AI News, BOOK REVIEW: Artificial Neural Networks/Feed-Forward Networks

Artificial Neural Networks/Feed-Forward Networks

In a feed-forward system PE are arranged into distinct layers with each layer receiving input from the previous layer and outputting to the next layer.

A network without all possible forward paths is known as a sparsely connected network, or a non-fully connected network.

The weights from each neuron in layer l - 1 to the neurons in layer l are arranged into a matrix wl.

If ρl is a vector of activation functions [σ1 σ2 … σn] that acts on each row of input and bl is an arbitrary offset vector (for generalization) then the total output of layer l is given as:

Feedforward neural network

A feedforward neural network is an artificial neural network wherein connections between the nodes do not form a cycle.[1]

The sum of the products of the weights and the inputs is calculated in each node, and if the value is above some threshold (typically 0) the neuron fires and takes the activated value (typically 1);

It calculates the errors between calculated output and sample output data, and uses this to create an adjustment to the weights, thus implementing a form of gradient descent.

in 1969 in a famous monograph entitled Perceptrons, Marvin Minsky and Seymour Papert showed that it was impossible for a single-layer perceptron network to learn an XOR function (nonetheless, it was known that multi-layer perceptrons are capable of producing any possible boolean function).

Although a single threshold unit is quite limited in its computational power, it has been shown that networks of parallel threshold units can approximate any continuous function from a compact interval of the real numbers into the interval [-1,1].

The universal approximation theorem for neural networks states that every continuous function that maps intervals of real numbers to some output interval of real numbers can be approximated arbitrarily closely by a multi-layer perceptron with just one hidden layer.

After repeating this process for a sufficiently large number of training cycles, the network will usually converge to some state where the error of the calculations is small.

For this, the network calculates the derivative of the error function with respect to the network weights, and changes the weights such that the error decreases (thus going downhill on the surface of the error function).

In general, the problem of teaching a network to perform well, even on samples that were not used as training samples, is a quite subtle issue that requires additional techniques.

To describe neural networks, we will begin by describing the simplest possible neural network, one which comprises a single “neuron.” We will use the following diagram to denote a single neuron: This “neuron” is a computational unit that takes as input x_1, x_2, x_3 (and a +1 intercept term), and outputs \textstyle h_{W,b}(x) = f(W^Tx) = f(\sum_{i=1}^3 W_{i}x_i +b), where f : \Re \mapsto \Re is called the activation function.

Although these notes will use the sigmoid function, it is worth noting that another common choice for f is the hyperbolic tangent, or tanh, function: Recent research has found a different activation function, the rectified linear function, often works better in practice for deep neural networks.

The rectified linear activation function is given by, Here are plots of the sigmoid, \tanh and rectified linear functions: The \tanh(z) function is a rescaled version of the sigmoid, and its output range is [-1,1] instead of [0,1].

Specifically, the computation that this neural network represents is given by: In the sequel, we also let z^{(l)}_i denote the total weighted sum of inputs to unit i in layer l, including the bias term (e.g., \textstyle z_i^{(2)} = \sum_{j=1}^n W^{(1)}_{ij} x_j + b^{(1)}_i), so that a^{(l)}_i = f(z^{(l)}_i).

Specifically, if we extend the activation function f(\cdot) to apply to vectors in an element-wise fashion (i.e., f([z_1, z_2, z_3]) = [f(z_1), f(z_2), f(z_3)]), then we can write the equations above more compactly as: We call this step forward propagation.

More generally, recalling that we also use a^{(1)} = x to also denote the values from the input layer, then given layer l’s activations a^{(l)}, we can compute layer l+1’s activations a^{(l+1)} as: By organizing our parameters in matrices and using matrix-vector operations, we can take advantage of fast linear algebra routines to quickly perform calculations in our network.

The most common choice is a \textstyle n_l-layered network where layer \textstyle 1 is the input layer, layer \textstyle n_l is the output layer, and each layer \textstyle l is densely connected to layer \textstyle l+1.

In this setting, to compute the output of the network, we can successively compute all the activations in layer \textstyle L_2, then layer \textstyle L_3, and so on, up to layer \textstyle L_{n_l}, using the equations above that describe the forward propagation step.

(For example, in a medical diagnosis application, the vector x might give the input features of a patient, and the different outputs y_i’s might indicate presence or absence of different diseases.) Suppose we have a fixed training set \{ (x^{(1)}, y^{(1)}), \ldots, (x^{(m)}, y^{(m)}) \} of m training examples.

If you’ve taken CS229 (Machine Learning) at Stanford or watched the course’s videos on YouTube, you may also recognize this weight decay as essentially a variant of the Bayesian regularization method you saw there, where we placed a Gaussian prior on the parameters and did MAP (instead of maximum likelihood) estimation.) The weight decay parameter \lambda controls the relative importance of the two terms.

To train our neural network, we will initialize each parameter W^{(l)}_{ij} and each b^{(l)}_i to a small random value near zero (say according to a {Normal}(0,\epsilon^2) distribution for some small \epsilon, say 0.01), and then apply an optimization algorithm such as batch gradient descent.

If all the parameters start off at identical values, then all the hidden layer units will end up learning the same function of the input (more formally, W^{(1)}_{ij} will be the same for all values of i, so that a^{(2)}_1 = a^{(2)}_2 = a^{(2)}_3 = \ldots for any input x).

Note that in this notation, ”\textstyle \Delta W^{(l)}” is a matrix, and in particular it isn’t ”\textstyle \Delta times \textstyle W^{(l)}.” We implement one iteration of batch gradient descent as follows: To train our neural network, we can now repeatedly take steps of gradient descent to reduce our cost function \textstyle J(W,b).

Deep Learning: Feedforward Neural Network

Visualising the two images in Fig 1 where the left image shows how multilayer neural network identify different object by learning different characteristic of object at each layer, for example at first hidden layer edges are detected, on second hidden layer corners and contours are identified.

As we can observe in Fig 2 that neural network has trained a model by combining different distribution in one desirable distribution considering the non linearity of the problem.

Unlike piecewise linear units, sigmoidal units saturate across most of their domain — they saturate to a high value when z is very positive, saturate to a low value when z is very negative, and are only strongly sensitive to their input when z is near 0.

Cybenko, 1989) states that a feedforward network with a linear output layer and at least one hidden layer with any “squashing” activation function (such as the sigmoid activation function) can approximate any Borel measurable function from one finite-dimensional space to another with any desired non-zero amount of error, provided that the network is given enough hidden units.

The universal approximation theorem says that there exists a network large enough to achieve any degree of accuracy we desire, but the theorem does not say how large this network will be.

Figure 6 illustrates how a network with absolute value rectification creates mirror images of the function computed on top of some hidden unit, with respect to the input of that hidden unit.

By composing these folding operations, we obtain an exponentially large number of piecewise linear regions which can capture all kinds of regular (e.g., repeating) patterns.

If we instead use a smooth cost function like the quadratic cost it turns out to be easy to figure out how to make small changes in the weights and biases so as to get an improvement in the cost.

Here, w denotes the collection of all weights in the network, b all the biases, n is the total number of training inputs, a is the vector of outputs from the network when x is input, and the sum is over all training inputs, x.

The largest difference between the linear models we have seen so far and neural networks is that the nonlinearity of a neural network causes most interesting loss functions to become non-convex.

This means that neural networks are usually trained by using iterative, gradient-based optimizers that merely drive the cost function to a very low value, rather than the linear equation solvers used to train linear regression models or the convex optimization algorithms with global convergence guarantees used to train logistic regression or SVMs.

To make these ideas more precise, stochastic gradient descent works by randomly picking out a small number m of randomly chosen training inputs.

Provided the sample size mm is large enough we expect that the average value of the ∇CXj will be roughly equal to the average over all ∇Cx, that is, This modification helps us in reducing a good amount of computational load.

Feedforward neural networks

A three layer feedforward neural network, using hyperbolic tangent and linear activation functions at the hidden and output layers, respectively, was employed.

In order to build NN empirical models, two independent data (training and validation), containing different data sets, were used (Figure 1).

For generalized empirical model building purposes, a three layer feedforward neural network, using hyperbolic tangent and linear activation functions at the hidden and output layers, respectively, was selected.

The existence of great quantity of mineral salts, especially sodium bicarbonate, in the culture medium and also to a certain increase in pH, during the process, both factors -high salinity and pH - favour an ideal condition for the selective growth of Spirulina.

Spurious solutions (Figure 4) were obtained in all NNs trained with insufficient data set, no matter the values used to initialize the NN parameters or neurons activation function.

The influence of the size of the training data set on the quality of the NN generalization capacity was analyzed in detail, using different initial guesses for the NN parameters.

For scarce experimental data set, all the quadratic average error functions analyzed led to multiple local minima.

In the section on linear classification we computed scores for different visual categories given the image using the formula \( s = W x \), where \(W\) was a matrix and \(x\) was an input column vector containing all pixel data of the image.

In the case of CIFAR-10, \(x\) is a [3072x1] column vector, and \(W\) is a [10x3072] matrix, so that the output scores is a vector of 10 class scores.

There are several choices we could make for the non-linearity (which we’ll study below), but this one is a common choice and simply thresholds all activations that are below zero to zero.

Notice that the non-linearity is critical computationally - if we left it out, the two matrices could be collapsed to a single matrix, and therefore the predicted class scores would again be a linear function of the input.

three-layer neural network could analogously look like \( s = W_3 \max(0, W_2 \max(0, W_1 x)) \), where all of \(W_3, W_2, W_1\) are parameters to be learned.

The area of Neural Networks has originally been primarily inspired by the goal of modeling biological neural systems, but has since diverged and become a matter of engineering and achieving good results in Machine Learning tasks.

Approximately 86 billion neurons can be found in the human nervous system and they are connected with approximately 10^14 - 10^15 synapses.

The idea is that the synaptic strengths (the weights \(w\)) are learnable and control the strength of influence (and its direction: excitory (positive weight) or inhibitory (negative weight)) of one neuron on another.

Based on this rate code interpretation, we model the firing rate of the neuron with an activation function \(f\), which represents the frequency of the spikes along the axon.

Historically, a common choice of activation function is the sigmoid function \(\sigma\), since it takes a real-valued input (the signal strength after the sum) and squashes it to range between 0 and 1.

An example code for forward-propagating a single neuron might look as follows: In other words, each neuron performs a dot product with the input and its weights, adds the bias and applies the non-linearity (or activation function), in this case the sigmoid \(\sigma(x) = 1/(1+e^{-x})\).

As we saw with linear classifiers, a neuron has the capacity to “like” (activation near one) or “dislike” (activation near zero) certain linear regions of its input space.

With this interpretation, we can formulate the cross-entropy loss as we have seen in the Linear Classification section, and optimizing it would lead to a binary Softmax classifier (also known as logistic regression).

The regularization loss in both SVM/Softmax cases could in this biological view be interpreted as gradual forgetting, since it would have the effect of driving all synaptic weights \(w\) towards zero after every parameter update.

The sigmoid non-linearity has the mathematical form \(\sigma(x) = 1 / (1 + e^{-x})\) and is shown in the image above on the left.

The sigmoid function has seen frequent use historically since it has a nice interpretation as the firing rate of a neuron: from not firing at all (0) to fully-saturated firing at an assumed maximum frequency (1).

Also note that the tanh neuron is simply a scaled sigmoid neuron, in particular the following holds: \( \tanh(x) = 2 \sigma(2x) -1 \).

Other types of units have been proposed that do not have the functional form \(f(w^Tx + b)\) where a non-linearity is applied on the dot product between the weights and the data.

TLDR: “What neuron type should I use?” Use the ReLU non-linearity, be careful with your learning rates and possibly monitor the fraction of “dead” units in a network.

For regular neural networks, the most common layer type is the fully-connected layer in which neurons between two adjacent layers are fully pairwise connected, but neurons within a single layer share no connections.

Working with the two example networks in the above picture: To give you some context, modern Convolutional Networks contain on orders of 100 million parameters and are usually made up of approximately 10-20 layers (hence deep learning).

The full forward pass of this 3-layer neural network is then simply three matrix multiplications, interwoven with the application of the activation function: In the above code, W1,W2,W3,b1,b2,b3 are the learnable parameters of the network.

Notice also that instead of having a single input column vector, the variable x could hold an entire batch of training data (where each input example would be a column of x) and then all examples would be efficiently evaluated in parallel.

Neural Networks work well in practice because they compactly express nice, smooth functions that fit well with the statistical properties of data we encounter in practice, and are also easy to learn using our optimization algorithms (e.g.

Similarly, the fact that deeper networks (with multiple hidden layers) can work better than a single-hidden-layer networks is an empirical observation, despite the fact that their representational power is equal.

As an aside, in practice it is often the case that 3-layer neural networks will outperform 2-layer nets, but going even deeper (4,5,6-layer) rarely helps much more.

We could train three separate neural networks, each with one hidden layer of some size and obtain the following classifiers: In the diagram above, we can see that Neural Networks with more neurons can express more complicated functions.

For example, the model with 20 hidden neurons fits all the training data but at the cost of segmenting the space into many disjoint red and green decision regions.

The subtle reason behind this is that smaller networks are harder to train with local methods such as Gradient Descent: It’s clear that their loss functions have relatively few local minima, but it turns out that many of these minima are easier to converge to, and that they are bad (i.e.

Conversely, bigger neural networks contain significantly more local minima, but these minima turn out to be much better in terms of their actual loss.

In practice, what you find is that if you train a small network the final loss can display a good amount of variance - in some cases you get lucky and converge to a good place but in some cases you get trapped in one of the bad minima.

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