Would someone be able to explain to me or point me to some resources of why (or situations where) more than one hidden layer would be necessary or useful in a neural network?
Basically more layers allow more functions to be represented. The standard book for AI courses, "Artificial Intelligence, A Modern Approach" by Russell and Norvig, goes into some detail of why multiple layers matter in Chapter 20.
One important point is that with a sufficiently large single hidden layer, you can represent every continuous function, but you will need at least 2 layers to be able to represent every discontinuous function.
In practice, though, a single layer is enough at least 99% of the time.
That's more similar to the way the brain works (which might not necessarily be a computational advantage, but a lot of people are researching NN to gain insight about the way the mind works, rather than to solve real world problems.
Its easier to achieve some kinds of invariance using more layers. For example, an image classifier that works regardless of where in the image the object is found, or the object's size. see Bouvrie, J. , L. Rosasco, and T. Poggio. "On Invariance in Hierarchical Models". Advances in Neural Information Processing Systems (NIPS) 22, 2009.
Each layer effectively raises the potential "complexity" of adaptation in an exponential fashion (as opposed to a multiplicative fashion of adding more nodes to a single layer).
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I have a dataset that overlaps a lot. So far my results with SVM are not good. Do you have any recomendations for a model that may be able to differ between these 2 datasets?
Scatter plot from both classes
It is easy to fit the dataset by interpolation of one of the classes and predicting the other one otherwise. The problem with this approach is though, that it will not generalize well. The question you have to ask yourself is, if you can predict the class of a point given its attributes. If not then every ML algorithm will also fail to do so.
Then the only reasonable thing you can do is to collect more data and more attributes for every point. Maybe by adding a third dimension you can seperate the data more easily.
If the data is overlapping so much, both should be of the same class, but we know they are not. So, there is/are some feature(s) or variable(s) that is/are separating these data points into two classes. Try to add more features for data.
And sometimes, just transforming the data into a different scale can help.
Both the classes need not be equally distributed, as skewed data distribution can be handled separately.
First of all, what is your criterion for "good results"? What style of SVM did you use? Simple linear will certainly fail for most concepts of "good", but a seriously convoluted Gaussian kernel might dredge something out of the handfuls of contiguous points in the upper regions of the plot.
I suggest that you run some basic statistics on the data you've presented, to see whether they're actually as separable as you'd want. I suggest a T-test for starters.
If you have other dimensions, I strongly recommend that you use them. Start with the greatest amount of input you can handle, and reduce from there (principal component analysis). Until we know the full shape and distribution of the data, there's not much hope of identifying a useful algorithm.
That said, I'll make a pre-emptive suggestion that you look into spectral clustering algorithms when you add the other dimensions. Some are good with density, some with connectivity, while others key on gaps.
I know how the algorithm works, but I'm not sure how it determines the clusters. Based on images I guess that it sees all the neurons that are connected by edges as one cluster. So that you might have two clusters of two groups of neurons each all connected. But is that really it?
I also wonder.. is GNG really a neural network? It doesn't have a propagation function or an activation function or weighted edges.. isn't it just a graph? I guess that depends on personal opinion a bit but I would like to hear them.
UPDATE:
This thesis www.booru.net/download/MasterThesisProj.pdf deals with GNG-clustering and on page 11 you can see an example of what looks like clusters of connected neurons. But then I'm also confused by the number of iterations. Let's say I have 500 data points to cluster. Once I put them all in, do I remove them and add them again to adapt die existing network? And how often do I do that?
I mean.. I have to re-add them at some point.. when adding a new neuron r, between two old neurons u and v then some data points formerly belonging to u should now belong to r because it's closer. But the algorithm does not contain changing the assignment of these data points. And even if I remove them after one iteration and add them all again, then the false assignment of the points for the rest of that first iteration changes the processing of the network doesn't it?
NG and GNG are a form of self-organizing maps (SOM), which are also referred to as "Kohonen neural networks".
These are based on older, much wider view of neutal networks when they were still inspired by nature rather than being driven by GPU capabilites of matrix operations. Back then, when you did not yet have massive-SIMD architectures yet, there was nothing bad about having neurons self-organize rather than being preorganized in strict layers.
I would not call them clustering although that term is commonly (ab-) used in related work. Because I don't see any strong propery of these "clusters".
SOMs are literally maps as in geography. A SOM is a set of nodes ("neurons") usually arranged in a 2d rectangular or hexagonal grid. (=the map). The positions in the input space are then optimized iteratively to fit the data. Because they influence their neighbors, they cannot move freely. Think of wrapping a net around a tree; the knots of the net are your neurons. NG and GNG appear to be pretty mich the same thing, but with a more flexible structure of nodes. But actually a nice property of SOMs is the 2d map that you can get.
The only approach I remember for clustering was to project the input data to the discrete 2d space of the SOM grid, then run k-means on this projection. It will probably work okayish (as in: it will perform similar to k-means), but I'm not convinced that it's theoretically well supported.
Are there known methods of continuous training and graceful degradation of a neural net while it shrinks or grows in size (by number of nodes, connections, whatever)?
To the best of my memory, everything I've read about neural networks is from a static perspective. You define the net and then train it.
If there is some neural network X with N nodes (neurons, whatever), is it possible to train the network (X) so that while N increases or decreases, the network is still useful and capable of performing?
In general, changing network architecture (adding new layers, adding more neurons into existing layers) once the network was already trained makes sense and a rather common operation in Deep Learning domain. One example is the dropout - during training half of the neurons randomly get switched off completely and only remaining half participates in training during specific iteration (each iteration or 'epoch' as it often is named has different random list of switched off neurons). Another example is transfer learning - where you learn network on one set of input data, cut off part of the outcoming layers, replace them with new layers and re-learn the model on another dataset.
To better explain why it makes sense lets step back for a moment. In deep networks, where you have lots of hidden layers each layer learns some abstraction from the incoming data. Each additional layer uses abstract representations learned by previous layer and builds upon them, combining such abstraction to form a higher level of the data representation. For instance, you could be trying to classify the images with DNN. First layer will learn rather simple concepts from images - like edges or points in data. Next layer could combine this simple concepts to learn primitives - like triangles or circles of squares. Next layer could drive it further and combine this primitives to represent some objects which you could find in images, like 'a car' or 'a house'and using softmax it calculates the probabilities of the answer you are looking for (what to actually output). I need to mention that these facts and learned representations could be actually checked. You could visualize the activation of your hidden layer and see what it learned. For example this was done with google's project 'inceptionism'. With that in mind let's get back to what I mentioned earlier.
Dropout is used to improve generalization of the network. It forces each neuron to 'not be so sure' that some pieces of the information from the previous layer will be available and makes it to try to learn the representations relying on less favorable and informative pieces of abstractions from previous layer. It forces it to consider all of the representations from previous layer to make decisions instead of putting all of its weight into couple of neurons it 'likes most of all'. By doing this the network is usually better prepared to new data where the input will be different from the training set.
Q: "As far as you're aware is the quality of the stored knowledge (whatever training has done to the net) still usable following the dropout? Maybe random halves could be substituted by random 10ths with a single 10th dropping, that might result in less knowledge loss during the transition period."
A: Unfortunately I can't properly answer why precisely half of the neurons is switched off and not 10% (or any other number). Maybe there is an explanation but I haven't seen it. In general it just works and that's it.
Also I need to mention that the task of dropout is to ensure that each neuron doesn't consider just several of the neurons from previous layer and is ready to make some decision even if neurons which usually helped it to make correct decision are not available. This is used for generalization only and helps the network to better cope with the data it haven't seen previously, nothing else is achieved with a dropout.
Now let's consider Transfer Learning again. Consider that you have a network with 4 layers. You train it to recognize specific objects in pictures (cat, dog, table, car etc). Than you cut off last layer, replace it with three additional layers and now you train the resulting 6-layered network on a dataset which, for instance, wrights short sentences about what is shown on this image ('a cat is on the car', 'house with windows and tree nearby' etc). What we did with such operation? Our original 4-layer network was capable to understand if some specific object is in the image we feed it with. Its first 3 layers learned good representations of the images - first layer learned about possible edges or points or some extremely primitive geometric shapes in images. Second layer learned some more elaborate geometric figures like 'circle' or 'square'. Last layer knows how to combine them to form some higher level objects - 'car', 'cat', 'house'. Now, we could just re-use this good representation which we learned in different domain and just add several more layers. Each of them will use abstractions from last (3rd) layer of original network and learn how combine them to create meaningful descriptions of images. While you will perform learning on new dataset with images as input and sentences as output it will adjust first 3 layers which we got from original network but these adjustments will be mostly minor, while 3 new layers will be adjusted by learning significantly. What we achieve with transfer learning is:
1) We can learn a much better data representations. We could create a network which is very good at specific task and than build upon that network to perform something different.
2) We can save training time - first layers of network will already be trained well enough so that your layers which are closer to output already get a rather good data representations. So the training should finish much faster using pre-trained first layers.
So the bottom line is that pre-training some network and than re-using part or whole network in another network makes perfect sense and is not something uncommon.
This is something I have seen in the likes of this video...
https://youtu.be/qv6UVOQ0F44
There are links to further resources in the video description.
And is based on a process called NEAT. Neuro Evolution of Augmenting Topologies.
It uses a genetic algorithm and evolutionary process to design and evolve a neural net from scratch with no prior assumptions of structure or complexity of the neural net.
I believe this is what you are looking for.
Many of the papers I have read so far have this mentioned "pre-training network could improve computational efficiency in terms of back-propagating errors", and could be achieved using RBMs or Autoencoders.
If I have understood correctly, AutoEncoders work by learning the
identity function, and if it has hidden units less than the size of
input data, then it also does compression, BUT what does this even have
anything to do with improving computational efficiency in propagating
error signal backwards? Is it because the weights of the pre
trained hidden units does not diverge much from its initial values?
Assuming data scientists who are reading this would by theirselves
know already that AutoEncoders take inputs as target values since
they are learning identity function, which is regarded as
unsupervised learning, but can such method be applied to
Convolutional Neural Networks for which the first hidden layer is
feature map? Each feature map is created by convolving a learned
kernel with a receptive field in the image. This learned kernel, how
could this be obtained by pre-training (unsupervised fashion)?
One thing to note is that autoencoders try to learn the non-trivial identify function, not the identify function itself. Otherwise they wouldn't have been useful at all. Well the pre-training helps moving the weight vectors towards a good starting point on the error surface. Then the backpropagation algorithm, which is basically doing gradient descent, is used improve upon those weights. Note that gradient descent gets stuck in the closes local minima.
[Ignore the term Global Minima in the image posted and think of it as another, better, local minima]
Intuitively speaking, suppose you are looking for an optimal path to get from origin A to destination B. Having a map with no routes shown on it (the errors you obtain at the last layer of the neural network model) kind of tells you where to to go. But you may put yourself in a route which has a lot of obstacles, up hills and down hills. Then suppose someone tells you about a route a a direction he has gone through before (the pre-training) and hands you a new map (the pre=training phase's starting point).
This could be an intuitive reason on why starting with random weights and immediately start to optimize the model with backpropagation may not necessarily help you achieve the performance you obtain with a pre-trained model. However, note that many models achieving state-of-the-art results do not use pre-training necessarily and they may use the backpropagation in combination with other optimization methods (e.g. adagrad, RMSProp, Momentum and ...) to hopefully avoid getting stuck in a bad local minima.
Here's the source for the second image.
I don't know a lot about autoencoder theory, but I've done a bit of work with RBMs. What RBMs do is they predict what the probability is of seeing the specific type of data in order to get the weights initialized to the right ball park- it is considered an (unsupervised) probabilistic model, so you don't correct using the known labels. Basically, the idea here is that having a learning rate that is too big will never lead to convergence but having one that is too small will take forever to train. Thus, by "pretraining" in this way you find out the ball park of the weights and then can set the learning rate to be small in order to get them down to the optimal values.
As for the second question, no, you don't generally prelearn kernels, at least not in an unsupervised fashion. I suspect that what is meant by pretraining here is a bit different than in your first question- this is to say, that what is happening is that they are taking a pretrained model (say from model zoo) and fine tuning it with a new set of data.
Which model you use generally depends on the type of data you have and the task at hand. Convnets I've found to train faster and efficiently, but not all data has meaning when convolved, in which case dbns may be the way to go. Unless say, you have a small amount of data then I'd use something other than neural networks entirely.
Anyways, I hope this helps clear some of your questions.
I am very new to the 'reservoir computing world', and I've heard that the Liquid State Machines (LSM) are a certain kind of spiking neuron network models (SNN). Exactly what is the difference in terms of the implementation between the two.
Another aspect on which I need some clarity is in regards to their counterpart the 'Leaky integrator models of Echo state network (ESN).
I found from another answer in the forum that 'as I see it (I could be wrong) the big difference between the two approaches is the individual unit. In liquid state machine use biological like neurons, and in the Echo state use more analog units. So in term of “very short term memory” the Liquid State approach each individual neuron remember its own history, where in the Echo state approach each individual neuron react base only on the current state, there for the memory stored in the activity between the units.
Please tell me if this is the correct and if not what is the actual concept behind them.
Spiking neurons is a neuron model. LSM on the other hand is a network model. So LSM is part of a group of network models with spiking neurons (also called graded response or analog). ESN has the same units like a normal Perceptron and is thus part of the other (more popular) paradigm where neurons fire at each propagation cycle. This gives a simple enough introduction. The basic idea is to not consider neurons to be binary/digital (on/off) but analog by decoding the time between spikes which is now thought to be main source of information transportation between neurons. Whether the human brain is actually analog or digital is unknown but there's evidence of both as well as the true mechanic being something completely different. So whether one model is in fact more realistic cannot really be said with certainty.
i also new to this field , liquid state Machine: consider a pound of water what happens when you threw a pebbles in it. A series concentric Circles are created which eventually vanish ,you can learn something about where and when those pebbles where dropped in water if you look at where ripples intersecting and in case of echo states machines the idea is similar like sound wave from echo sort of interfering with one another is similar to the way that multiple inputs would interfere with one another and they are also past instantiations interference with one another within high dimension network