Super high level of deep nets, skip if you already know:

Deep nets are good at taking a vector of size N and transforming it to a vector of size M, where M is perhaps a more general, abstract, or "useful" representation.

e.g.

Your N-vector might be a length 786 vector of floating point values representing black-intensity in a 28x28 grayscale image. This is the case in the MNIST dataset (lots of 28x28 images of the digits 0-9), a classic dataset in machine learning.

After a layer or two of a deep net, this N-vector might be transformed into an M-vector where each component represents some particular edge, curve, or blip within the source image.

So you've gone from the representation of "pixel 0 is gray, pixel 1 is dark gray, pixel 2 is white..." to a representation of "There is a vertical edge on the central lefthand side of the image, there is an upwards facing curve in the central top part of the image....".

It's clear that the latter representation is more compact and useful for the purpose of digit recognition.

It's also worth noting this representation is specific to the problem at hand. The edges and curves you have learned would probably be unable to accurately reproduce say, letters of the alphabet, as they are specialized to reproducing digits. The net has learned a more compact representation by using statistics to figure out that most of the information is redundant. There are only 10 possible outputs, but the input space is 256 grayscale values ^ 768 pixels.

In a traditional deep net, your output layer for this particular problem (digit recognition) might be a vector of length 10, where element of the vector is the probability of that digit being the one shown. So a result of <0.1, 0.1, 0.998, 0.0 ... 0.0> would indicate that the net thought the digit was a 2.

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Super high level of decision trees and forests, skip if you already know:

A decision tree is somewhat self-explanatory -- it's kind of like a flow chart for making judgments. Here is an example, classifying cool vs. uncool based on 3 attributes.

My dataset:

| bow_tie | socks | sandals | cool
------|---------|-------|---------|------
Alice | true | true | true | true
Bob | false | true | true | false
Carol | false | true | false | true
Doug | true | false | true | true
Ella | false | false | false | false

A possible decision tree:

if bow_tie == true
return true
else
if socks == true
if sandals == true
return false
else
return true
else
if sandals == true
return true
else
return false

It's also worth noting that decisions trees can be equal with different representation. The following will always return the same value as the above for the elements of the dataset:

if socks == true
if sandals == true
if bow_tie == true
return true
else
return false
else
return true
else
if sandals == true
return true
else
if bow_tie == true
return true
else
return false

The difference is how you pick the divisions. The first one uses a more entropy-reducing strategy -- we notice that the bow_tie division is a simple, hard rule. Bowties are cool. The second is more of a random decision decision tree, so it's less "efficient" in that it must make potentially more judgments.

Why would we ever want to be less efficient? It turns out if you train several (hundreds, thousands, etc) decision trees on subsets of the data, and then average their results together, they are alarmingly good classifiers. Extremely simple to code, train, and use. This is called a decision forest. A decision tree on its own is often weak, but decision forests are a powerful tool.

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High level of why this work is interesting:

The traditional means of training a deep neural net is with gradient descent. The most common form of this is some method of "backpropagation". You run a training example through your network, calculate the error between the result and expected result, and then propagate this error gradient back through the network to tune the transformations to produce closer to what you want. This method often requires the functions you use within the deep network to be differentiable.

As mentioned above, there are several training strategies for decision trees, but the most common is some form of "mostly random".

To extend my example from the first section, one could use the deep neural net to transform the 768 grayscale pixel values into perhaps 30 higher level edge/curve features. Then one could use this length 30 vector as the input to train a decision forest.

This might end up getting better results than either strategy by itself. You use the neural net to do the abstracting and the decision forest to make the final decision. This uses both of their advantages in tandem -- deep neural nets are great at generating more abstract and general features, and decision forests are quite good at producing accurate classifications given high-quality, lower-dimensional input data.

This idea of multi-tiered systems isn't particularly new. What this paper does, though, is introduce a differentiable decision tree. This means that they can train their decision trees with gradient descent, the same way they train the neural network. This means that, rather than training the two tiers of their system individually, they can train them together, producing even better results.