Entropy, Probability and Deep Learning

Most day to day life comes down to the management of entropy - creating order from disorder. Mostly things revert to a disordered state without some human input - washing/cleaning, repairing, trimming, replacing, refueling.

Mess often comes in the form of the natural spreading out of things, and tidying is moving them back to their 'proper' place. Brushing dirt off the path, putting toys back in the box, or moving the logic in an algorithm into well defined methods and behind well defined interface points.

How this relates to entropy rather depends on the context, but broadly, you put the toys in the box so that you can (A) Find them easily next time they're needed, (B) So they're not in boxes of unrelated items causing delay searching for other things. You separate out cutlery into forks, knives, spoons, teaspoons for the same reason. It's all about efficiency. Consider the extreme alternative where everything just lies about the house with no order - want a fork? I think I saw one under the tennis shoe, but you might want to give it a thorough clean first!! OK, where's the tennis shoe? Life would be spent looking for things and cleaning them. Separating things into groups of the same thing results in much less mental effort tracking where everything is, it reduces searches from O(N) to O(log N). And when N is large that's the difference between being able to live or not.

In the software and business worlds the phrase 'technical debt' is banded about, and it amounts to the same issue - are things where they need to be to make the software development process efficient? And because software doesn't have the constraints or visibility of items in the physical world the situation is probably much worse. i.e. if your house starts to get messy it becomes unbearable and you make an effort to sort it out, but if your house was as disordered as many software projects then you wouldn't be able to live. Often efficiency is worse than O(N), perhaps O(N^2) or worse, due to dependencies that shouldn't exist, or perhaps overly complex logic that performs some simple task almost as a side effect to some incomprehensible evolutionary process.

How bad or good things are structured depends on what you want to do, i.e. whether some system is ordered sort of depends on the question - with respect to what? But in reality, if things that are similar are grouped together, that is a general concept that works for pretty much any goal or task. There might be very specific structures that work really well for specific tasks but the grouping of things is a general strategy that works well everywhere, i.e. is not awful for any specific system/task.

So then we turn to information theory, here the concept of entropy takes the form of Shannon entropy - basically the amount of information (e.g. measured in bits, or nats) required to describe some thing, such as a picture or some time series data, a video, whatever.

Consider pictures. If I create an image file where all the pixels are randomly generated, then the image has high entropy, there is no more compact way of representing it than just describing the RGB of each pixel. If however we point a camera anywhere...inside or outside, day or night, zoomed in or out, attached to a telescope or a microscope - there will be lots of structure in all of those images (apart from the ones that are just pure white/grey/black which obviously are easy to compress), because there is lots of structure in the real world. Our naive encoding by describing each pixel is now highly suboptimal. This is where GIFs, PNGs and JPEGs, etc. came in. There are some really obvious ways of describing images in more compact ways, e.g. most pixels are very similar in colour and shade to adjacent pixels. You could say that those image encodings were a form of early AI, because the encoding schemes embody something that is known about the system that produced the images (the world, the laws of physics in general, and/or the human visual system), and use that knowledge to define a compact representation.

JPEGs in particular are pretty neat. They break an image into little squares and represent each square as the composite of lots of predefined image components, e.g. a block with a light shading on one side, dark on the other. You find the component with the closest match, extract it, then find the next closest match in the 'residual', extract it, and so on. You can get a smaller file by simply limiting how many components you extract from each block, but you are left with varying amounts of residuals, i.e. JPEG artifacts (NB. This isn't quite how JPEG works, but it's close enough for this discussion). It turns out that this works really well compared to those other 'naive' encodings, and it's simple enough that it has become ubiquitous and will be hard to beat.

What's this got to do with entropy? Well, we took lots of pixels and found a compact representation for them, i.e. we found that we could represent the image with far less information than that suggested by the raw pixels. We did this with a process very similar to putting toys in a box, i.e. we grouped things that were similar. In this case the things were little square blocks of pixels with various patterns, the patterns that occur a lot are given a code so we can say... pattern #1 looks like this, so we don't have to describe the pixels in full each time we see that pattern.

It turns out that you can take this concept further to a sort of logical conclusion. E.g. JPEG breaking images into blocks seems like an unnatural and artificial approach, as does the choice of image components. Could we just learn the optimal components and the optimal 'chunks'/areas/sections of image to work on? Sure, that's basically what deep learning is doing, but the core concept is that of 'modeling the system', i.e. here is some system (e.g. a set of images), what is the most compact way of representing the entire set? What are the common features? Rather than a human observing that adjacent pixels are mostly similar and creating a trivial encoding scheme around that, we could define an automatic process for finding/discovering useful representations. And probably the broadest general model in this area is the notion of hierarchies of features i.e. images are made up of lines, that make up shaped edges, that make up objects, etc. So e.g. we might have a node or nodes in a hierarchy that represent 'ball shape' and 'wheel shape', those nodes feed down to activate the pixels to create a ball shape with the appropriate size, orientation, shading, etc., but how do we do all of those things?

The answer is that we expand the notion of a hierarchy (i.e a tree) to that of a network (a directed acyclic graph or DAG). You can think of a DAG as being lots of trees all connected to each other, so now a node can have a parent that says 'I want a ball shape', and another parent that says 'red', and so on, such that we cover all of the relevant concepts (orientation, texture, whatever).

If we define such a model and train it then we it should be able to learn all relevant concepts automatically. The relevant training metric here is entropy, or the amount of information required to describe the images being trained on. What does that mean if we have a DAG with a fixed topology (say)? Minimising entropy amounts to maximising the probability that our model will generate the images in the training set. So e.g. at the highest level, if our set contains a 50/50 split between images of cats and boats then we would expect the model to have somewhere within it a 50/50 flip between a 'cat mode' and 'boat mode'

When using gradient descent learning we can maximise the probability of generating the training image set by presenting each image in the set and following the gradient that maximises each image's probability of being generated. The key is finding models with easily computable gradients to follow. Contrastive Divergence is one method of calculating such a gradient, but is usually used in a mode that calculates a crude estimate in order to be fast enough to be practical.

In principle following a learning gradient that maximises probability if generating the training data can learn all relevant visual concepts - size, orientation, colour, shade, shiny, fluffy, animal, rock, whatever. The key is that internally each node is settling on representing one concept, and the concepts chosen are the ones that explain the most variance, are the most prominent features (principal features), that result in the model generating the training set with the highest probability, which in turn is equivalent to representing the training set with the least amount of information (see Shannon Entropy for more on the link between probability and amount of information).

Colin,

July 2015