Do you ever get tired of reading walls of text, and just want a nice video or 10 to explain to you the magic of logistic regression and how to program it with Python?

This section will focus on artificial neural networks (ANNs) by building upon the logistic regression model we learned about last time. It’ll be a little shorter because we already built the foundation for some very important topics in part 1 – namely the objective / error function and gradient descent.

We will focus on 2 main functions of ANNs – the forward pass (prediction) and backpropagation (learning). Your sci-kit learn analogues would be model.predict() and model.fit().

As with logistic regression, we have some set of training samples, X1, …, Xn, and we will use gradient descent to learn the weights of our model. We then test our model by computing predicted outputs given some test inputs (the forward pass) and comparing them to the true outputs.

This topic is covered in-depth in my course, Data Science: Deep Learning in Python. We derive all the equations by hand, step-by-step, and we implement everything using Numpy and Python. To solidify the concepts, we apply the method to some real-world problems, including an e-commerce dataset and facial expression recognition.

Prediction

As with logistic regression, we will start with a diagram / schematic of a neural network.

We call the column of x’s the “input layer”, the column of z’s the “hidden layer”, and the column of y’s the “output layer”.

As in part 1, we will only use one y (binary classification) for most of the tutorial. Recall that the only difference is that when you have more than one output, you use the “softmax” output function. The methods (calculating the gradients for gradient descent) remain the same.

Each of the variables can be computed as follows:

z1 = sigma( x1*w(1,1) + x2*w(2,1) )

z2 = sigma( x1*w(1,2) + x2*w(2,2) )

y = sigma( z1*v(1) + z2*v(2) )

We can combine each of the weights w(i,j) into a matrix W – this is useful for coding in languages like Python and MATLAB where matrix and vector operations are much faster than for-loops. The size of W will be N x M where N is the number of x’s and M is the number of z’s.

Similarly, v(j) can be combined into a vector V of size M.

If we had more than one output for y, V would be a matrix of size M x P, where P is the number of y’s.

As in part 1, “sigma” refers to the sigmoid function, but other functions may be used. The hyperbolic tangent, or “tanh” is sometimes used – it is just a vertically scaled version of the sigmoid. Both make it relatively easy to compute the derivatives for gradient descent later on.

If you look at how we compute z1, z2, and y closely – you’ll recognize that these are all just the logistic regression formula. In fact, an artificial neural network is just a combination of multiple logistic regression units put together.

This is the neural network with one logistic unit highlighted:

One way we interpret this is that z1 is some “feature” extracted from (x1,x2), weighted by (w(1,1),w(1,2)), and similarly for z2.

Then y is a logistic regression on (z1,z2) – the features learned from the input.

This all begs the question – why use neural networks in the first place if we are just going to add a bunch of parameters and make it look more complicated?

Recall that logistic regression only worked on linearly separable problems. For example, you couldn’t train a logistic regression unit to learn the XOR function because you can’t draw a line between the classes.

What you could do if you really wanted to use logistic regression, is create another input x3 = x1*x2. As an exercise, convince yourself that this works. Hint: try [w0,w1,w2,w3] = [-0.5,1,1,-2].

The problem with the above approach is that you had to come up with the extra feature (x3) manually. You don’t know ahead of time what will work and what won’t. Real machine learning problems can have hundreds or thousands of inputs – you can’t try every combination possible. We haven’t even considered other functions. What about sin(x)? x^2 or x^3? log(x)? There are infinitely many features we could extract.

The beauty of neural networks is that they learn these features automatically. As an exercise, try manually assigning weights to a neural network with 3 hidden units that can compute the XOR function at y.

Another way of stating what we have just learned – artificial neural networks can learn nonlinear functions.

Learning aka. Backpropagation

Learning the weights for a neural network is very similar to logistic regression. We will follow the same method here – write out the objective function we want to minimize, calculate its derivative with respect to the parameter we want to update, and use the gradient descent algorithm to perform the weight update.

In fact, the steps remain the same:

for i = 1…number of epochs:
error = negative log-likelihood aka. -L(Y|X,W,V)
w = w - learning rate * error gradient wrt w
v = v - learning rate * error gradient wrt v

The only difference now is that the likelihood depends on W (which was 1-D for logistic regression and is now 2-D) and V – since Y depends on W and V.

Even the objective function J remains the same as with logistic regression – it only depends on the output y and the target t – and will be the squared error or cross-entropy depending on the problem.

Calculating the gradient for any v(j) is simple because y depends directly on V and by the chain rule:

Here we’ve assumed we’re using the cross-entropy error, R is the total number of training samples and we index them by r – running out of letters!

The gradient for W is a little more complicated because it involves calculating the “total derivative”. If you have more than one output y(k), k=1…P – then the objective function will depend on all the y’s. At the same time, each y(k) will depend on the same w(i,j).

In general, if you have a function f(x,y) where x(t) is a function of t and y(t) is a function of t, you can write the “total derivative” of f(x,y) as:

For a vector x with N components, the above can be generalized to:

If we replace f() with the objective function J(), t with the weight w(i,j), and each component x(i) with the outputs of the neural network y(k), k = 1…P, we get the following:

Note that we can expand the right-most derivative so that we take the derivative of y(k) with respect to z(j), multiplied by the derivative of z(j) with respect to w(i,j). The latter term does not depend on k, so it can be removed from the summation.

Although this may seem now like a straightforward application of vector calculus – don’t be fooled – it took researchers many years to figure out how to solve this problem. Read more on Wikipedia.

Multi-layer neural networks

So far we have looked at neural networks with only one hidden layer, but neural networks can have any number of hidden layers, with any number of dimensions per layer. (You will need to apply the total derivative rule recursively for each layer going backward).

You may want to do your own research as to what type of architectures will work best for your problem.

Neural networks almost give us too many choices – how many layers should I have? 1? 3? 100? How many units per layer? 500? 10000? 10001?

Of course, adding layers and units will only increase the time in takes to train your neural network. Every layer you add will result in an increase of N1 x N2 parameters to your model – where N1 is the number of inputs into the layer and N2 is the number of units in the layer that receives the inputs.

Thus neural networks can be very prone to overfitting. Suppose we are training a network with one hidden layer, where the input is a 32 x 32 image, the hidden layer has 500 units (i.e. 500 features extracted), and the output is 10 (because the images are handwritten digits from 0 to 9).

That’s 32 x 32 x 500 parameters for W, and 500 x 10 parameters for V. That’s 517 000 parameters!

One “rule of thumb” I’ve seen is that you want the number of training samples to be at least 10x the number of parameters. So for the example above, you’d want at least approximately 5.2 million samples to train from.

So you don’t want to needlessly add more layers and more units to your neural network just to make it more expressive.

This topic is covered in-depth in my course, Data Science: Deep Learning in Python. We derive all the equations by hand, step-by-step, and we implement everything using Numpy and Python. To solidify the concepts, we apply the method to some real-world problems, including an e-commerce dataset and facial expression recognition.

This is part 1/3 of a series on deep learning and deep belief networks. I’ve wanted to do this for a long time because learning about neural networks introduces a lot of useful topics and algorithms that are useful in machine learning in general.

Unfortunately, while the material I’ve read focusing on logistic regression and the multiple layer perceptron (building blocks of the deep belief network) are great and accessible to a wide audience, I’ve found most of the material I’ve encountered about deep learning are highly technical and hard to follow.

So, I’ve decided to create this series in order to teach the most practical aspects of deep learning and neural networks – enough so that you can implement one yourself, but not so much that you’ll get bogged down by all the theory.

Part 1 will focus on logistic regression. Part 2 will focus on the multilayer perceptron (a.k.a. artificial neural network) and backpropagation. Part 3 will focus on restricted Boltzmann machines and deep networks. Each is designed to be a stepping stone to the next.

The topic of this post (logistic regression) is covered in-depth in my online course, Deep Learning Prerequisites: Logistic Regression in Python. We derive all the equations step-by-step, and fully implement all the code in Python and Numpy. To solidify the concepts, we apply the method to real world datasets, including an e-commerce dataset and facial expression recognition.

Let us begin.

Logistic Regression doesn’t do Regression

Despite its name, Logistic Regression is actually a classification algorithm.

This means the output gives us a label, not a real number.

HOWEVER: the methods you read about in this series can be applied to both regression and classification. Just the equations for the outputs and the error function differ. I will note these differences where appropriate, but the tutorials will focus on classification.

Diagram of how Logistic Regression works

I’ve included a few pictures here so you get used to looking at how we visualize a neural network.

Here’s one where X (input) is 3-dimensional and Y (output) is 2-dimensional.

Here’s one where the weights use the symbol theta and the summation operation and sigmoid function are shown explicitly.

Here’s one where the weights use the variable “w” and the bias is explicitly shown as “b”. Here the sigmoid function uses the Greek letter “phi”, but more often you see the letter “sigma”.

A Little Math

So what do these diagrams mean about how we calculate the output from a set of inputs?

Notice first that we can have more than one output Y.

For K classes/labels, as in the digit recognition problem, we would have K outputs, and Y(k) = 1 if the label is the kth digit, otherwise it is 0.

The only exception is the 2-class case. In this situation, we only need 1 output because Y = 1 is the first class and Y = 0 is the second class.

We’ll focus on this scenario first.

The equation in its compact form is this:

The inside part is the dot product of the weights and the input:

As in linear regression we assume there is an x0 and that it is 1.

The “sigma” part is the sigmoid function:

If we graph the sigmoid, it looks like this:

There are 2 things we can tell from the above equation:

1) For logistic regression to work, the classes must be linearly separable. This is because the dot product between “w” and “x” is a line/plane.

(i.e. ax + by + c = 0)

w0 + w1x1 + w2x2 + … = 0 is the plane (more correctly, hyperplane) here.

So here is a situation where logistic regression would work well:

Here is a situation where it wouldn’t work well.

But we will cover that more in parts 2 and 3.

2) The sigmoid means the output Y is between 0 and 1.

So if w*x = 0, we land right on the hyperplane, and Y = 0.5.

If w*x > 0, we get Y > 0.5, and vice versa for w*x < 0.

As w*x approaches infinity, Y approaches 1, and vice versa.

Probabilistic Interpretation

Because Y is between 0 and 1, we can interpret it as a probability.

This makes more sense if you consider the following:

If we fall right on the barrier/plane between the two classes – our probability of being in either class is 0.5.

If we are further away from that barrier, the probability of being in either class increases.

We usually denote Y as P(Y=1|X) and P(Y=0|X).

Note that while we use some probabilistic concepts here, the way in which we use them is different than for say, a Bayesian classifier.

Also note that P(Y=0|X) = 1 – P(Y=1|X).

Maximizing the Likelihood

We have seen squared error used as an error function before, as with linear regression.

In fact, if we were doing regression, we could use the same thing here.

For classification, we take a different approach.

You may have seen this error function before:

t is the target and y is the output of the network/model. (This introduces some ambiguity because we usually write p(y=1 | x) as the output and y as the target).

This is called the cross-entropy error.

Where does this come from?

Let us go back to first principles.

Instead of minimizing error, we maximize likelihood. This seems like a logical place to start – maximizing the probability that our model parameters are correct.

Consider N IID (independent and identically distributed) training samples and corresponding labels (we’ll call them “t” here).

The likelihood of the model given the entire dataset can be represented by this equation. We can use the product rule because each sample is independent.

(Sidenote 1: This is the same thing we do when we want to say, find the maximum likelihood estimate for the mean. We calculate the joint probability aka. likelihood P(data|mean) and find the “argmax” mean that gives us the highest likelihood, hence the term – “maximum likelihood”)

(Sidenote 2: This is the same likelihood you see when we do Bayesian inference – posterior ~ likelihood x prior or P(param | data) ~ P(data | param) P(param))

(Sidenote 3: If you wanted to do regression, you would simply not have a sigmoid at the end, and you would use the squared error. The exponential of the squared error is a Gaussian, because in regression we often assume the error is Gaussian distributed. By making these 2 changes, we would just be doing linear regression.)

Recall y = P(y=1|x).

The target t can be 1 or 0.

When t is 1, only the left part of the product matters (the right side evaluates to 1). All the y’s here are the probability that the output of the network is 1. Given that the target is 1, we want to maximize this probability.

When t is 0, only the right part of the product matters. Recall that 1-y is the probability that the output of the network is 0. So when t = 0, we want to maximize this probability.

Since each sample is independent, we can get the joint probability by multiplying all the individual probabilities together.

2 key points:

1) There is no analytic solution, we must use iterative methods. In this tutorial we will cover gradient descent, but there are others (such as conjugate gradient, and L-BFGS).

The added advantage of learning gradient descent now is that it is also used to train neural networks.

2) As is usual with these ML problems, we will work with the log likelihood instead of the likelihood. Just try taking the derivative of both, and you will see why.

If you take the log of the above expression, notice you’ll get the same error function we started with!

We take the negative because we want something to minimize. We call this the “error” or “cost” function.

Maximizing the likelihood is equivalent to minimizing the negative likelihood.

It is also equivalent to minimizing the negative log-likelihood. This is because log() is a monotonically increasing function.

Gradient Descent

How do we actually minimize the negative log-likelihood if we can’t simply set the derivative = 0 and solve for the weights?

This is where gradient descent enters the picture.

Note that gradient descent is just a numerical method – it can be applied whenever you want to solve for the minima of a function, not just for machine learning.

Here is a picture of what we’re trying to do:

We start at some random weight, w = random().

Then we update the weight by going in the direction of the derivative of the error function (slope), which we have previously stated is the negative log-likelihood.

With squared error it is easy to see that the error function is quadratic, and so we are descending down a parabola in that case. The minimum is global.

With log-likelihood the extremum is also global.

It may help to plot the function E(y,t) = tlog(y) + (1-t)log(1-y) to see why.

The equation for updating the weights is:

Here j indexes the dimension, so j = 1…D.

t indexes the iteration number (not to be confused with the other t, which was the target).

“Eta” is called the “learning rate”. This hyperparameter determines how far along the error surface we travel on each iteration. Bigger values mean we go further, which means our weights might converge to the final solution faster, but it also means we may “overshoot” that solution.

Since w is a vector, we can usually speed up our code by doing vector operations (i.e. in MATLAB or Python). In this case, we can use this equation:

The full training algorithm is:

for i = 1…number of epochs:
error = negative log-likelihood ( -L(Y|X,w) )
w = w – learning rate * error gradient

The number of epochs is yet another hyperparameter. There are many ways to determine when to stop the gradient descent process.

Some other methods you may want to look into:

Stopping when the gradient is small enough

Stopping when the training error is no longer decreasing or approaching 0

Stopping when the error on a held out test set starts to increase (overfitting)

We call things like learning rate and epochs “hyperparameters”. These are parameters that are not part of the model itself, but can still be optimized, perhaps via cross-validation.

Biological Inspiration

In computational neuroscience, a logistic regression unit is sometimes referred to as a “neuron”. How are the two related?

Here is a diagram of a typical neuron.

Some notable components:

Dendrites: These are the “inputs” into the neuron – they take electrical signals from other neurons’ axons.

Cell body / Nucleus: This part of the neuron “sums up” all the inputs and propagates this summed signal to the axon.

Axon: This is the “output” of the neuron. It sends the signal from this neuron to other neurons’ dendrites.

So dendrites are our logistic unit’s X, and axons are the Y.

The brain is essentially a network of neurons, or rather, a neural network. An artificial neuron network, which is the topic discussed in Part 2 of this tutorial, is a network of connected logistic regression units.

Another notable feature of neurons is the behavior of the “action potential”.

Observe a typical amplitude/potential (voltage) vs. time signal:

Notice how the potential rises gradually and then spikes. We call this the “all-or-nothing” principle. If the sum of the inputs to the neuron is high enough, a spike is generated. Otherwise, the voltage stays relatively low.

This is reflected in the logistic units’ binary output. The output if a sigmoid is interpreted as P(Y=1|X) – the probability of being “on”, or in other words, the probability that a spike is generated.

Inhibitory vs. Excitatory neurons:

It is well-known that the signal a neuron sends can either “excite” or “inhibit” the receiving neuron. These are reflected in the logistic model by the weights. A positive weight is excitatory. A negative weight is inhibitory.

Researchers have tried to create models with “spiking” neurons, however, it has been difficult to get them to actually learn anything.

Where can I learn more?

The topic of this post (logistic regression) is covered in-depth in my online course, Deep Learning Prerequisites: Logistic Regression in Python. We derive all the equations step-by-step, and fully implement all the code in Python and Numpy. To solidify the concepts, we apply the method to real world datasets, including an e-commerce dataset and facial expression recognition.