinpython/?couponCode=JUL123
Advanced Natural Language Processing: Deep Learning in Python
https://www.udemy.com/naturallanguageprocessingwithdeeplearninginpython/?couponCode=JUL123
Advanced AI: Deep Reinforcement Learning in Python
https://www.udemy.com/deepreinforcementlearninginpython/?couponCode=JUL123
Easy Natural Language Processing in Python
https://www.udemy.com/datasciencenaturallanguageprocessinginpython/?couponCode=JUL123
Cluster Analysis and Unsupervised Machine Learning in Python
https://www.udemy.com/clusteranalysisunsupervisedmachinelearningpython/?couponCode=JUL123
Unsupervised Machine Learning: Hidden Markov Models in Python
https://www.udemy.com/unsupervisedmachinelearninghiddenmarkovmodelsinpython/?couponCode=JUL123
Data Science: Supervised Machine Learning in Python
https://www.udemy.com/datasciencesupervisedmachinelearninginpython/?couponCode=JUL123
Bayesian Machine Learning in Python: A/B Testing
https://www.udemy.com/bayesianmachinelearninginpythonabtesting/?couponCode=JUL123
Ensemble Machine Learning in Python: Random Forest and AdaBoost
https://www.udemy.com/machinelearninginpythonrandomforestadaboost/?couponCode=JUL123
Artificial Intelligence: Reinforcement Learning in Python
https://www.udemy.com/artificialintelligencereinforcementlearninginpython/?couponCode=JUL123
SQL for Newbs and Marketers
https://www.udemy.com/sqlformarketersdataanalyticsdatasciencebigdata/?couponCode=JUL123
RECENT COURSE UPDATES
You guys know I’m always working hard to improve my courses with new and fresh content. Here’s what I’ve done recently:
– Hidden Markov Models: added Tensorflow implementation and improved Theano implementation
– Hidden Markov Models: the Tensorflow scan function
– Unsupervised Deep Learning: added Tensorflow version of Autoencoder, RBM, and Unsupervised PreTraining
– Deep Learning in Python (pt1): How to interpret the weights
– Deep Learning in Theano / Tensorflow (pt2): How to improve your Theano and Tensorflow skills
– Deep Learning in Theano / Tensorflow (pt2): Some ideas about combining big data + deep learning
PREREQUISITE COURSE COUPONS
And last but not least, $10 coupons for some helpful prerequisite courses. You NEED to know this stuff before you study machine learning:
General (sitewide): http://bit.ly/2oCY14Z
Python http://bit.ly/2pbXxXz
Calc 1 http://bit.ly/2okPUib
Calc 2 http://bit.ly/2oXnhpX
Calc 3 http://bit.ly/2pVU0gQ
Linalg 1 http://bit.ly/2oBBir1
Linalg 2 http://bit.ly/2q5SGEE
Probability (option 1) http://bit.ly/2prFQ7o
Probability (option 2) http://bit.ly/2p8kcC0
Probability (option 3) http://bit.ly/2oXa2pb
Probability (option 4) http://bit.ly/2oXbZSK
Remember, these links will selfdestruct on July 20 (9 days). Act NOW!
Go to commentsJune 14, 2017
Today, Udemy has decided to do yet another AMAZING $10 promo.
As usual, I’m providing $10 coupons for all my courses in the links below. Please use these links and share them with your friends!
The $10 promo doesn’t come around often, so make sure you pick up everything you are interested in, or could become interested in later this year. The promo goes until June 21. Don’t wait!
At the end of this post, I’m going to provide you with some additional links to get machine learning prerequisites (calculus, linear algebra, Python, etc…) for $10 too!
If you don’t know what order to take the courses in, please check here: https://deeplearningcourses.com/course_order
Here are the links for my courses:
Deep Learning Prerequisites: Linear Regression in Python
https://www.udemy.com/datasciencelinearregressioninpython/?couponCode=JUN456
Deep Learning Prerequisites: Logistic Regression in Python
https://www.udemy.com/datasciencelogisticregressioninpython/?couponCode=JUN456
Deep Learning in Python
https://www.udemy.com/datasciencedeeplearninginpython/?couponCode=JUN456
Practical Deep Learning in Theano and TensorFlow
https://www.udemy.com/datasciencedeeplearningintheanotensorflow/?couponCode=JUN456
Deep Learning: Convolutional Neural Networks in Python
https://www.udemy.com/deeplearningconvolutionalneuralnetworkstheanotensorflow/?couponCode=JUN456
Unsupervised Deep Learning in Python
https://www.udemy.com/unsuperviseddeeplearninginpython/?couponCode=JUN456
Deep Learning: Recurrent Neural Networks in Python
https://www.udemy.com/deeplearningrecurrentneuralnetworksinpython/?couponCode=JUN456
Advanced Natural Language Processing: Deep Learning in Python
https://www.udemy.com/naturallanguageprocessingwithdeeplearninginpython/?couponCode=JUN456
Advanced AI: Deep Reinforcement Learning in Python
https://www.udemy.com/deepreinforcementlearninginpython/?couponCode=JUN456
Easy Natural Language Processing in Python
https://www.udemy.com/datasciencenaturallanguageprocessinginpython/?couponCode=JUN456
Cluster Analysis and Unsupervised Machine Learning in Python
https://www.udemy.com/clusteranalysisunsupervisedmachinelearningpython/?couponCode=JUN456
Unsupervised Machine Learning: Hidden Markov Models in Python
https://www.udemy.com/unsupervisedmachinelearninghiddenmarkovmodelsinpython/?couponCode=JUN456
Data Science: Supervised Machine Learning in Python
https://www.udemy.com/datasciencesupervisedmachinelearninginpython/?couponCode=JUN456
Bayesian Machine Learning in Python: A/B Testing
https://www.udemy.com/bayesianmachinelearninginpythonabtesting/?couponCode=JUN456
Ensemble Machine Learning in Python: Random Forest and AdaBoost
https://www.udemy.com/machinelearninginpythonrandomforestadaboost/?couponCode=JUN456
Artificial Intelligence: Reinforcement Learning in Python
https://www.udemy.com/artificialintelligencereinforcementlearninginpython/?couponCode=JUN456
SQL for Newbs and Marketers
https://www.udemy.com/sqlformarketersdataanalyticsdatasciencebigdata/?couponCode=JUN456
And last but not least, $10 coupons for some helpful prerequisite courses. You NEED to know this stuff before you study machine learning:
General (sitewide): http://bit.ly/2oCY14Z
Python http://bit.ly/2pbXxXz
Calc 1 http://bit.ly/2okPUib
Calc 2 http://bit.ly/2oXnhpX
Calc 3 http://bit.ly/2pVU0gQ
Linalg 1 http://bit.ly/2oBBir1
Linalg 2 http://bit.ly/2q5SGEE
Probability (option 1) http://bit.ly/2prFQ7o
Probability (option 2) http://bit.ly/2p8kcC0
Probability (option 3) http://bit.ly/2oXa2pb
Probability (option 4) http://bit.ly/2oXbZSK
Remember, these links will selfdestruct on June 21 (7 days). Act NOW!
Go to commentsJune 5, 2017
Today, Udemy has decided to do yet another AMAZING $10 promo.
As usual, I’m providing $10 coupons for all my courses in the links below. Please use these links and share them with your friends!
The $10 promo doesn’t come around often, so make sure you pick up everything you are interested in, or could become interested in later this year. The promo goes until June 10. Don’t wait!
At the end of this post, I’m going to provide you with some additional links to get machine learning prerequisites (calculus, linear algebra, Python, etc…) for $10 too!
If you don’t know what order to take the courses in, please check here: https://deeplearningcourses.com/course_order
Here are the links for my courses:
Deep Learning Prerequisites: Linear Regression in Python
https://www.udemy.com/datasciencelinearregressioninpython/?couponCode=JUN123
Deep Learning Prerequisites: Logistic Regression in Python
https://www.udemy.com/datasciencelogisticregressioninpython/?couponCode=JUN123
Deep Learning in Python
https://www.udemy.com/datasciencedeeplearninginpython/?couponCode=JUN123
Practical Deep Learning in Theano and TensorFlow
https://www.udemy.com/datasciencedeeplearningintheanotensorflow/?couponCode=JUN123
Deep Learning: Convolutional Neural Networks in Python
https://www.udemy.com/deeplearningconvolutionalneuralnetworkstheanotensorflow/?couponCode=JUN123
Unsupervised Deep Learning in Python
https://www.udemy.com/unsuperviseddeeplearninginpython/?couponCode=JUN123
Deep Learning: Recurrent Neural Networks in Python
https://www.udemy.com/deeplearningrecurrentneuralnetworksinpython/?couponCode=JUN123
Advanced Natural Language Processing: Deep Learning in Python
https://www.udemy.com/naturallanguageprocessingwithdeeplearninginpython/?couponCode=JUN123
Advanced AI: Deep Reinforcement Learning in Python
https://www.udemy.com/deepreinforcementlearninginpython/?couponCode=JUN123
Easy Natural Language Processing in Python
https://www.udemy.com/datasciencenaturallanguageprocessinginpython/?couponCode=JUN123
Cluster Analysis and Unsupervised Machine Learning in Python
https://www.udemy.com/clusteranalysisunsupervisedmachinelearningpython/?couponCode=JUN123
Unsupervised Machine Learning: Hidden Markov Models in Python
https://www.udemy.com/unsupervisedmachinelearninghiddenmarkovmodelsinpython/?couponCode=JUN123
Data Science: Supervised Machine Learning in Python
https://www.udemy.com/datasciencesupervisedmachinelearninginpython/?couponCode=JUN123
Bayesian Machine Learning in Python: A/B Testing
https://www.udemy.com/bayesianmachinelearninginpythonabtesting/?couponCode=JUN123
Ensemble Machine Learning in Python: Random Forest and AdaBoost
https://www.udemy.com/machinelearninginpythonrandomforestadaboost/?couponCode=JUN123
Artificial Intelligence: Reinforcement Learning in Python
https://www.udemy.com/artificialintelligencereinforcementlearninginpython/?couponCode=JUN123
SQL for Newbs and Marketers
https://www.udemy.com/sqlformarketersdataanalyticsdatasciencebigdata/?couponCode=JUN123
And last but not least, $10 coupons for some helpful prerequisite courses. You NEED to know this stuff before you study machine learning:
General (sitewide): http://bit.ly/2oCY14Z
Python http://bit.ly/2pbXxXz
Calc 1 http://bit.ly/2okPUib
Calc 2 http://bit.ly/2oXnhpX
Calc 3 http://bit.ly/2pVU0gQ
Linalg 1 http://bit.ly/2oBBir1
Linalg 2 http://bit.ly/2q5SGEE
Probability (option 1) http://bit.ly/2prFQ7o
Probability (option 2) http://bit.ly/2p8kcC0
Probability (option 3) http://bit.ly/2oXa2pb
Probability (option 4) http://bit.ly/2oXbZSK
Remember, these links will selfdestruct on June 10 (5 days). Act NOW!
Go to commentsAugust 9, 2016
[Scroll to the bottom for the early bird discount if you already know what this course is about]
In this course we are going to look at advanced NLP using deep learning.
Previously, you learned about some of the basics, like how many NLP problems are just regular machine learning and data science problems in disguise, and simple, practical methods like bagofwords and termdocument matrices.
These allowed us to do some pretty cool things, like detect spam emails, write poetry, spin articles, and group together similar words.
In this course I’m going to show you how to do even more awesome things. We’ll learn not just 1, but 4 new architectures in this course.
First up is word2vec.
In this course, I’m going to show you exactly how word2vec works, from theory to implementation, and you’ll see that it’s merely the application of skills you already know.
Word2vec is interesting because it magically maps words to a vector space where you can find analogies, like:
We are also going to look at the GLoVe method, which also finds word vectors, but uses a technique called matrix factorization, which is a popular algorithm for recommender systems.
Amazingly, the word vectors produced by GLoVe are just as good as the ones produced by word2vec, and it’s way easier to train.
We will also look at some classical NLP problems, like partsofspeech tagging and named entity recognition, and use recurrent neural networks to solve them. You’ll see that just about any problem can be solved using neural networks, but you’ll also learn the dangers of having too much complexity.
Lastly, you’ll learn about recursive neural networks, which finally help us solve the problem of negation in sentiment analysis. Recursive neural networks exploit the fact that sentences have a tree structure, and we can finally get away from naively using bagofwords.
All of the materials required for this course can be downloaded and installed for FREE. We will do most of our work in Numpy and Matplotlib,and Theano. I am always available to answer your questions and help you along your data science journey.
See you in class!
UPDATE: New coupon if the above is sold out:
#deep learning #GLoVe #natural language processing #nlp #python #recursive neural networks #tensorflow #theano #word2vecApril 25, 2016
This article will be of interest to you if you want to learn about recommender systems and predicting movie ratings (or book ratings, or product ratings, or any other kind of rating).
Contests like the $1 million Netflix Challenge are an example of what collaborative filtering can be used for.
Let’s use the “users rating movies” example for this tutorial. After some Internet searching, we can determine that there are approximately 500, 000 movies in existence. Let’s also suppose that your very popular movie website has 1 billion users (Facebook has 1.6 billion users as of 2015, so this number is plausible).
How many possible usermovie ratings can you have? That is \( 10^9 \times 5 \times 10^5 = 5 \times 10^{14} \). That’s a lot of ratings! Way too much to fit into your RAM, in fact.
But that’s just one problem.
How many movies have you seen in your life? Of those movies, what percentage of them have you rated? The number is miniscule. In fact, most users have not rated most movies.
This is why recommender systems exist in the first place – so we can recommend you movies that you haven’t seen yet, that we know you’ll like.
So if you were to create a usermovie matrix of movie ratings, most of it would just have missing values.
However, that’s not to say there isn’t a pattern to be found.
Suppose we look at a subset of movie ratings, and we find the following:
Where we’ve used N/A to show that a movie has not yet been rated by a user.
If we used the “cosine distance” ( \( \frac{u^T v}{ uv } \) ) on the vectors created by looking at only the common movies, we could see that Guy A and Guy B have similar tastes. We could then surmise, based on this closeness, that Guy A might rate the Batman movie a “4”, and Guy B might rate Batman Returns a “4”. And since this is a pretty high rating, we might want to recommend these movies to these users.
This is the idea behind collaborative filtering.
Matrix factorization solves the above problems by reducing the number of free parameters (so the total number of parameters is much smaller than #users times #movies), and by fitting these parameters to the data (ratings) that do exist.
What is matrix factorization?
Think of factorization in general:
15 = 3 x 5 (15 is made up of the factors 3 and 5)
\( x^2 + x = x(x + 1) \)
We can do the same thing with matrices:
$$\left( \begin{matrix}3 & 4 & 5 \\ 6 & 8 & 10 \end{matrix} \right) = \left( \begin{matrix}1 \\ 2 \end{matrix} \right) \left( \begin{matrix}3 & 4 & 5 \end{matrix} \right) $$
In fact, this is exactly what we do in matrix factorization. We “pretend” the big ratings matrix (the one that can’t fit into our RAM) is actually made up of 2 smaller matrices multiplied together.
Remember that to do a valid matrix multiply, the inner dimensions must match. What is the size of this dimension? We call it “K”. It is unknown, but we can choose it via possibly crossvalidation so that our model generalizes well.
If we have \( M \) users and \( N \) ratings, then the total number of parameters in our model is \( MK + NK \). If we set \( K = 10 \), the total number of parameters we’d have for the usermovie problem would be \( 10^{10} + 5 \times 10^6 \), which is still approximately \( 10^{10} \), which is a factor of \( 10^4 \) smaller than before.
This is a big improvement!
So now we have:
$$ A \simeq \hat{ A } = UV $$
If you were to picture the matrices themselves, they would look like this:
Because I am lazy and took this image from elsewhere on the Internet, the “d” here is what I am calling “K”. And their “R” is my “A”.
You know that with any machine learning algorithm we have 2 procedures – the fitting procedure and the prediction procedure.
For the fitting procedure, we want every known \( A_{ij} \) to be as close to \( \hat{A}_{ij} = u_i^Tv_j \) as possible. \( u_i \) is the ith row of \( U \). \( v_j \) is the jth column of \( V \).
For the prediction procedure, we won’t have an \( A_{ij} \), but we can use \( \hat{A}_{ij} = u_i^Tv_j \) to tell us what user i might rate movie j given the existing patterns.
A natural cost function for this problem is the squared error. Think of it as a regression. This is just:
$$ J = \sum_{(i, j) \in \Omega} (A_{ij} – \hat{A}_{ij})^2 $$
Where \( \Omega \) is the set of all pairs \( (i, j) \) where user i has rated movie j.
Later, we will use \( \Omega_i \) to be the set of all j’s (movies) that user i has rated, and we will use \( \Omega_j \) to be the set of all i’s (users) that have rated movie j.
What do you do when you want to minimize a function? Take the derivative and set it to 0, of course. No need to use anything more complicated if the simple approach is solvable and performs well. It is also possible to use gradient descent on this problem by taking the derivative and then taking small steps in that direction.
You will notice that there are 2 derivatives to take here. The first is \( \partial{J} / \partial{u} \).
The other is \( \partial{J} / \partial{v} \). After calculating the derivatives and solving for \( u \) and \( v \), you get:
$$ u_i = ( \sum_{j \in \Omega_i} v_j v_j^T )^{1} \sum_{j \in \Omega_i} A_{ij} v_j $$
$$ v_j = ( \sum_{i \in \Omega_j} u_i u_i^T )^{1} \sum_{i \in \Omega_j} A_{ij} u_i $$
So you take both derivatives. You set both to 0. You solve for the optimal u and v. Now what?
The answer is: coordinate descent.
You first update \( u \) using the current setting of \( v \), then you update \( v \) using the current setting of \( u \). The order doesn’t matter, just that you alternate between the two.
There is a mathematical guarantee that J will improve on each iteration.
This technique is also known as alternating least squares. (This makes sense because we’re minimizing the squared error and updating \( u \) and \( v \) in an alternating fashion.)
As with other methods like linear regression and logistic regression, we can add bias parameters to our model to improve accuracy. In this case our model becomes:
$$ \hat{A}_{ij} = u_i^T v_j + b_i + c_j + \mu $$
Where \( \mu \) is the global mean (average of all known ratings).
You can interpret \( b_i \) as the bias of a user. A negative bias means this user just hates movies more than the average person. A positive bias would mean the opposite. Similarly, \( c_j \) is the bias of a movie. A positive bias would mean, “Wow, this movie is good, regardless of who is watching it!” A negative bias would be a movie like Avatar: The Last Airbender.
We can recalculate the optimal settings for each parameter (again by taking the derivatives and setting them to 0) to get:
$$ u_i = ( \sum_{j \in \Omega_i} v_j v_j^T )^{1} \sum_{j \in \Omega_i} (A_{ij} – b_i – c_j – \mu )v_j $$
$$ v_j = ( \sum_{i \in \Omega_j} u_i u_i^T )^{1} \sum_{i \in \Omega_j}(A_{ij} – b_i – c_j – \mu )u_i $$
$$ b_i = \frac{1}{ \Omega_i }\sum_{j \in \Omega_i} A_{ij} – u_i^Tv_j – c_j – \mu $$
$$ c_j= \frac{1}{ \Omega_j }\sum_{i \in \Omega_j} A_{ij} – u_i^Tv_j – b_i – \mu $$
With the above model, you may encounter what is called the “singular covariance” problem. This is what happens when you can’t invert the matrix that appears in the updates for \( u \) and \( v \).
The solution is again, similar to what you would do in linear regression or logistic regression: Add a squared error term with a weight \( \lambda \) that keeps the parameters small.
In terms of the likelihood, the previous formulation assumes that the difference between \( A_{ij} \) and \( \hat{A}_{ij} \) is normally distributed, while the cost function with regularization is like adding a normallydistributed prior on each parameter centered at 0.
i.e. \( u_i, v_j, b_i, c_j \sim N(0, 1/\lambda) \).
So the cost function becomes:
$$ J = \sum_{(i, j) \in \Omega} (A_{ij} – \hat{A}_{ij})^2 + \lambda(U_F + V_F + b^2 + c^2) $$
Where \( X_F \) is the Frobenius norm of \( X \).
For each parameter, setting the derivative with respect to that parameter, setting it to 0 and solving for the optimal value yields:
$$ u_i = ( \sum_{j \in \Omega_i} v_j v_j^T + \lambda{I})^{1} \sum_{j \in \Omega_i} (A_{ij} – b_i – c_j – \mu )v_j $$
$$ v_j = ( \sum_{i \in \Omega_j} u_i u_i^T + \lambda{I})^{1} \sum_{i \in \Omega_j}(A_{ij} – b_i – c_j – \mu )u_i $$
$$ b_i = \frac{1}{ \Omega_i  +\lambda}\sum_{j \in \Omega_i} A_{ij} – u_i^Tv_j – c_j – \mu $$
$$ c_j= \frac{1}{ \Omega_j  +\lambda}\sum_{i \in \Omega_j} A_{ij} – u_i^Tv_j – b_i – \mu $$
The simplest way to implement the above formulas would be to just code them directly.
Initialize your parameters as follows:
U = np.random.randn(M, K) / K V = np.random.randn(K, N) / K B = np.zeros(M) C = np.zeros(N)
Next, you want \( \Omega_i \) and \( \Omega_j \) to be easily accessible, so create dictionaries “ratings_by_i” where “i” is the key, and the value is an array of all the (j, r) pairs that user i has rated (r is the rating). Do the same for “ratings_by_j”.
Then, your updates would be as follows:
for t in xrange(T): # update B for i in xrange(M): if i in ratings_by_i: accum = 0 for j, r in ratings_by_i[i]: accum += (r  U[i,:].dot(V[:,j])  C[j]  mu) B[i] = accum / (len(ratings_by_i[i]) + reg) # update U for i in xrange(M): if i in ratings_by_i: matrix = np.zeros((K, K)) + reg*np.eye(K) vector = np.zeros(K) for j, r in ratings_by_i[i]: matrix += np.outer(V[:,j], V[:,j]) vector += (r  B[i]  C[j]  mu)*V[:,j] U[i,:] = np.linalg.solve(matrix, vector) # update C for j in xrange(N): if j in ratings_by_j: accum = 0 for i, r in ratings_by_j[j]: accum += (r  U[i,:].dot(V[:,j])  B[i]  mu) C[j] = accum / (len(ratings_by_j[j]) + reg) # update V for j in xrange(N): if j in ratings_by_j: matrix = np.zeros((K, K)) + reg*np.eye(K) vector = np.zeros(K) for i, r in ratings_by_j[j]: matrix += np.outer(U[i,:], U[i,:]) vector += (r  B[i]  C[j]  mu)*U[i,:] V[:,j] = np.linalg.solve(matrix, vector)
And that’s all there is to it!
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