What does a neural network actually do?
moalquraishi.wordpress.com
moalquraishi.wordpress.com
This is a fundamental Question: Can we really say and predict what a neural network does?
Contrary to an engineered / constructed algorithm a neural network is 'trained'.
Whenever we will present a 'known' input pattern it will reposnd with a 'learned' response.
This however introduces interesting problems: How can we _debug_ a neural network? How can we debug a correlation? Sure we can tune its paramters, we can train it some more to again show the desired response. But reaching this point we just abandoned knowing how the intrinsic algorithm works in favor to just focussing on the result.
Okay - now if we follow this argument - this would lead to: If we simulate the whole brain by simulating the neural network, we won't gain any knowlege about the intrinsic workings of the brain. We won't find any enlightenment about the innermost algorithm _represented_ by the neural network we call our brain.
An example of doing facial reconstruction: http://deeplearning4j.org/facial-reconstruction-tutorial.htm...
To debug a program you actually verify that it's logic is correct (at least the good kind of debugging).
Consider a spectrum:
1. Natural language - we humans combine fragments of natural language easily and on an ad-hoc basis. We can get a fairly amount of use from reusing Shakespeare quotes and neologisms while spending rather little effort.
2. Trained programmers can reuse and combine general purpose libraries - with difficulty and often after considerable debugging.
3. AI algorithms like Neural Networks. These are just plopped in and tweaked, not combining seems possible.
It seem like "intelligent behavior" should be going more towards #1 but the process of Machine Learning seems to move things more towards #3. The "learn once, understand never" approach means that for each significant case, you'll need to do a re-tweaking and re-learning. The potential to get harder rather than easier over time might well be there.
Is a human brain intelligent? I believe so.
Admittedly, all this is in manner of speaking but still, I would claim that most if not all the times you debug a program you are also debugging your mental concept of what the program does. By that fact that we can change our concepts, our minds are very "debuggable."
Neural networks, Support Vector Machines, Hidden Markov Models and other stuff (Markov Networks, etc) do something like linear regression on some huge space - they draw a curve/plane between groups of things on this feature space. The tendency is for this division to make sense and to correspond to our common sense categorization of these things.
The problem is that once that happens, you can't really reason about the division you've drawn. It's just there. You can tweak for various purposes but that's a manual process.
You categorize animals by shape or particular adaptation or by genetic makeup. You can teach one of these algorithm each of these categorizations. But you can't do something like have the thing categorize for one purpose and then tell it to "change it's outlook" and categorize for a different purpose.
In this sense, despite seeming impressive, the products of these processes are dead-ends that we can't reason about, that lack the flexible intelligence of a human being.
In other words, the incomprehensibility of a modern AI model is not a failing of AI, it is a failing of (AI) psychology and (AI) neuroscience.
The artificially constructed intelligence works whether or not we understand how. The frontier of AI science is now open to AI Psychology. Psychologists and Neuroscientists will replace the data scientists! Such fun!
The artificially constructed intelligence works whether or not we understand how
The operation of human intelligence is very dependent on the fact that we human have operational understanding of each other's mental processes. Moreover, it is well known that human beings process language and can re-evaluate past experience in light of present understand.
Oppositely, we know enough about the properties of the various non-linear recognizers to know that constructs can't do and won't ever be able to do these things.
I use the word "traditional" to describe the NN architecture discussed in the article. Recent NN research has been promising [1], but this article strictly discusses traditional NN's. I don't really have much experience with the newer NN algorithms, so I'm not sure to what extent they suffer from the same problems as traditional NN's.
[1] http://en.wikipedia.org/wiki/Neural_network#Recent_improveme...
[1] http://arxiv.org/pdf/1207.0580.pdf [2] http://cs.nyu.edu/~wanli/dropc/
The fuzzing creates a very similar effect to convolutional nets where it can learn different poses of an image.
For me ANN's always seem to get stuck on not very helpful local minima— they're not one of the first tools in my bags of tricks by far.
Often I associate them as being the sort of thing that someone who doesn't really know what they're talking about talks about. (Esp. if its clear that in their minds NN have magical powers. :) maybe they'll also mention something about "genetic algorithms")
And correct me if I'm wrong, but isn't the cost function for a feed forward neural networks that uses a sigmoid activation function convex wrt the parameters being trained, i.e. gradient descent is guaranteed to find the global minimum when small enough of a step size is used?
Take a simple classifier network with one input, one hidden unit and one output and no biases. To make things even simpler, tie the two weights, i.e. make the first weight equal to the second. Now, mathematically the output of the network can be written: z=f(w * f(w * x)) where f() is the sigmoid.
Next, consider a dataset with two items: [(x_1, y_1), (x_2, y_2)] where x_i is the input and y_i is the class label, 0 or 1. Take as values: [(0.9, 1), (0.1,0)]. The cost function (loglikelihood in this case) is:
L(w) = sum_i { y_i * log( f(w * f(w * x_i)) ) + (1-y_i) * log( 1-f(w * f(w * x_i)) ) }
or
L(w) = log( f(w * f(w * 0.9)) ) + log( 1-f(w * f(w * 0.1)) )
Plot that last guy replacing f with the sigmoid, and you'll see the result is non-convex - there's a kink near zero.
If it models the function over the input domain, then it is properly trained. If it is trained to a local minima then it doesn't model the underlying function well over the whole input domain. If you have good/representative training and validation sets you will be able to tell.
> Esp. if its clear that in their minds NN have magical powers
I know that type. When dealing with ANN's you realize quickly (just like in all data science) that all of the "magic" relies on the manual work and thought that goes into washing and adapting the data. Not very sexy work, and work that requires a fair bit of knowledge about the problem domain.
> For me ANN's always seem to get stuck on not very helpful local minima
That isn't the ANN that gets stuck, it's the training algorithm (using gradient descent) that gets stuck :) Training is orthogonal to the operation of the network itself (which is just a nonlinear function in the end!). Gradient descent via error backpropagation is the most common training method for MLP's, but you could imagine doing a random/brute force algorithm that is significantly simpler to implement, but slower. Since a network is often trained once and then used repeatedly, it is often plausible to train it for several weeks if needed! A pure random search is usually not feasible, but adding randomization to a gradient descent will help. There are many ways to avoid local minima for a gradient desccent, if you have time to wait.
> maybe they'll also mention something about "genetic algorithms"
The simple error backpropagation methods only work well for normal feed-forward networks. Other topologies e.g. recurrent networks require more exotic methods. In my (limited) experience genetic algorithms are rarely efficient as a training method though.
Coding a digit recognizer using a neural network is an extremely rewarding exercise and there's a lot of help on the web to get you started.
Perhaps a simpler way to look at it is to understand that a feed-forward ANN is basically just a really fancy transformation matrix.
OK, so unless you know linear algebra, you're probably now asking what's a transformation matrix? Without the full explanation, the important understanding is why they are so important in 3D graphics: they can perform essentially arbitrary operations (translation, rotation, scaling) on points/vectors. Once you have set up your matrix, it will dutifully perform the same transformations on every point/vector you give it. In graphics programming, we use 4x4 matrices to perform these transformations on 3D points (vertices) but the same principle works in any number of dimensions - you just need a matrix that is one bigger than the number of dimensions in your data*.
Edit: For NNs the matrices don't always have to be square. For instance you might want your output space to have far fewer dimensions that your input. If you want a simple yes/no decision then your output space is one-dimensional. The only reason the matrices are square in 3D graphics is because the vertices are always 3-dimensional.
What a neural network does is take a bunch of "points" (the input data) in some arbitrary, high number of dimensions and performs the same transformation on all of them, so as to distort that space. The reason it does this is so that the points go from being some complex intertwining that might appear random or intractable, into something where the points are linearly separable: i.e., we can now draw a series of planes in between the data that segments it into the classifications we care about.
The only difference between a transformation matrix and a neural network is that a neural network has at least two layers. In other words, it is two (or more) transformation matrices bolted together. For reasons that are a bit too complex to get into here, allows an NN to perform more complex transformations than a single matrix can. In fact, it turns out that an arbitrarily large NN can perform any polynomial-based transformation on the data.
The reason this is often seen as somewhat magic is that although you can tell what transformations a neural network is doing in trivial cases, NNs are generally used where the number of dimensions is so large that reasoning about what it is doing is difficult. Different training methods can give wildly different networks that seemingly give much the same results, or fairly similar networks that give wildly different results. How easy it is to understand the various convolutions that are taking place rather depends on what the input data represents. In the case of computer vision it can be quite easy to visualise the features that each neuron in the hidden layer is looking for. In cases where the data is more arbitrary, it can be much harder to reason about, so if your training algorithm isn't performing as you'd like, it can be difficult to understand why it isn't working, even if you already understand that the basic principle of a feed-forward network is just a bunch of simple algebra.
It's good to remember the ANN's input offset comes as vector data. The ANN isn't directly transforming those vectors directly, rather it is transforms these input to a higher dimensional "feature" space and performs the linear transform. If you take the separating plane that's drawn in the feature space and reverse the map, you'll the ANN has drawn complex surface between the points it want to recognize and those it rejects.
So it's basically a heuristic and no more intelligent than Taylor's series.
Nice explanation. I need one clarification, though. Isn't matrix multiplication associative? Isn't thus any transformation defined by two matrices representable by a single matrix that is the product of the two matrices?
I am probably misunderstanding how NNs bolt matrices together.
I suspect some kind of transformation could be used to make a two level NN into a one level one. The thing is the resulting one level network might be more complex and less useful than the original two level network. Still, I think this does illustrate the limitations of multilevel networks.
Another way to see this is to notice that NNs and SVMs[1] are (approximately or exactly) equivalent [2] because they both involve the fairly simple linear and non-linear transformations we've been looking at.
[1] http://en.wikipedia.org/wiki/Support_vector_machine [2] http://www.staff.ncl.ac.uk/peter.andras/PAnpl2002.pdf
- define an input vector space (i.e choose dimensions you want to operate on with input data)
- define your categories in another space (or another basis in the same space?)
- set up a transformation pipeline between the two spaces (with at least two stages)
- devise an algorithm that takes categorised elements and produces new transformation matrices
- train the NN (i.e feed input and categorise the result so that through some algorithm the transformation matrices converge)