August 24, 2018
Go to commentsAugust 14, 2018





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Go to commentsAugust 2, 2018
I’ve finally gotten around to adding a section on PyTorch basics to my course, Modern Deep Learning in Python (which already goes indepth on Theano and Tensorflow).
As you recall, this course focuses on modern deep learning techniques such as adaptive learning rates and momentum, modern deep learning frameworks and GPU acceleration, and modern regularization techniques like dropout and batch normalization.
Check out the new videos here:
May 7, 2018
I’m hard at work at my next course, so guess what that means? Everything on sale!
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Go to commentsApril 10, 2018
A lot of students come up to me and ask about when they’re going to learn the latest and greatest new deep learning algorithm.
Sometimes, it’s easy to forget how applicable even the most basic of tools are.
As you know, I consider Linear Regression to be the best starting point for deep learning and machine learning in general.
And wouldn’t you know, here it is being used in the most advanced, stateoftheart video codec we have today:
next generation video: Introducing AV1
Check it out!
This new stateoftheart video codec is based on research done by multiple big companies, such as Google, Cisco, and Mozilla.
As you can see, the final equation is just a line (\( y = mx + b \)).
$$ CfL(\alpha) = \alpha L^{AC} + DC $$
Go to commentsMay 15, 2017
This month, Udemy is having a special event called the “Udemy Learn Fest”, and you know I watch these things like a hawk so that when Udemy has their best deals I can bring the news to you as soon as they happen.
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Go to commentsAugust 25, 2016
For some reason Udemy announced a promotion but when you go to the site it doesn’t appear.
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Go to commentsApril 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:
$$ u_i = ( \sum_{j \in \Omega_i} v_j v_j^T + \lambda{I})^{1} \sum_{j \in \Omega_i} (A_{ij} – b_i – c_j – \mu )v_j $$
$$ v_j = ( \sum_{i \in \Omega_j} u_i u_i^T + \lambda{I})^{1} \sum_{i \in \Omega_j}(A_{ij} – b_i – c_j – \mu )u_i $$
$$ b_i = \frac{1}{1 + \lambda} \frac{1}{ \Omega_i }\sum_{j \in \Omega_i} A_{ij} – u_i^Tv_j – c_j – \mu $$
$$ c_j= \frac{1}{1 + \lambda} \frac{1}{ \Omega_j }\sum_{i \in \Omega_j} A_{ij} – u_i^Tv_j – b_i – \mu $$
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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April 2, 2016
I was aiming to get this course out before the end of March, and it is now April. So you know I put it some extra work to make it as awesome as possible.
Course summary (scroll down for coupons):
This is the 3rd part in my Data Science and Machine Learning series on Deep Learning in Python. At this point, you already know a lot about neural networks and deep learning, including not just the basics like backpropagation, but how to improve it using modern techniques like momentum and adaptive learning rates. You’ve already written deep neural networks in Theano and TensorFlow, and you know how to run code using the GPU.
This course is all about how to use deep learning for computer vision using convolutional neural networks. These are the state of the art when it comes to image classification and they beat vanilla deep networks at tasks like MNIST.
In this course we are going to up the ante and look at the StreetView House Number (SVHN) dataset – which uses larger color images at various angles – so things are going to get tougher both computationally and in terms of the difficulty of the classification task. But we will show that convolutional neural networks, or CNNs, are capable of handling the challenge!
Because convolution is such a central part of this type of neural network, we are going to go indepth on this topic. It has more applications than you might imagine, such as modeling artificial organs like the pancreas and the heart. I’m going to show you how to build convolutional filters that can be applied to audio, like the echo effect, and I’m going to show you how to build filters for image effects, like the Gaussian blur and edge detection.
We will also do some biology and talk about how convolutional neural networks have been inspired by the animal visual cortex.
After describing the architecture of a convolutional neural network, we will jump straight into code, and I will show you how to extend the deep neural networks we built last time (in part 2) with just a few new functions to turn them into CNNs. We will then test their performance and show how convolutional neural networks written in both Theano and TensorFlow can outperform the accuracy of a plain neural network on the StreetView House Number dataset.
All the materials for this course are FREE. You can download and install Python, Numpy, Scipy, Theano, and TensorFlow with simple commands shown in previous courses.
Coupons:
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Go to commentsMarch 30, 2016
Quick note to announce 2 free books out this week (3/28 – 4/1):
Deep Learning Prerequisites https://kdp.amazon.com/amazondpaction/us/bookshelf.marketplacelink/B01D7GDRQ2
Do you find deep learning difficult?
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Perhaps you’ve already tried to read some tutorials about deep learning, and were just left scratching your head because you did not understand any of it. This book is for you.
This book was designed to contain all the prerequisite information you need for my next book, Deep Learning in Python: Master Data Science and Machine Learning with Modern Neural Networks written in Python, Theano, and TensorFlow.
There are many techniques that you should be comfortable with before diving into deep learning. For example, the “backpropagation” algorithm is just gradient descent, which is the same technique that is used to solve logistic regression.
SQL for Marketers https://kdp.amazon.com/amazondpaction/us/bookshelf.marketplacelink/B01D42UBP4
Do you want to know how to optimize your sales funnel using SQL, look at the seasonal trends in your industry, and run a SQL query on Hadoop? Then join me now in my new book, SQL for marketers: Dominate data analytics, data science, and big data.
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