First published: May 6, 2018, 2:01pm -0600
Last edited: May 6, 2018, 2:01pm -0600

Multinomial Logistic Classification

This article was originally a problem I wrote for a coding competition I hosted, Vivint’s 2017 Game of Codes (now offline). The goal of this problem was not only to be a fun challenge but also to teach contestants almost everything they needed to know to build a neural network from scratch. I thought it might be neat to revive on my site! If machine learning is still scary sounding and foreign to you, you should feel much more at ease after working through this problem. I left out the details of back-propagation, and a single-layer neural network isn’t really a neural network, but in this problem you can learn how to train and run a complete model! There’s lots of maybe scary-looking math but honestly if you can multiply matrices you should be fine.

In this problem, you’re going to build and train a machine learning model… from scratch! Don’t be intimidated - it will be much easier than it sounds!

What is machine learning?

Machine learning is a broad and growing range of topics, but essentially the idea is to teach the computer how to find patterns in large amounts of data, then use those patterns to make predictions. Surprisingly, the techniques that have been developed allow computers to translate languages, drive cars, recognize cats, synthesize voice, understand your music tastes, cure diseases, and even adjust your thermostat!

You might be surprised to learn that since about 2010, the entire artificial intelligence and machine learning community has reorganized around a surprisingly small and common toolbox for all of these problems. So, let’s dive in to this toolbox!


One of the most fundamental ways of solving problems in machine learning is by recasting problems as classification problems. In other words, if you can describe a problem as data that needs labels, you can use machine learning!

Machine learning will go through a phase of training, where data and existing labels are provided to the system. As a motivating example, imagine you have a large collection of photos that either contain hot dogs or don’t. Some of your photos have already been labeled if they contain a hot dog or not, but the other photos we want to build a system that will automatically label them “hotdog” or “nothotdog.” During training, we attempt to build a model of what exactly the essence of each label is. In this case, we will run all of our existing labeled photos through the system so it can learn what makes a hot dog a hot dog.

After training, we run the unseen photos through the model and use the model to generate classifications. If you provide a new photo to your hotdog/nothotdog model, your model should be able to tell you if the photo contains a hot dog, assuming your model had a good training data set and was able to capture the core concept of what a hot dog is.

Many different types of problems can be described as classification problems. As an example, perhaps you want to predict which word comes next in a sequence. Given four input words, a classifier can label those four words as “likely the fourth word follows the last three words” or “not likely.” Alternatively, the classification label for three words could be the most likely word to follow those three.

How I learned to stop worrying and love multinomial logistic classification

Okay, let’s do the simplest thing we can think of to take input data and classify it.

Let’s imagine our data that we want to classify is a big list of values. If what we have is a 16 by 16 pixel picture, we’re going to just put all the pixels in one big row so we have 256 pixel values in a row. So we’ll say \(\mathbf{x}\) is a vector in 256 dimensions, and each dimension is the pixel value.

We have two labels, “hotdog” and “nothotdog.” Just like any other machine learning system, our system will never be 100% confident with a classification, so we will need to output confidence probabilities. The output of our system will be a two-dimensional vector, \(\mathbf{p}\). \(p_0\) will represent the probability that the input should be labeled “hotdog” and \(p_1\) will represent the probability that the input should be labeled “nothotdog.”

How do we take a vector in 256 (or \(\dim(\mathbf{x})\)) dimensions and make something in just 2 (or \(\dim(\mathbf{p})\)) dimensions? Why, matrix multiplication of course! If you have a matrix with 2 rows and 256 columns, multiplying it by a 256-dimensional vector will result in a 2-dimensional one.

Surprisingly, this is actually really close to the final construction of our classifier, but there are two problems:

  1. If one of the input \(\mathbf{x}\)s is all zeros, the output will have to be zeros. But we need one of the output dimensions to not be zero!
  2. There’s nothing guaranteeing the probabilities in the output will be non-negative and all sum to 1.

The first problem is easy, we add a bias vector \(\mathbf{b}\), turning our matrix multiplication into a standard linear equation of the form \(\mathbf{W}\cdot\mathbf{x}+\mathbf{b}=\mathbf{y}\).

The second problem can be solved by using the softmax function. For a given vector \(\mathbf{v}\), softmax is defined as:

In case the \(\sum\) scares you, \(\sum_{j=0}^{n-1}\) is basically a math “for loop.” All it’s saying is that we’re going to add together everything that comes after it (\(e^{v_j}\)) for every \(j\) value from 0 to \(n-1\).

Softmax is a neat function! The output will be a vector where the largest dimension in the input will be the closest number to 1, no dimensions will be less than zero, and all dimensions sum to 1. Here are some examples:

Unbelievably, these are all the building blocks you need for a linear model! Let’s put all the blocks together. If you already have \(\mathbf{W}\cdot\mathbf{x}+\mathbf{b}=\mathbf{y}\), your prediction \(\mathbf{p}\) can be found as \(\text{softmax}\left(\mathbf{y}\right)\). More fully, given an input \(\mathbf{x}\) and a trained model \(\left(\mathbf{W},\mathbf{b}\right)\), your prediction \(\mathbf{p}\) is:

Once again, in this context, \(p_0\) is the probability given the model that the input should be labeled “hotdog” and \(p_1\) is the probability given the model that the input should be labeled “nothotdog.”

It’s kind of amazing that all you need for good success with things even as complex as handwriting recognition is a linear model such as this one.


How do we find \(\mathbf{W}\) and \(\mathbf{b}\)? It might surprise you but we’re going to start off by guessing some random numbers and then changing them until we aren’t predicting things too badly (via a process known as gradient descent). But what does “too badly” mean?

Recall that we have data that we’ve already labeled. We already have photos labeled “hotdog” and “nothotdog” in what’s called our training set. For each photo, we’re going to take whatever our current model is (\(\mathbf{W}\) and \(\mathbf{b}\)) and find \(\mathbf{p}\). Perhaps for one photo (that really is of a hot dog) our \(\mathbf{p}\) looks like this:

This isn’t great! Our model says that the photo should be labeled “nothotdog” with 60% probability, but it is a hot dog.

We need a bit more terminology. So far, we’ve only talked about one sample, one label, and one prediction at a time, but obviously we have lots of samples, lots of labels, and lots of predictions, and we want to score how our model does not just on one sample, but on all of our training samples. Assume we have \(s\) training samples, each sample has \(d\) dimensions, and there are \(l\) labels. In the case of our 16 by 16 pixel hot dog photos, \(d = 256\) and \(l = 2\). We’ll refer to sample \(i\) as \(\mathbf{x}^{(i)}\), our prediction for sample \(i\) as \(\mathbf{p}^{(i)}\), and the correct label vector for sample \(i\) as \(\mathbf{L}^{(i)}\). \(\mathbf{L}^{(i)}\) is a vector that is all zeros except for the dimension corresponding to the correct label, where that dimension is a 1. In other words, we have \(\mathbf{W}\cdot\mathbf{x}^{(i)}+\mathbf{b} = \mathbf{p}^{(i)}\) and we want \(\mathbf{p}^{(i)}\) to be as close to \(\mathbf{L}^{(i)}\) as possible, for all \(s\) samples.

To score our model, we’re going to compute something called the average cross entropy loss. In general, loss is used to mean how off the mark a machine learning model is. While there are many ways of calculating loss, we’re going to use average cross entropy because it has some nice properties.

Here’s the definition of the average cross entropy loss across all samples:

All we need to do is find \(\mathbf{W}\) and \(\mathbf{b}\) that make this loss smallest. How do we do that?


As we said before, we will start \(\mathbf{W}\) and \(\mathbf{b}\) off with random values. For each value, choose a floating-point random number between -1 and 1.

Of course, we’ll need to correct these values given the training data, and we now have enough information to describe how we will back-propagate corrections.

The plan is to process all of the training data enough times that the loss drops to an “acceptable level.” Each time through the training data we’ll collect all of the predictions, and at the end we’ll update \(\mathbf{W}\) and \(\mathbf{b}\) with the information we’ve found.

One problem that can occur is that your model might overcorrect after each run. A simple way to limit overcorrection some is to add a “learning rate”, usually designated \(\alpha\), which is some small fraction. You get to choose the learning rate! A good default choice for \(\alpha\) is 0.1.

At the end of each run through all of the training data, here’s how you update \(\mathbf{W}\) and \(\mathbf{b}\):

Just because this syntax is starting to get out of hand, let’s refresh what each symbol means.

If you’re curious how we got these equations, we applied the chain rule to calculate partial derivatives of the total loss. It’s hairy, and this problem description is already too long!

Anyway, once you’ve updated your \(\mathbf{W}\) and \(\mathbf{b}\), you start the whole process over!

When do we stop?

Knowing when to stop is a hard problem. How low your loss goes is a function of your learning rate, how many iterations you run over your training data, and a huge number of other factors. On the flip side, if you train your model so your loss is too low, you run the risk of overfitting your model to your training data, so it won’t work well on data it hasn’t seen before.

One of the more common ways of deciding when to stop training is to have a separate validation set of samples we check our success on and stop when we stop improving. But for this problem, to keep things simple what we’re going to do is just keep track of how our loss changes and stop when the loss stops changing as much.

After the first 10 iterations, your loss will have changed 9 times (there was no change from the first time since it was the first time). Take the average of those 9 changes and stop training when your loss change is less than a hundredth the average loss change.

Tie it all together

Alright! If you’ve stuck with me this far, you’ve learned to implement a multinomial logistic classifier using gradient descent, back-propagation, and one-hot encoding. Good job!

You should now be able to write a program that takes labeled training samples, trains a model, then takes unlabeled test samples and predicts labels for them!

Your program

As input your program should take vectors of floating-point values, followed by a label. Some of the labels will be question marks. Your program should output the correct label for all of the question marks it sees. The label your program should output will always be one it has seen training examples of.

Your program will pass the tests if it labels 75% or more of the unlabeled data correctly.

Where to learn more

If you want to learn more or dive deeper into optimizing your solution, you may be interested in the first section of Udacity’s free course on Deep Learning, or Dom Luma’s tutorial on building a mini-TensorFlow.



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