Our first competition, cont. (Wrapping it all up - actually)

So this is our final post on the handwriting recognition competition! It was swell, but now it’s time to say goodbye...


In summary: out of the four algorithms we tried, k-nearest neighbors was the most accurate (95%), followed by our neural network (93%). Our logistic classifier was the next most accurate (81%), followed closely by our linear average classifier (80%). We tried parameter optimization for the logistic classifier and the neural network, and received 0.2% and 2% accuracy boosts, respectively.


Did we try all we could have tried? No. If we were really serious about the performance of our algorithms, there’s a lot we could have investigated - better optimizations, performance enhancements, and more advanced techniques. For us, the purpose of this competition was to introduce us to machine learning, and this was definitely accomplished. We implemented 4 different algorithms from scratch, learned how to test algorithmic performance, and even tackled some machine learning system design with parameter optimization. I’d say our goal was more than accomplished. For our next project, we’ll be sure to try much more advanced techniques, think more carefully about system design, and try to test out any hypothesis we may have for improving our system.


Here are some major things we learned along the way:

• More complicated does not mean better. We saw this with the k-nearest neighbor classifier - which was conceptually simple and easy to implement. It ended up outperforming all our other algorithms, despite other algorithms being more complex (especially neural networks).

• The 80-20 rule really applies. We coded our first algorithm in under an hour, and right away that gave us 80% accuracy. The rest of our time was spent trying to improve on this. When we tried optimizing parameters, we didn’t get much improvement either. However, there’s a huge difference between 95% accuracy and 80% accuracy. Consider how the USPS uses handwriting recognition to read addresses off of letters. Imagine if their classifier only had 80% accuracy - that would be a lot of letters delivered to wrong addresses. Even if it had 99% accuracy - if they deliver millions of letters a day that’s tens of thousands of letters that get incorrectly sent. If precision matters, every tenth or hundredth of a percent counts.

• Visualizing results is helpful. When dealing with big data, little bugs can go unnoticed, especially if they don’t seem to have much effect on the system. Sometimes the best solution is to slow down and watch as the computer trains or tests each sample individually. With digit recognition, it helped a lot to display the picture of each training or test sample as the algorithm classified it.

If we were to make our recognizer more accurate, here’s some things we would try:

• Optimize better. We could try increasing the number of different values we test over for each parameter, or we could try optimizing over different parameters.

• Image preprocessing. We tried a bit of this at the end, but without much success. To decrease the variation between samples, we could rotate each sample to be facing the same direction, remove any border on the samples, etc. Less variation means more accurate detection.

• Introduce artificial training samples. More training samples means more accuracy. Often this will produce better results than trying to improve the algorithm. Though we used all the Kaggle training samples, we could have made some of our own. One way to do this is to make modified copies of the existing samples. For example, it is possible that some digits had thicker line widths than others. If we take each sample, make 3 copies of it with varying line widths and use all of these samples for training, we could multiply the our training sample size by 4. All of the artificial examples would be plausible as real examples, so this could increase our accuracy.

Well, that’s all we have on handwriting recognition. Next up: identifying dark matter halos!!!
(And in case you’re wondering, we are currently placed 429th for the handwriting recognition competition. It’s a popular one!)

Our first competition, cont. (Part 4: Neural Networks)

At this point we’ve attempted fairly simple machine learning algorithms. Time to bring out the big guns...

Neural networks are one of the most commonly used algorithms for handwriting recognition. The basic idea behind a neural network is actually fairly simple (though the implementation is fairly difficult).

Backtracking for a second: with the logistic regression, we had input variables (one for each pixel) and output variables (one for each digit 0-9). In other words, we had a layer of input variables and a layer of output variables.

Example logistic classifier

Neural networks are similar, but with a bunch of other stuff thrown in the middle. In between the input and output layers are 1 or more hidden layers, which do internal processing of the data. For a given layer, each node is connected with all the nodes in adjacent layers. The network is supposed to be similar to actual neurons in the brain.

Example neural network

Every connection in the neural network has a weight associated with it. Each layer is learned in succession, starting off with the first hidden layer and ending with the output layer. The value of a node is determined by the weighted sum of all the nodes in the previous layer, which is then run through the logistic function. It ends up being like a bunch of logistic classifiers chained together.

Similarly to the logistic regression, training the neural network consists of finding the weights that minimize a cost function. The cost function is minimized with an iterative algorithm that requires knowing the cost function gradient (where each weight is a variable). Finding the gradient requires an algorithm for this is known as back propagation, which is an incredibly complicated and unintuitive process. The implementation of it is very prone to errors, so this was the cause of much stress.

To start off, we created a neural network consisting of three layers. The first layer - the input layer - had 784 neurons, one to represent each of the 28x28 pixels in the image. The final layer - the output layer - had 10 neurons, one to represent each of the 10 digits that were being read in. This leaves the middle layer, or the hidden layer. To our surprise, the number of neurons in this layer don’t actually represent anything. The Coursera course - and other resources from a quick Google search - had some black magic rules for determining the size of this hidden layer. For starters, we used a 20 neuron hidden layer, and later we tried to find an optimal size. (Also, per suggestion of the coursera course, every layer except the output layer included a biasing neuron. This neuron, for each layer, always had a value of 1. It was connected to every neuron in subsequent layers, but there were no connections to it from the previous layer.)

We used all 30k training samples when setting up the neural network. Because the network is so complex, training takes a long time (an hour or two). Overall, our network ended up making predictions on the Kaggle test set with 91% accuracy. Not as high as we were hoping, but not bad either!

Since neural nets are supposed to be very powerful, we knew there had to be a way to improve its accuracy – and again we turned to parameter optimization. In particular, we tried varying the number of nodes in our hidden layer and increasing the number of hidden layers. We tried 100, 150, and 200 hidden layer nodes for networks with one hidden layer and two hidden layers. (In a perfect world we would have liked to try more values of the parameters. However, when it takes up to 2 hours to train a configuration, 6 different configurations is already pushing the limits of our computers). The conventional wisdom with neural nets is that bigger is better – so we predicted that more hidden layers with more nodes would improve our accuracy. However, sometimes this heuristic doesn’t hold, so we wanted to be sure we weren’t overshooting. We trained each of these configurations with 24k samples, leaving 6k samples for cross-validation. Results from cross-validation for each configuration are plotted below.

Cross-validation results from parameter optimization

As far as hidden layer size goes, bigger does mean better (for the most part). However, what is surprising is that the 2 hidden layer network was less accurate than the 1 layer network. One possible explanation for this is that the cost function minimization algorithm did not run enough iterations. Because there are a lot more parameters to optimize for a 2 hidden layer network, it would make sense that it should take longer to arrive at a true minimum. Thus the parameters that the 2 hidden layer network were trained with may not have been optimal.

Using the most accurate network configuration (1 hidden layer with 200 nodes), we obtained 93% accuracy with the Kaggle test set. That’s a 2 percent accuracy boost from our first neural net attempt. It’s a bit sad that the simple k-nearest neighbor algorithm worked better than this more complex one, but perhaps neural networks would be better if we did further optimizations to increase accuracy. There are other parameters we could choose to look at for optimization – such as the regularization parameter. We could also run cost function minimization for more iterations to ensure that our weights have converged on their optimal values (especially useful for the 2 hidden layer network). Nevertheless, the simplest way to improve accuracy would be to have more training samples. 30k is definitely a lot, but neural networks perform even better when there’s on the order of 100k samples.

Check out our neural network implementation here: https://github.com/rachelbobbins/machinelearning/tree/master/neural_network

Well, that covers all the handwriting recognition algorithms we tried. In our next entry we’ll recap and talk about future steps. Then it’s onto our next project!

Our first competition, cont. (Part 3: k-Nearest Neighbors)

This algorithm was a last-ditch* attempt to break the 90% success rate as we wrapped up our effort on the digit recognition challenge. The kNN benchmark on Kaggle had a 96.667% success rate, and it used an algorithm not covered in Coursera. So, doing my own implementation seemed like a) a way to learn about a new algorithm and b) a likely way of beating the 90% mark.

The k-Nearest Neighbor (kNN) algorithm is surprisingly simple for such an accurate classifier. Given a a test sample "t", it:

1. Determines the euclidean distance between t and every individual training example.
2. Finds the k closest training examples, and their known values/labels (in this case, 0-9)
3. Classifies "t" as the label of its mode of the k nearest neighbors.

So, for sample "t" and k=10, let's say the nearest training examples have labels [1, 1, 2, 1, 1, 7, 1, 1, 0, 2]. The algorithm then labels "t" as 1.

We ran the algorithm with an arbitrarily chosen k=15. It took forever (~16 hours) to run on my MacBook Air, with an Intel i5 processor and 4 GB of RAM. It was 95.8% accurate. This was slightly less accurate than Kaggle's success rate, and probably could have been attributed to the fact that they used a different value of k (10).

Curious about how k's value would impact the algorithm's accuracy, I wrote an optimization function that would try values of k from 1-20, and return the value of k which gave the highest accuracy. It split the training data into two groups - one for training, and one for cross validation. It used 28k training examples to predict the label for 2k training examples, then compared its predictions to the actual labels. Surprisingly, different values of k did not impact the algorithm's accuracy - it generally achieved between 95 and 97% accuracy. Because the kNN algorithm is so time-intensive, I did not run it again using the best k.

(*Chronologically speaking, we did this after working on a neural networks approach... but that post hasn't been published yet)