New course out today – Recurrent Neural Networks in Python: Deep Learning part 5.

If you already know what the course is about (recurrent units, GRU, LSTM), grab your 50% OFF coupon and go!:

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Like the course I just released on Hidden Markov Models, Recurrent Neural Networks are all about learning sequences – but whereas Markov Models are limited by the Markov assumption, Recurrent Neural Networks are not – and as a result, they are more expressive, and more powerful than anything we’ve seen on tasks that we haven’t made progress on in decades.

Sequences appear everywhere – stock prices, language, credit scoring, and webpage visits.

Recurrent neural networks have a history of being very hard to train. It hasn’t been until recently that we’ve found ways around what is called the vanishing gradient problem, and since then, recurrent neural networks have become one of the most popular methods in deep learning.

If you took my course on Hidden Markov Models, we are going to go through a lot of the same examples in this class, except that our results are going to be a lot better.

Our classification accuracies will increase, and we’ll be able to create vectors of words, or word embeddings, that allow us to visualize how words are related on a graph.

We’ll see some pretty interesting results, like that our neural network seems to have learned that all religions and languages and numbers are related, and that cities and countries have hierarchical relationships.

If you’re interested in discovering how modern deep learning has propelled machine learning and data science to new heights, this course is for you.

I’ll see you in class.

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#data science #deep learning #gru #lstm #machine learning #word vectors
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EARLY BIRD 50% OFF COUPON: CLICK HERE

**Hidden Markov Models** are all about learning sequences.

A lot of the data that would be very useful for us to model is in sequences. **Stock prices** are sequences of prices. Language is a sequence of words. **Credit scoring** involves sequences of borrowing and repaying money, and we can use those sequences to predict whether or not you’re going to default. In short, sequences are everywhere, and being able to analyze them is an important skill in your **data science** toolbox.

The easiest way to appreciate the kind of information you get from a sequence is to consider what you are reading right now. If I had written the previous sentence backwards, it wouldn’t make much sense to you, even though it contained all the same words. So order is important.

While the current fad in **deep learning **is to use **recurrent neural networks** to model sequences, I want to first introduce you guys to a machine learning algorithm that has been around for several decades now – the Hidden Markov Model.

This course follows directly from my first course in **Unsupervised Machine Learning for Cluster Analysis**, where you learned how to measure the **probability distribution** of a **random variable**. In this course, you’ll learn to measure the probability distribution of a sequence of random variables.

You guys know how much I love **deep learning**, so there is a little twist in this course. We’ve already covered **gradient descent** and you know how central it is for solving deep learning problems. I claimed that gradient descent could be used to optimize any objective function. In this course I will show you how you can use gradient descent to solve for the optimal parameters of an HMM, as an alternative to the popular **expectation-maximization** algorithm.

We’re going to do it in Theano, which is a popular library for deep learning. This is also going to teach you how to work with sequences in Theano, which will be very useful when we cover **recurrent neural networks** and **LSTMs**.

This course is also going to go through the many practical applications of Markov models and hidden Markov models. We’re going to look at a model of sickness and health, and calculate how to predict how long you’ll stay sick, if you get sick. We’re going to talk about how Markov models can be used to analyze how people interact with your website, and fix problem areas like high **bounce rate**, which could be affecting your **SEO**. We’ll build language models that can be used to identify a writer and even generate text – imagine a machine doing your writing for you.

We’ll look at what is possibly the most recent and prolific application of Markov models – **Google’s PageRank** algorithm. And finally we’ll discuss even more practical applications of Markov models, including generating images, **smartphone** **autosuggestions**, and using HMMs to answer one of the most fundamental questions in **biology** – how is **DNA**, the code of life, translated into physical or behavioral attributes of an organism?

All of the materials of this course can be downloaded and installed for FREE. We will do most of our work in **Numpy** and **Matplotlib**, along with a little bit of **Theano**. I am always available to answer your questions and help you along your data science journey.

Sign up now and get 50% off by clicking HERE

#data science #deep learning #hidden markov models #machine learning #recurrent neural networks #theano
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This course is the next logical step in my **deep learning, data science,** and **machine learning** series. I’ve done a lot of courses about deep learning, and I just released a course about **unsupervised learning**, where I talked about **clustering** and **density estimation**. So what do you get when you put these 2 together? Unsupervised deep learning!

In these course we’ll start with some very basic stuff – **principal components analysis (PCA)**, and a popular nonlinear dimensionality reduction technique known as **t-SNE (t-distributed stochastic neighbor embedding)**.

Next, we’ll look at a special type of unsupervised neural network called the **autoencoder**. After describing how an autoencoder works, I’ll show you how you can link a bunch of them together to form a deep stack of autoencoders, that leads to better performance of a supervised** deep neural network**. Autoencoders are like a non-linear form of PCA.

Last, we’ll look at **restricted Boltzmann machines (RBMs)**. These are yet another popular unsupervised neural network, that you can use in the same way as autoencoders to **pretrain** your supervised deep neural network. I’ll show you an interesting way of training restricted Boltzmann machines, known as **Gibbs sampling**, a special case of **Markov Chain Monte Carlo,** and I’ll demonstrate how even though this method is only a rough approximation, it still ends up reducing other cost functions, such as the one used for autoencoders. This method is also known as **Contrastive Divergence** or **CD-k**. As in physical systems, we define a concept called **free energy** and attempt to minimize this quantity.

Finally, we’ll bring all these concepts together and I’ll show you visually what happens when you use PCA and t-SNE on the features that the autoencoders and RBMs have learned, and we’ll see that even without labels the results suggest that a pattern has been found.

All the materials used in this course are FREE. Since this course is the 4th in the deep learning series, I will assume you already know calculus, linear algebra, and **Python** coding. You’ll want to install **Numpy** and**Theano** for this course. These are essential items in your **data analytics** toolbox.

If you are interested in deep learning and you want to learn about modern deep learning developments beyond just plain **backpropagation**, including using unsupervised neural networks to interpret what features can be automatically and hierarchically learned in a deep learning system, this course is for you.

Get your EARLY BIRD coupon for 50% off here: https://www.udemy.com/unsupervised-deep-learning-in-python/?couponCode=EARLYBIRD

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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.

## Problem Setup

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 user-movie 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 user-movie 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:

Batman

Batman Returns

Batman Begins

The Dark Knight

Batman v. Superman

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” ( \( \frac{u^T v}{ |u||v| } \) ) 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.

## Enter Matrix Factorization

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 cross-validation 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 user-movie 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.

## The Cost Function

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.

## Coordinate Descent

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.)

## Bias Parameters

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 re-calculate the optimal settings for each parameter (again by taking the derivatives and setting them to 0) to get:

$$ u_i = ( \sum_{j \in \Omega_i} v_j v_j^T )^{-1} \sum_{j \in \Omega_i} (A_{ij} – b_i – c_j – \mu )v_j $$

$$ v_j = ( \sum_{i \in \Omega_j} u_i u_i^T )^{-1} \sum_{i \in \Omega_j}(A_{ij} – b_i – c_j – \mu )u_i $$

$$ b_i = \frac{1}{| \Omega_i |}\sum_{j \in \Omega_i} A_{ij} – u_i^Tv_j – c_j – \mu $$

$$ c_j= \frac{1}{| \Omega_j |}\sum_{i \in \Omega_j} A_{ij} – u_i^Tv_j – b_i – \mu $$

## Regularization

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 normally-distributed 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:

$$ J = \sum_{(i, j) \in \Omega} (A_{ij} – \hat{A}_{ij})^2 + \lambda(||U||_F^2 + ||V||_F^2 + ||b||^2 + ||c||^2) $$

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:

$$ u_i = ( \sum_{j \in \Omega_i} v_j v_j^T + \lambda{I})^{-1} \sum_{j \in \Omega_i} (A_{ij} – b_i – c_j – \mu )v_j $$

$$ v_j = ( \sum_{i \in \Omega_j} u_i u_i^T + \lambda{I})^{-1} \sum_{i \in \Omega_j}(A_{ij} – b_i – c_j – \mu )u_i $$

$$ b_i = \frac{1}{1 + \lambda} \frac{1}{| \Omega_i |}\sum_{j \in \Omega_i} A_{ij} – u_i^Tv_j – c_j – \mu $$

$$ c_j= \frac{1}{1 + \lambda} \frac{1}{| \Omega_j |}\sum_{i \in \Omega_j} A_{ij} – u_i^Tv_j – b_i – \mu $$

## Python Code

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 / (1 + reg) / len(ratings_by_i[i])
# 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 / (1 + reg) / len(ratings_by_j[j])
# 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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