Given a target function f: R^4 -> R^2, can you draw me(give me an example) an Artificial Neural Network , lets say with two layers, and 3 nodes in the hidden layer.
Now, I think I understand how an ANN works when a function is like [0,1]^5 ->[0,1], but I am not quite sure how to do an example from R4 to R2.
I am new to machine learning, and it's a little bit diffult to catch up with all this concepts.
Thanks in advance.
First, you need two neurons in the output layer. Each neuron would correspond to one dimension of your output space.
Neurons in the output layer don't need an activation function that limits their values in the [0,1] interval (e.g. the logistic function). And even if you scale your output space in the interval [0,1], don't use a sigmoid function for activation.
Although your original data is not in [0,1]^4, you should do some preprocessing to scale and shift them to have mean zero and variance 1. You must apply same preprocessing to all your examples (training and test).
This should give you something to build up on.
Related
I am try to write a neural network class but I don't fully understand some aspects of it. I have two questions on the folling design.
Am I doing this correctly? Does the bias neuron need to connect to all of neurons (except those in the input layer) or just those in the hidden layer?
My second question is about calculation the output value. I'm using the equation below to calculate the output value of the neurons.
HiddenLayerFirstNeuron.Value =
(input1.Value * weight) + (input2.Value * weight) + (Bias.Value * weight)
After this equation, I'm calculating the activation and the result send the output. And output neurons doing same.
I'm not sure what I am do and I want to clear up problems.
Take a look at: http://deeplearning.net/tutorial/contents.html in theano. This explains everything you need to know for multi layer perceptron using theano (symbolic mathematic library).
The bias is usually connected to all hidden and output units.
Yes, you compute the input of activation function like summation of weight*output of previous layer neuron.
Good luck with development ;)
There should be a separate bias neuron for each hidden and the output layer. Think of the layers as a function applied to a first order polynomials such as f(m*x+b)=y where y is your output and f(x) your activation function. If you look at the the linear term you will recognize the b. This represents the bias and it behaves similar with neural network as with this simplification: It shifts the hyperplane up and down the in the space. Keep in mind that you will have one bias per layer connected to all neurons of that layer f((wi*xi+b)+...+(wn*xn+b)) with an initial value of 1. When it comes to gradient descent, you will have to train this neuron like a normal weight.
In my opinion should you apply the activation function to the output layer as well. This is how it's usually done with multilayer perceptrons. But it actually depends of what you want. If you, for example, use the logistic function as activation function and you want an output in the interval (0,1), then you have to apply your activation function to the output as well. Since a basic linear combination, as it is in your example, can theoretically go above the boundaries of the previously mentioned Intervall.
Most examples of neural networks for classification tasks I've seen use the a softmax layer as output activation function. Normally, the other hidden units use a sigmoid, tanh, or ReLu function as activation function. Using the softmax function here would - as far as I know - work out mathematically too.
What are the theoretical justifications for not using the softmax function as hidden layer activation functions?
Are there any publications about this, something to quote?
I haven't found any publications about why using softmax as an activation in a hidden layer is not the best idea (except Quora question which you probably have already read) but I will try to explain why it is not the best idea to use it in this case :
1. Variables independence : a lot of regularization and effort is put to keep your variables independent, uncorrelated and quite sparse. If you use softmax layer as a hidden layer - then you will keep all your nodes (hidden variables) linearly dependent which may result in many problems and poor generalization.
2. Training issues : try to imagine that to make your network working better you have to make a part of activations from your hidden layer a little bit lower. Then - automaticaly you are making rest of them to have mean activation on a higher level which might in fact increase the error and harm your training phase.
3. Mathematical issues : by creating constrains on activations of your model you decrease the expressive power of your model without any logical explaination. The strive for having all activations the same is not worth it in my opinion.
4. Batch normalization does it better : one may consider the fact that constant mean output from a network may be useful for training. But on the other hand a technique called Batch Normalization has been already proven to work better, whereas it was reported that setting softmax as activation function in hidden layer may decrease the accuracy and the speed of learning.
Actually, Softmax functions are already used deep within neural networks, in certain cases, when dealing with differentiable memory and with attention mechanisms!
Softmax layers can be used within neural networks such as in Neural Turing Machines (NTM) and an improvement of those which are Differentiable Neural Computer (DNC).
To summarize, those architectures are RNNs/LSTMs which have been modified to contain a differentiable (neural) memory matrix which is possible to write and access through time steps.
Quickly explained, the softmax function here enables a normalization of a fetch of the memory and other similar quirks for content-based addressing of the memory. About that, I really liked this article which illustrates the operations in an NTM and other recent RNN architectures with interactive figures.
Moreover, Softmax is used in attention mechanisms for, say, machine translation, such as in this paper. There, the Softmax enables a normalization of the places to where attention is distributed in order to "softly" retain the maximal place to pay attention to: that is, to also pay a little bit of attention to elsewhere in a soft manner. However, this could be considered like to be a mini-neural network that deals with attention, within the big one, as explained in the paper. Therefore, it could be debated whether or not Softmax is used only at the end of neural networks.
Hope it helps!
Edit - More recently, it's even possible to see Neural Machine Translation (NMT) models where only attention (with softmax) is used, without any RNN nor CNN: http://nlp.seas.harvard.edu/2018/04/03/attention.html
Use a softmax activation wherever you want to model a multinomial distribution. This may be (usually) an output layer y, but can also be an intermediate layer, say a multinomial latent variable z. As mentioned in this thread for outputs {o_i}, sum({o_i}) = 1 is a linear dependency, which is intentional at this layer. Additional layers may provide desired sparsity and/or feature independence downstream.
Page 198 of Deep Learning (Goodfellow, Bengio, Courville)
Any time we wish to represent a probability distribution over a discrete variable with n possible values, we may use the softmax function. This can be seen as a generalization of the sigmoid function which was used to represent a probability
distribution over a binary variable.
Softmax functions are most often used as the output of a classifier, to represent the probability distribution over n different classes. More rarely, softmax functions can be used inside the model itself, if we wish the model to choose between one of n different options for some internal variable.
Softmax function is used for the output layer only (at least in most cases) to ensure that the sum of the components of output vector is equal to 1 (for clarity see the formula of softmax cost function). This also implies what is the probability of occurrence of each component (class) of the output and hence sum of the probabilities(or output components) is equal to 1.
Softmax function is one of the most important output function used in deep learning within the neural networks (see Understanding Softmax in minute by Uniqtech). The Softmax function is apply where there are three or more classes of outcomes. The softmax formula takes the e raised to the exponent score of each value score and devide it by the sum of e raised the exponent scores values. For example, if I know the Logit scores of these four classes to be: [3.00, 2.0, 1.00, 0.10], in order to obtain the probabilities outputs, the softmax function can be apply as follows:
import numpy as np
def softmax(x):
z = np.exp(x - np.max(x))
return z / z.sum()
scores = [3.00, 2.0, 1.00, 0.10]
print(softmax(scores))
Output: probabilities (p) = 0.642 0.236 0.087 0.035
The sum of all probabilities (p) = 0.642 + 0.236 + 0.087 + 0.035 = 1.00. You can try to substitute any value you know in the above scores, and you will get a different values. The sum of all the values or probabilities will be equal to one. That’s makes sense, because the sum of all probability is equal to one, thereby turning Logit scores to probability scores, so that we can predict better. Finally, the softmax output, can help us to understand and interpret Multinomial Logit Model. If you like the thoughts, please leave your comments below.
I am learning neural networks for the first time. I was trying to understand how using a single hidden layer function approximation can be performed. I saw this example on stackexchange but I had some questions after going through one of the answers.
Suppose I want to approximate a sine function between 0 and 3.14 radians. So will I have 1 input neuron? If so, then next if I assume K neurons in the hidden layer and each of which uses a sigmoid transfer function. Then in the output neuron(if say it just uses a linear sum of results from hidden layer) how can be output be something other than sigmoid shape? Shouldn't the linear sum be sigmoid as well? Or in short how can a sine function be approximated using this architecture in a Neural network.
It is possible and it is formally stated as the universal approximation theorem. It holds for any non-constant, bounded, and monotonically-increasing continuous activation function
I actually don't know the formal proof but to get an intuitive idea that it is possible I recommend the following chapter: A visual proof that neural nets can compute any function
It shows that with the enough hidden neurons and the right parameters you can create step functions as the summed output of the hidden layer. With step functions it is easy to argue how you can approximate any function at least coarsely. Now to get the final output correct the sum of the hidden layer has to be since the final neuron then outputs: . And as already said, we are be able to approximate this at least to some accuracy.
I am finding a very hard time to visualize how the activation function actually manages to classify non-linearly separable training data sets.
Why does the activation function (e.g tanh function) work for non-linear cases? What exactly happens mathematically when the activation function projects the input to output? What separates training samples of different classes, and how does this work if one had to plot this process graphically?
I've tried looking for numerous sources, but what exactly makes the activation function actually work for classifying training samples in a neural network, I just cannot grasp easily and would like to be able to picture this in my mind.
Mathematical result behind neural networks is Universal Approximation Theorem. Basically, sigmoidal functions (those which saturate on both ends, like tanh) are smooth almost-piecewise-constant approximators. The more neurons you have – the better your approximation is.
This picture was taked from this article: A visual proof that neural nets can compute any function. Make sure to check that article, it has other examples and interactive applets.
NNs actually, at each level, create new features by distorting input space. Non-linear functions allow you to change "curvature" of target function, so further layers have chance to make it linear-separable. If there were no non-linear functions, any combination of linear function is still linear, thus no benefit from multi-layerness. As a graphical example consider
this animation
This pictures where taken from this article. Also check out that cool visualization applet.
Activation functions have very little to do with classifying non-linearly separable sets of data.
Activation functions are used as a way to normalize signals at every step in your neural network. They typically have an infinite domain and a finite range. Tanh, for example, has a domain of (-∞,∞) and a range of (-1,1). The sigmoid function maps the same domain to (0,1).
You can think of this as a way of enforcing equality across all of your learned features at a given neural layer (a.k.a. feature scaling). Since the input domain is not known before hand it's not as simple as regular feature scaling (for linear regression) and thusly activation functions must be used. The effects of the activation function are compensated for when computing errors during back-propagation.
Back-propagation is a process that applies error to the neural network. You can think of this as a positive reward for the neurons that contributed to the correct classification and a negative reward for the neurons that contributed to an incorrect classification. This contribution is often known as the gradient of the neural network. The gradient is, effectively, a multi-variable derivative.
When back-propagating the error, each individual neuron's contribution to the gradient is the activations function's derivative at the input value for that neuron. Sigmoid is a particularly interesting function because its derivative is extremely cheap to compute. Specifically s'(x) = 1 - s(x); it was designed this way.
Here is an example image (found by google image searching: neural network classification) that demonstrates how a neural network might be superimposed on top of your data set:
I hope that gives you a relatively clear idea of how neural networks might classify non-linearly separable datasets.
I understand neural networks with any number of hidden layers can approximate nonlinear functions, however, can it approximate:
f(x) = x^2
I can't think of how it could. It seems like a very obvious limitation of neural networks that can potentially limit what it can do. For example, because of this limitation, neural networks probably can't properly approximate many functions used in statistics like Exponential Moving Average, or even variance.
Speaking of moving average, can recurrent neural networks properly approximate that? I understand how a feedforward neural network or even a single linear neuron can output a moving average using the sliding window technique, but how would recurrent neural networks do it without X amount of hidden layers (X being the moving average size)?
Also, let us assume we don't know the original function f, which happens to get the average of the last 500 inputs, and then output a 1 if it's higher than 3, and 0 if it's not. But for a second, pretend we don't know that, it's a black box.
How would a recurrent neural network approximate that? We would first need to know how many timesteps it should have, which we don't. Perhaps a LSTM network could, but even then, what if it's not a simple moving average, it's an exponential moving average? I don't think even LSTM can do it.
Even worse still, what if f(x,x1) that we are trying to learn is simply
f(x,x1) = x * x1
That seems very simple and straightforward. Can a neural network learn it? I don't see how.
Am I missing something huge here or are machine learning algorithms extremely limited? Are there other learning techniques besides neural networks that can actually do any of this?
The key point to understand is compact:
Neural networks (as any other approximation structure like, polynomials, splines, or Radial Basis Functions) can approximate any continuous function only within a compact set.
In other words the theory states that, given:
A continuous function f(x),
A finite range for the input x, [a,b], and
A desired approximation accuracy ε>0,
then there exists a neural network that approximates f(x) with an approximation error less than ε, everywhere within [a,b].
Regarding your example of f(x) = x2, yes you can approximate it with a neural network within any finite range: [-1,1], [0, 1000], etc. To visualise this, imagine that you approximate f(x) within [-1,1] with a Step Function. Can you do it on paper? Note that if you make the steps narrow enough you can achieve any desired accuracy. The way neural networks approximate f(x) is not much different than this.
But again, there is no neural network (or any other approximation structure) with a finite number of parameters that can approximate f(x) = x2 for all x in [-∞, +∞].
The question is very legitimate and unfortunately many of the answers show how little practitioners seem to know about the theory of neural networks. The only rigorous theorem that exists about the ability of neural networks to approximate different kinds of functions is the Universal Approximation Theorem.
The UAT states that any continuous function on a compact domain can be approximated by a neural network with only one hidden layer provided the activation functions used are BOUNDED, continuous and monotonically increasing. Now, a finite sum of bounded functions is bounded by definition.
A polynomial is not bounded so the best we can do is provide a neural network approximation of that polynomial over a compact subset of R^n. Outside of this compact subset, the approximation will fail miserably as the polynomial will grow without bound. In other words, the neural network will work well on the training set but will not generalize!
The question is neither off-topic nor does it represent the OP's opinion.
I am not sure why there is such a visceral reaction, I think it is a legitimate question that is hard to find by googling it, even though I think it is widely appreciated and repeated outloud. I think in this case you are looking for the actually citations showing that a neural net can approximate any function. This recent paper explains it nicely, in my opinion. They also cite the original paper by Barron from 1993 that proved a less general result. The conclusion: a two-layer neural network can represent any bounded degree polynomial, under certain (seemingly non-restrictive) conditions.
Just in case the link does not work, it is called "Learning Polynomials with Neural Networks" by Andoni et al., 2014.
I understand neural networks with any number of hidden layers can approximate nonlinear functions, however, can it approximate:
f(x) = x^2
The only way I can make sense of that question is that you're talking about extrapolation. So e.g. given training samples in the range -1 < x < +1 can a neural network learn the right values for x > 100? Is that what you mean?
If you had prior knowledge, that the functions you're trying to approximate are likely to be low-order polynomials (or any other set of functions), then you could surely build a neural network that can represent these functions, and extrapolate x^2 everywhere.
If you don't have prior knowledge, things are a bit more difficult: There are infinitely many smooth functions that fit x^2 in the range -1..+1 perfectly, and there's no good reason why we would expect x^2 to give better predictions than any other function. In other words: If we had no prior knowledge about the function we're trying to learn, why would we want to learn x -> x^2? In the realm of artificial training sets, x^2 might be a likely function, but in the real world, it probably isn't.
To give an example: Let's say the temperature on Monday (t=0) is 0°, on Tuesday it's 1°, on Wednesday it's 4°. We have no reason to believe temperatures behave like low-order polynomials, so we wouldn't want to infer from that data that the temperature next Monday will probably be around 49°.
Also, let us assume we don't know the original function f, which happens to get the average of the last 500 inputs, and then output a 1 if it's higher than 3, and 0 if it's not. But for a second, pretend we don't know that, it's a black box.
How would a recurrent neural network approximate that?
I think that's two questions: First, can a neural network represent that function? I.e. is there a set of weights that would give exactly that behavior? It obviously depends on the network architecture, but I think we can come up with architectures that can represent (or at least closely approximate) this kind of function.
Question two: Can it learn this function, given enough training samples? Well, if your learning algorithm doesn't get stuck in a local minimum, sure: If you have enough training samples, any set of weights that doesn't approximate your function gives a training error greater that 0, while a set of weights that fit the function you're trying to learn has a training error=0. So if you find a global optimum, the network must fit the function.
A network can learn x|->x * x if it has a neuron that calculates x * x. Or more generally, a node that calculates x**p and learns p. These aren't commonly used, but the statement that "no neural network can learn..." is too strong.
A network with ReLUs and a linear output layer can learn x|->2*x, even on an unbounded range of x values. The error will be unbounded, but the proportional error will be bounded. Any function learnt by such a network is piecewise linear, and in particular asymptotically linear.
However, there is a risk with ReLUs: once a ReLU is off for all training examples it ceases learning. With a large domain, it will turn on for some possible test examples, and give an erroneous result. So ReLUs are only a good choice if test cases are likely to be within the convex hull of the training set. This is easier to guarantee if the dimensionality is low. One work around is to prefer LeakyReLU.
One other issue: how many neurons do you need to achieve the approximation you want? Each ReLU or LeakyReLU implements a single change of gradient. So the number needed depends on the maximum absolute value of the second differential of the objective function, divided by the maximum error to be tolerated.
There are theoretical limitations of Neural Networks. No neural network can ever learn the function f(x) = x*x
Nor can it learn an infinite number of other functions, unless you assume the impractical:
1- an infinite number of training examples
2- an infinite number of units
3- an infinite amount of time to converge
NNs are good in learning low-level pattern recognition problems (signals that in the end have some statistical pattern that can be represented by some "continuous" function!), but that's it!
No more!
Here's a hint:
Try to build a NN that takes n+1 data inputs (x0, x1, x2, ... xn) and it will return true (or 1) if (2 * x0) is in the rest of the sequence. And, good luck.
Infinite functions especially those that are recursive cannot be learned. They just are!