I recently made my first neural network simulation which also uses a genetic evolution algorithm. It's simple software that just simulates simple organisms collecting food, and they evolve, as one would expect, from organisms with random and sporadic movements into organisms with controlled, food-seeking movements. Since this kind of organism is so simple, I only used a few hidden layer neurons and a few input and output neurons. I understand that more complex neural networks could be made by simply adding more neurons, but can't you add more layers? Or would this create some kind of redundancy? All of the pictures of diagrams of neural networks, such as this one http://mechanicalforex.com/wp-content/uploads/2011/06/NN.png, always have one input layer, one hidden layer, and one output layer. Couldn't a more complex neural network be made if you just added a bunch of hidden layers? Of course this would make processing the neural network harder, but would it create any sort of advantage, or would it be just the same as adding more neurons to a single layer?
You can include as many hidden layers you want, starting from zero (--that case is called perceptron).
The ability to represent unknown functions, however, does -- in principle -- not increase. Single-hidden layer neural networks already possess a universal representation property: by increasing the number of hidden neurons, they can fit (almost) arbitrary functions. You can't get more than this. And particularly not by adding more layers.
However, that doesn't mean that multi-hidden-layer ANN's can't be useful in practice. Yet, as you get another dimension in your parameter set, people usually stuck with the single-hidden-layer version.
Related
I'm quite new to neural network and I recently built neural network for number classification in vehicle license plate. It has 3 layers: 1 input layer for 16*24(382 neurons) number image with 150 dpi , 1 hidden layer(199 neurons) with sigmoid activation function, 1 softmax output layer(10 neurons) for each number 0 to 9.
I'm trying to expand my neural network to also classify letters in license plate. But I'm worried if I just simply add more classes into output, for example add 10 letters into classification so total 20 classes, it would be hard for neural network to separate feature from each class. And also, I think it might cause problem when input is one of number and neural network wrongly classifies as one of letter with biggest probability, even though sum of probabilities of all number output exceeds that.
So I wonder if it is possible to build hierarchical neural network in following manner:
There are 3 neural networks: 'Item', 'Number', 'Letter'
'Item' neural network classifies whether input is numbers or letters.
If 'Item' neural network classifies input as numbers(letters), then input goes through 'Number'('Letter') neural network.
Return final output from Number(Letter) neural network.
And learning mechanism for each network is below:
'Item' neural network learns all images of numbers and letters. So there are 2 output.
'Number'('Letter') neural network learns images of only numbers(letter).
Which method should I pick to have better classification? Just simply add 10 more classes or build hierarchical neural networks with method above?
I'd strongly recommend training only a single neural network with outputs for all the kinds of images you want to be able to detect (so one output node per letter you want to be able to recognize, and one output node for every digit you want to be able to recognize).
The main reason for this is because recognizing digits and recognizing letters is really kind of exactly the same task. Intuitively, you can understand a trained neural network with multiple layers as performing the recognition in multiple steps. In the hidden layer it may learn to detect various kinds of simple, primitive shapes (e.g. the hidden layer may learn to detect vertical lines, horizontal lines, diagonal lines, certain kinds of simple curved shapes, etc.). Then, in the weights between hidden and output layers, it may learn how to recognize combinations of multiple of these primitive shapes as a specific output class (e.g. a vertical and a horizontal line in roughly the correct locations may be recoginzed as a capital letter L).
Those "things" it learns in the hidden layer will be perfectly relevant for digits as well as letters (that vertical line which may indicate an L may also indicate a 1 when combined with other shapes). So, there are useful things to learn that are relevant for both ''tasks'', and it will probably be able to learn these things more easily if it can learn them all in the same network.
See also a this answer I gave to a related question in the past.
I'm trying to expand my neural network to also classify letters in license plate. But i'm worried if i just simply add more classes into output, for example add 10 letters into classification so total 20 classes, it would be hard for neural network to separate feature from each class.
You're far from where it becomes problematic. ImageNet has 1000 classes and is commonly done in a single network. See the AlexNet paper. If you want to learn more about CNNs, have a look at chapter 2 of "Analysis and Optimization of
Convolutional Neural Network Architectures". And when you're on it, see chapter 4 for hirarchical classification. You can read the summary for ... well, a summary of it.
I searched for the reason a lot but I didn't get it clear, May someone explain it in some more detail please?
In theory you do not have to attach a fully connected layer, you could have a full stack of convolutions till the very end, as long as (due to custom sizes/paddings) you end up with the correct number of output neurons (usually number of classes).
So why people usually do not do that? If one goes through the math, it will become visible that each output neuron (thus - prediction wrt. to some class) depends only on the subset of the input dimensions (pixels). This would be something among the lines of a model, which only decides whether an image is an element of class 1 depending on first few "columns" (or, depending on the architecture, rows, or some patch of the image), then whether this is class 2 on a few next columns (maybe overlapping), ..., and finally some class K depending on a few last columns. Usually data does not have this characteristic, you cannot classify image of the cat based on a few first columns and ignoring the rest.
However, if you introduce fully connected layer, you provide your model with ability to mix signals, since every single neuron has a connection to every single one in the next layer, now there is a flow of information between each input dimension (pixel location) and each output class, thus the decision is based truly on the whole image.
So intuitively you can think about these operations in terms of information flow. Convolutions are local operations, pooling are local operations. Fully connected layers are global (they can introduce any kind of dependence). This is also why convolutions work so well in domains like image analysis - due to their local nature they are much easier to train, even though mathematically they are just a subset of what fully connected layers can represent.
note
I am considering here typical use of CNNs, where kernels are small. In general one can even think of MLP as a CNN, where the kernel is of the size of the whole input with specific spacing/padding. However these are just corner cases, which are not really encountered in practise, and not really affecting the reasoning, since then they end up being MLPs. The whole point here is simple - to introduce global relations, if one can do it by using CNNs in a specific manner - then MLPs are not needed. MLPs are just one way of introducing this dependence.
Every fully connected (FC) layer has an equivalent convolutional layer (but not vice versa). Hence it is not necessary to add FC layers. They can always be replaced by convolutional layers (+ reshaping). See details.
Why do we use FC layers then?
Because (1) we are used to it (2) it is simpler. (1) is probably the reason for (2). For example, you would need to adjust the loss fuctions / the shape of the labels / add a reshape add the end if you used a convolutional layer instead of a FC layer.
I found this answer by Anil-Sharma on Quora helpful.
We can divide the whole network (for classification) into two parts:
Feature extraction:
In the conventional classification algorithms, like SVMs, we used to extract features from the data to make the classification work. The convolutional layers are serving the same purpose of feature extraction. CNNs capture better representation of data and hence we don’t need to do feature engineering.
Classification:
After feature extraction we need to classify the data into various classes, this can be done using a fully connected (FC) neural network. In place of fully connected layers, we can also use a conventional classifier like SVM. But we generally end up adding FC layers to make the model end-to-end trainable.
The CNN gives you a representation of the input image. To learn the sample classes, you should use a classifier (such as logistic regression, SVM, etc.) that learns the relationship between the learned features and the sample classes. Fully-connected layer is also a linear classifier such as logistic regression which is used for this reason.
Convolution and pooling layers extract features from image. So this layer doing some "preprocessing" of data. Fully connected layrs perform classification based on this extracted features.
I just started to learn about neural networks and so far my knowledge of machine learning is simply linear and logistic regression. from my understanding of the latter algorithms, is that given multiples of inputs the job of the learning algorithm is to come up with appropriate weights for each input so that eventually I have a polynomial that either describes the data which is the case of linear regression or separates it as in the case of logistic regression.
if I was to represent the same mechanism in neural network, according to my understanding, it would look something like this,
multiple nodes at the input layer and a single node in the output layer. where I can back propagate the error proportionally to each input. so that also eventually I arrive to a polynomial X1W1 + X2W2+....XnWn that describes the data. to me having multiple nodes per layer, aside from the input layer, seems to make the learning process parallel, so that I can arrive to the result faster. it's almost like running multiple learning algorithms each with different starting points to see which one converges faster. and as for the multiple layers I'm at a lose of what mechanism and advantage does it have on the learning outcome.
why do we have multiple layers and multiple nodes per layer in a neural network?
We need at least one hidden layer with a non-linear activation to be able to learn non-linear functions. Usually, one thinks of each layer as an abstraction level. For computer vision, the input layer contains the image and the output layer contains one node for each class. The first hidden layer detects edges, the second hidden layer might detect circles / rectangles, then there come more complex patterns.
There is a theoretical result which says that an MLP with only one hidden layer can fit every function of interest up to an arbitrary low error margin if this hidden layer has enough neurons. However, the number of parameters might be MUCH larger than if you add more layers.
Basically, by adding more hidden layers / more neurons per layer you add more parameters to the model. Hence you allow the model to fit more complex functions. However, up to my knowledge there is no quantitative understanding what adding a single further layer / node exactly makes.
It seems to me that you might want a general introduction into neural networks. I recommend chapter 4.3 and 4.4 of [Tho14a] (my bachelors thesis) as well as [LBH15].
[Tho14a]
M. Thoma, “On-line recognition of handwritten mathematical symbols,”
Karlsruhe, Germany, Nov. 2014. [Online]. Available: https://arxiv.org/abs/1511.09030
[LBH15]
Y. LeCun,
vol. 521,
Y. Bengio,
no. 7553,
and G. Hinton,
pp. 436–444,
“Deep learning,”
Nature,
May 2015. [Online]. Available:
http://www.nature.com/nature/journal/v521/n7553/abs/nature14539.html
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.
I Wrote a simple recurrent neural network (7 neurons, each one is initially connected to all the neurons) and trained it using a genetic algorithm to learn "complicated", non-linear functions like 1/(1+x^2). As the training set, I used 20 values within the range [-5,5] (I tried to use more than 20 but the results were not changed dramatically).
The network can learn this range pretty well, and when given examples of other points within this range, it can predict the value of the function. However, it can not extrapolate correctly and predicting the values of the function outside the range [-5,5]. What are the reasons for that and what can I do to improve its extrapolation abilities?
Thanks!
Neural networks are not extrapolation methods (no matter - recurrent or not), this is completely out of their capabilities. They are used to fit a function on the provided data, they are completely free to build model outside the subspace populated with training points. So in non very strict sense one should think about them as an interpolation method.
To make things clear, neural network should be capable of generalizing the function inside subspace spanned by the training samples, but not outside of it
Neural network is trained only in the sense of consistency with training samples, while extrapolation is something completely different. Simple example from "H.Lohninger: Teach/Me Data Analysis, Springer-Verlag, Berlin-New York-Tokyo, 1999. ISBN 3-540-14743-8" shows how NN behave in this context
All of these networks are consistent with training data, but can do anything outside of this subspace.
You should rather reconsider your problem's formulation, and if it can be expressed as a regression or classification problem then you can use NN, otherwise you should think about some completely different approach.
The only thing, which can be done to somehow "correct" what is happening outside the training set is to:
add artificial training points in the desired subspace (but this simply grows the training set, and again - outside of this new set, network's behavious is "random")
add strong regularization, which will force network to create very simple model, but model's complexity will not guarantee any extrapolation strength, as two model's of exactly the same complexity can have for example completely different limits in -/+ infinity.
Combining above two steps can help building model which to some extent "extrapolates", but this, as stated before, is not a purpose of a neural network.
As far as I know this is only possible with networks which do have the echo property. See Echo State Networks on scholarpedia.org.
These networks are designed for arbitrary signal learning and are capable to remember their behavior.
You can also take a look at this tutorial.
The nature of your post(s) suggests that what you're referring to as "extrapolation" would be more accurately defined as "sequence recognition and reproduction." Training networks to recognize a data sequence with or without time-series (dt) is pretty much the purpose of Recurrent Neural Network (RNN).
The training function shown in your post has output limits governed by 0 and 1 (or -1, since x is effectively abs(x) in the context of that function). So, first things first, be certain your input layer can easily distinguish between negative and positive inputs (if it must).
Next, the number of neurons is not nearly as important as how they're layered and interconnected. How many of the 7 were used for the sequence inputs? What type of network was used and how was it configured? Network feedback will reveal the ratios, proportions, relationships, etc. and aid in the adjustment of network weight adjustments to match the sequence. Feedback can also take the form of a forward-feed depending on the type of network used to create the RNN.
Producing an 'observable' network for the exponential-decay function: 1/(1+x^2), should be a decent exercise to cut your teeth on RNNs. 'Observable', meaning the network is capable of producing results for any input value(s) even though its training data is (far) smaller than all possible inputs. I can only assume that this was your actual objective as opposed to "extrapolation."