AI News, Artificial Neural Networks for Beginners 5

Artificial Neural Networks for Beginners 5

Deep Learning is a very hot topic these days especially in computer vision applications and you probably see it in the news and get curious.

Today's guest blogger, Toshi Takeuchi, gives us a quick tutorial on artificial neural networks as a starting point for your study of deep learning.

In the remaining columns, a row represents a 28 x 28 image of a handwritten digit, but all pixels are placed in a single row, rather than in the original rectangular form.

The app expects two sets of data: The labels range from 0 to 9, but we will use '10' to represent '0' because MATLAB is indexing is 1-based.

Then you will partition the data so that you hold out 1/3 of the data for model evaluation, and you will only use 2/3 for training our artificial neural network model.

Individual neurons in the hidden layer look like this - 784 inputs and corresponding weights, 1 bias unit, and 10 activation outputs.

If you look inside myNNfun.m, you see variables like IW1_1 and x1_step1_keep that represent the weights your artificial neural network model learned through training.

The general rule of thumb is to pick a number between the number of input neurons, 784 and the number of output neurons, 10, and I just picked 100 arbitrarily.

It looks like you get the best result around 250 neurons and the best score will be around 0.96 with this basic artificial neural network model.

As you can see, you gain more accuracy if you increase the number of hidden neurons, but then the accuracy decreases at some point (your result may differ a bit due to random initialization of weights).

As you increase the number of neurons, your model will be able to capture more features, but if you capture too many features, then you end up overfitting your model to the training data and it won't do well with unseen data.

You now have some intuition on artificial neural networks - a network automatically learns the relevant features from the inputs and generates a sparse representation that maps to the output labels.

In this example we focused on getting a high level intuition on artificial neural network using a concrete example of handwritten digit recognition.

Using neural nets to recognize handwritten digits

Simple intuitions about how we recognize shapes - 'a 9 has a loop at the top, and a vertical stroke in the bottom right' - turn out to be not so simple to express algorithmically.

As a prototype it hits a sweet spot: it's challenging - it's no small feat to recognize handwritten digits - but it's not so difficult as to require an extremely complicated solution, or tremendous computational power.

But along the way we'll develop many key ideas about neural networks, including two important types of artificial neuron (the perceptron and the sigmoid neuron), and the standard learning algorithm for neural networks, known as stochastic gradient descent.

Today, it's more common to use other models of artificial neurons - in this book, and in much modern work on neural networks, the main neuron model used is one called the sigmoid neuron.

A perceptron takes several binary inputs, $x_1, x_2, \ldots$, and produces a single binary output: In the example shown the perceptron has three inputs, $x_1, x_2, x_3$.

The neuron's output, $0$ or $1$, is determined by whether the weighted sum $\sum_j w_j x_j$ is less than or greater than some threshold value.

To put it in more precise algebraic terms: \begin{eqnarray} \mbox{output} & = & \left\{ \begin{array}{ll} 0 & \mbox{if } \sum_j w_j x_j \leq \mbox{ threshold} \\ 1 & \mbox{if } \sum_j w_j x_j > \mbox{ threshold} \end{array} \right.

And it should seem plausible that a complex network of perceptrons could make quite subtle decisions: In this network, the first column of perceptrons - what we'll call the first layer of perceptrons - is making three very simple decisions, by weighing the input evidence.

The first change is to write $\sum_j w_j x_j$ as a dot product, $w \cdot x \equiv \sum_j w_j x_j$, where $w$ and $x$ are vectors whose components are the weights and inputs, respectively.

Using the bias instead of the threshold, the perceptron rule can be rewritten: \begin{eqnarray} \mbox{output} = \left\{ \begin{array}{ll} 0 & \mbox{if } w\cdot x + b \leq 0 \\ 1 & \mbox{if } w\cdot x + b > 0 \end{array} \right.

This requires computing the bitwise sum, $x_1 \oplus x_2$, as well as a carry bit which is set to $1$ when both $x_1$ and $x_2$ are $1$, i.e., the carry bit is just the bitwise product $x_1 x_2$: To get an equivalent network of perceptrons we replace all the NAND gates by perceptrons with two inputs, each with weight $-2$, and an overall bias of $3$.

Note that I've moved the perceptron corresponding to the bottom right NAND gate a little, just to make it easier to draw the arrows on the diagram: One notable aspect of this network of perceptrons is that the output from the leftmost perceptron is used twice as input to the bottommost perceptron.

(If you don't find this obvious, you should stop and prove to yourself that this is equivalent.) With that change, the network looks as follows, with all unmarked weights equal to -2, all biases equal to 3, and a single weight of -4, as marked: Up to now I've been drawing inputs like $x_1$ and $x_2$ as variables floating to the left of the network of perceptrons.

In fact, it's conventional to draw an extra layer of perceptrons - the input layer - to encode the inputs: This notation for input perceptrons, in which we have an output, but no inputs, is a shorthand.

Then the weighted sum $\sum_j w_j x_j$ would always be zero, and so the perceptron would output $1$ if $b > 0$, and $0$ if $b \leq 0$.

Instead of explicitly laying out a circuit of NAND and other gates, our neural networks can simply learn to solve problems, sometimes problems where it would be extremely difficult to directly design a conventional circuit.

If it were true that a small change in a weight (or bias) causes only a small change in output, then we could use this fact to modify the weights and biases to get our network to behave more in the manner we want.

In fact, a small change in the weights or bias of any single perceptron in the network can sometimes cause the output of that perceptron to completely flip, say from $0$ to $1$.

We'll depict sigmoid neurons in the same way we depicted perceptrons: Just like a perceptron, the sigmoid neuron has inputs, $x_1, x_2, \ldots$.

Instead, it's $\sigma(w \cdot x+b)$, where $\sigma$ is called the sigmoid function* *Incidentally, $\sigma$ is sometimes called the logistic function, and this new class of neurons called logistic neurons.

\tag{3}\end{eqnarray} To put it all a little more explicitly, the output of a sigmoid neuron with inputs $x_1,x_2,\ldots$, weights $w_1,w_2,\ldots$, and bias $b$ is \begin{eqnarray} \frac{1}{1+\exp(-\sum_j w_j x_j-b)}.

In fact, there are many similarities between perceptrons and sigmoid neurons, and the algebraic form of the sigmoid function turns out to be more of a technical detail than a true barrier to understanding.

var data = d3.range(sample).map(function(d){ return { x: x1(d), y: s(x1(d))};

var y = d3.scale.linear() .domain([0, 1]) .range([height, 0]);

}) var graph = d3.select('#sigmoid_graph') .append('svg') .attr('width', width + m[1] + m[3]) .attr('height', height + m[0] + m[2]) .append('g') .attr('transform', 'translate(' + m[3] + ',' + m[0] + ')');

var xAxis = d3.svg.axis() .scale(x) .tickValues(d3.range(-4, 5, 1)) .orient('bottom') graph.append('g') .attr('class', 'x axis') .attr('transform', 'translate(0, ' + height + ')') .call(xAxis);

var yAxis = d3.svg.axis() .scale(y) .tickValues(d3.range(0, 1.01, 0.2)) .orient('left') .ticks(5) graph.append('g') .attr('class', 'y axis') .call(yAxis);

graph.append('text') .attr('class', 'x label') .attr('text-anchor', 'end') .attr('x', width/2) .attr('y', height+35) .text('z');

graph.append('text') .attr('x', (width / 2)) .attr('y', -10) .attr('text-anchor', 'middle') .style('font-size', '16px') .text('sigmoid function');

var data = d3.range(sample).map(function(d){ return { x: x1(d), y: s(x1(d))};

var y = d3.scale.linear() .domain([0,1]) .range([height, 0]);

}) var graph = d3.select('#step_graph') .append('svg') .attr('width', width + m[1] + m[3]) .attr('height', height + m[0] + m[2]) .append('g') .attr('transform', 'translate(' + m[3] + ',' + m[0] + ')');

var xAxis = d3.svg.axis() .scale(x) .tickValues(d3.range(-4, 5, 1)) .orient('bottom') graph.append('g') .attr('class', 'x axis') .attr('transform', 'translate(0, ' + height + ')') .call(xAxis);

var yAxis = d3.svg.axis() .scale(y) .tickValues(d3.range(0, 1.01, 0.2)) .orient('left') .ticks(5) graph.append('g') .attr('class', 'y axis') .call(yAxis);

graph.append('text') .attr('class', 'x label') .attr('text-anchor', 'end') .attr('x', width/2) .attr('y', height+35) .text('z');

graph.append('text') .attr('x', (width / 2)) .attr('y', -10) .attr('text-anchor', 'middle') .style('font-size', '16px') .text('step function');

If $\sigma$ had in fact been a step function, then the sigmoid neuron would be a perceptron, since the output would be $1$ or $0$ depending on whether $w\cdot x+b$ was positive or negative* *Actually, when $w \cdot x +b = 0$ the perceptron outputs $0$, while the step function outputs $1$.

The smoothness of $\sigma$ means that small changes $\Delta w_j$ in the weights and $\Delta b$ in the bias will produce a small change $\Delta \mbox{output}$ in the output from the neuron.

In fact, calculus tells us that $\Delta \mbox{output}$ is well approximated by \begin{eqnarray} \Delta \mbox{output} \approx \sum_j \frac{\partial \, \mbox{output}}{\partial w_j} \Delta w_j + \frac{\partial \, \mbox{output}}{\partial b} \Delta b, \tag{5}\end{eqnarray} where the sum is over all the weights, $w_j$, and $\partial \, \mbox{output} / \partial w_j$ and $\partial \, \mbox{output} /\partial b$ denote partial derivatives of the $\mbox{output}$ with respect to $w_j$ and $b$, respectively.

While the expression above looks complicated, with all the partial derivatives, it's actually saying something very simple (and which is very good news): $\Delta \mbox{output}$ is a linear function of the changes $\Delta w_j$ and $\Delta b$ in the weights and bias.

If it's the shape of $\sigma$ which really matters, and not its exact form, then why use the particular form used for $\sigma$ in Equation (3)\begin{eqnarray} \sigma(z) \equiv \frac{1}{1+e^{-z}} \nonumber\end{eqnarray}$('#margin_778862672352_reveal').click(function() {$('#margin_778862672352').toggle('slow', function() {});});?

In fact, later in the book we will occasionally consider neurons where the output is $f(w \cdot x + b)$ for some other activation function $f(\cdot)$.

The main thing that changes when we use a different activation function is that the particular values for the partial derivatives in Equation (5)\begin{eqnarray} \Delta \mbox{output} \approx \sum_j \frac{\partial \, \mbox{output}}{\partial w_j} \Delta w_j + \frac{\partial \, \mbox{output}}{\partial b} \Delta b \nonumber\end{eqnarray}$('#margin_726336021933_reveal').click(function() {$('#margin_726336021933').toggle('slow', function() {});});

It turns out that when we compute those partial derivatives later, using $\sigma$ will simplify the algebra, simply because exponentials have lovely properties when differentiated.

But in practice we can set up a convention to deal with this, for example, by deciding to interpret any output of at least $0.5$ as indicating a '9', and any output less than $0.5$ as indicating 'not a 9'.

Exercises Sigmoid neurons simulating perceptrons, part I $\mbox{}$ Suppose we take all the weights and biases in a network of perceptrons, and multiply them by a positive constant, $c > 0$.

Show that the behaviour of the network doesn't change.Sigmoid neurons simulating perceptrons, part II $\mbox{}$ Suppose we have the same setup as the last problem - a network of perceptrons.

Suppose the weights and biases are such that $w \cdot x + b \neq 0$ for the input $x$ to any particular perceptron in the network.

Now replace all the perceptrons in the network by sigmoid neurons, and multiply the weights and biases by a positive constant $c > 0$.

Suppose we have the network: As mentioned earlier, the leftmost layer in this network is called the input layer, and the neurons within the layer are called input neurons.

The term 'hidden' perhaps sounds a little mysterious - the first time I heard the term I thought it must have some deep philosophical or mathematical significance - but it really means nothing more than 'not an input or an output'.

For example, the following four-layer network has two hidden layers: Somewhat confusingly, and for historical reasons, such multiple layer networks are sometimes called multilayer perceptrons or MLPs, despite being made up of sigmoid neurons, not perceptrons.

If the image is a $64$ by $64$ greyscale image, then we'd have $4,096 = 64 \times 64$ input neurons, with the intensities scaled appropriately between $0$ and $1$.

The output layer will contain just a single neuron, with output values of less than $0.5$ indicating 'input image is not a 9', and values greater than $0.5$ indicating 'input image is a 9 '.

A trial segmentation gets a high score if the individual digit classifier is confident of its classification in all segments, and a low score if the classifier is having a lot of trouble in one or more segments.

So instead of worrying about segmentation we'll concentrate on developing a neural network which can solve the more interesting and difficult problem, namely, recognizing individual handwritten digits.

As discussed in the next section, our training data for the network will consist of many $28$ by $28$ pixel images of scanned handwritten digits, and so the input layer contains $784 = 28 \times 28$ neurons.

The input pixels are greyscale, with a value of $0.0$ representing white, a value of $1.0$ representing black, and in between values representing gradually darkening shades of grey.

A seemingly natural way of doing that is to use just $4$ output neurons, treating each neuron as taking on a binary value, depending on whether the neuron's output is closer to $0$ or to $1$.

The ultimate justification is empirical: we can try out both network designs, and it turns out that, for this particular problem, the network with $10$ output neurons learns to recognize digits better than the network with $4$ output neurons.

In a similar way, let's suppose for the sake of argument that the second, third, and fourth neurons in the hidden layer detect whether or not the following images are present:

Of course, that's not the only sort of evidence we can use to conclude that the image was a $0$ - we could legitimately get a $0$ in many other ways (say, through translations of the above images, or slight distortions).

Assume that the first $3$ layers of neurons are such that the correct output in the third layer (i.e., the old output layer) has activation at least $0.99$, and incorrect outputs have activation less than $0.01$.

We'll use the MNIST data set, which contains tens of thousands of scanned images of handwritten digits, together with their correct classifications.

To make this a good test of performance, the test data was taken from a different set of 250 people than the original training data (albeit still a group split between Census Bureau employees and high school students).

For example, if a particular training image, $x$, depicts a $6$, then $y(x) = (0, 0, 0, 0, 0, 0, 1, 0, 0, 0)^T$ is the desired output from the network.

We use the term cost function throughout this book, but you should note the other terminology, since it's often used in research papers and other discussions of neural networks.

\tag{6}\end{eqnarray} 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, $x$.

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.

Even given that we want to use a smooth cost function, you may still wonder why we choose the quadratic function used in Equation (6)\begin{eqnarray} C(w,b) \equiv \frac{1}{2n} \sum_x \|

This is a well-posed problem, but it's got a lot of distracting structure as currently posed - the interpretation of $w$ and $b$ as weights and biases, the $\sigma$ function lurking in the background, the choice of network architecture, MNIST, and so on.

And for neural networks we'll often want far more variables - the biggest neural networks have cost functions which depend on billions of weights and biases in an extremely complicated way.

We could do this simulation simply by computing derivatives (and perhaps some second derivatives) of $C$ - those derivatives would tell us everything we need to know about the local 'shape' of the valley, and therefore how our ball should roll.

So rather than get into all the messy details of physics, let's simply ask ourselves: if we were declared God for a day, and could make up our own laws of physics, dictating to the ball how it should roll, what law or laws of motion could we pick that would make it so the ball always rolled to the bottom of the valley?

To make this question more precise, let's think about what happens when we move the ball a small amount $\Delta v_1$ in the $v_1$ direction, and a small amount $\Delta v_2$ in the $v_2$ direction.

Calculus tells us that $C$ changes as follows: \begin{eqnarray} \Delta C \approx \frac{\partial C}{\partial v_1} \Delta v_1 + \frac{\partial C}{\partial v_2} \Delta v_2.

To figure out how to make such a choice it helps to define $\Delta v$ to be the vector of changes in $v$, $\Delta v \equiv (\Delta v_1, \Delta v_2)^T$, where $T$ is again the transpose operation, turning row vectors into column vectors.

We denote the gradient vector by $\nabla C$, i.e.: \begin{eqnarray} \nabla C \equiv \left( \frac{\partial C}{\partial v_1}, \frac{\partial C}{\partial v_2} \right)^T.

In fact, it's perfectly fine to think of $\nabla C$ as a single mathematical object - the vector defined above - which happens to be written using two symbols.

With these definitions, the expression (7)\begin{eqnarray} \Delta C \approx \frac{\partial C}{\partial v_1} \Delta v_1 + \frac{\partial C}{\partial v_2} \Delta v_2 \nonumber\end{eqnarray}$('#margin_832985330775_reveal').click(function() {$('#margin_832985330775').toggle('slow', function() {});});

\tag{9}\end{eqnarray} This equation helps explain why $\nabla C$ is called the gradient vector: $\nabla C$ relates changes in $v$ to changes in $C$, just as we'd expect something called a gradient to do.

In particular, suppose we choose \begin{eqnarray} \Delta v = -\eta \nabla C, \tag{10}\end{eqnarray} where $\eta$ is a small, positive parameter (known as the learning rate).

\nabla C \|^2 \geq 0$, this guarantees that $\Delta C \leq 0$, i.e., $C$ will always decrease, never increase, if we change $v$ according to the prescription in (10)\begin{eqnarray} \Delta v = -\eta \nabla C \nonumber\end{eqnarray}$('#margin_39079991636_reveal').click(function() {$('#margin_39079991636').toggle('slow', function() {});});.

to compute a value for $\Delta v$, then move the ball's position $v$ by that amount: \begin{eqnarray} v \rightarrow v' = v -\eta \nabla C.

To make gradient descent work correctly, we need to choose the learning rate $\eta$ to be small enough that Equation (9)\begin{eqnarray} \Delta C \approx \nabla C \cdot \Delta v \nonumber\end{eqnarray}$('#margin_663076476028_reveal').click(function() {$('#margin_663076476028').toggle('slow', function() {});});

In practical implementations, $\eta$ is often varied so that Equation (9)\begin{eqnarray} \Delta C \approx \nabla C \cdot \Delta v \nonumber\end{eqnarray}$('#margin_362932327599_reveal').click(function() {$('#margin_362932327599').toggle('slow', function() {});});

Then the change $\Delta C$ in $C$ produced by a small change $\Delta v = (\Delta v_1, \ldots, \Delta v_m)^T$ is \begin{eqnarray} \Delta C \approx \nabla C \cdot \Delta v, \tag{12}\end{eqnarray} where the gradient $\nabla C$ is the vector \begin{eqnarray} \nabla C \equiv \left(\frac{\partial C}{\partial v_1}, \ldots, \frac{\partial C}{\partial v_m}\right)^T.

\tag{13}\end{eqnarray} Just as for the two variable case, we can choose \begin{eqnarray} \Delta v = -\eta \nabla C, \tag{14}\end{eqnarray} and we're guaranteed that our (approximate) expression (12)\begin{eqnarray} \Delta C \approx \nabla C \cdot \Delta v \nonumber\end{eqnarray}$('#margin_398945612724_reveal').click(function() {$('#margin_398945612724').toggle('slow', function() {});});

This gives us a way of following the gradient to a minimum, even when $C$ is a function of many variables, by repeatedly applying the update rule \begin{eqnarray} v \rightarrow v' = v-\eta \nabla C.

The rule doesn't always work - several things can go wrong and prevent gradient descent from finding the global minimum of $C$, a point we'll return to explore in later chapters.

But, in practice gradient descent often works extremely well, and in neural networks we'll find that it's a powerful way of minimizing the cost function, and so helping the net learn.

It can be proved that the choice of $\Delta v$ which minimizes $\nabla C \cdot \Delta v$ is $\Delta v = - \eta \nabla C$, where $\eta = \epsilon / \|\nabla C\|$ is determined by the size constraint $\|\Delta v\|

Hint: If you're not already familiar with the Cauchy-Schwarz inequality, you may find it helpful to familiarize yourself with it.

If there are a million such $v_j$ variables then we'd need to compute something like a trillion (i.e., a million squared) second partial derivatives* *Actually, more like half a trillion, since $\partial^2 C/ \partial v_j \partial v_k = \partial^2 C/ \partial v_k \partial v_j$.

The idea is to use gradient descent to find the weights $w_k$ and biases $b_l$ which minimize the cost in Equation (6)\begin{eqnarray} C(w,b) \equiv \frac{1}{2n} \sum_x \|

In other words, our 'position' now has components $w_k$ and $b_l$, and the gradient vector $\nabla C$ has corresponding components $\partial C / \partial w_k$ and $\partial C / \partial b_l$.

Writing out the gradient descent update rule in terms of components, we have \begin{eqnarray} w_k & \rightarrow & w_k' = w_k-\eta \frac{\partial C}{\partial w_k} \tag{16}\\ b_l & \rightarrow & b_l' = b_l-\eta \frac{\partial C}{\partial b_l}.

In practice, to compute the gradient $\nabla C$ we need to compute the gradients $\nabla C_x$ separately for each training input, $x$, and then average them, $\nabla C = \frac{1}{n} \sum_x \nabla C_x$.

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 $m$ is large enough we expect that the average value of the $\nabla C_{X_j}$ will be roughly equal to the average over all $\nabla C_x$, that is, \begin{eqnarray} \frac{\sum_{j=1}^m \nabla C_{X_{j}}}{m} \approx \frac{\sum_x \nabla C_x}{n} = \nabla C, \tag{18}\end{eqnarray} where the second sum is over the entire set of training data.

Swapping sides we get \begin{eqnarray} \nabla C \approx \frac{1}{m} \sum_{j=1}^m \nabla C_{X_{j}}, \tag{19}\end{eqnarray} confirming that we can estimate the overall gradient by computing gradients just for the randomly chosen mini-batch.

Then stochastic gradient descent works by picking out a randomly chosen mini-batch of training inputs, and training with those, \begin{eqnarray} w_k & \rightarrow & w_k' = w_k-\frac{\eta}{m} \sum_j \frac{\partial C_{X_j}}{\partial w_k} \tag{20}\\ b_l & \rightarrow & b_l' = b_l-\frac{\eta}{m} \sum_j \frac{\partial C_{X_j}}{\partial b_l}, \tag{21}\end{eqnarray} where the sums are over all the training examples $X_j$ in the current mini-batch.

And, in a similar way, the mini-batch update rules (20)\begin{eqnarray} w_k & \rightarrow & w_k' = w_k-\frac{\eta}{m} \sum_j \frac{\partial C_{X_j}}{\partial w_k} \nonumber\end{eqnarray}$('#margin_255037324417_reveal').click(function() {$('#margin_255037324417').toggle('slow', function() {});});

and (21)\begin{eqnarray} b_l & \rightarrow & b_l' = b_l-\frac{\eta}{m} \sum_j \frac{\partial C_{X_j}}{\partial b_l} \nonumber\end{eqnarray}$('#margin_141169455106_reveal').click(function() {$('#margin_141169455106').toggle('slow', function() {});});

We can think of stochastic gradient descent as being like political polling: it's much easier to sample a small mini-batch than it is to apply gradient descent to the full batch, just as carrying out a poll is easier than running a full election.

For example, if we have a training set of size $n = 60,000$, as in MNIST, and choose a mini-batch size of (say) $m = 10$, this means we'll get a factor of $6,000$ speedup in estimating the gradient!

Of course, the estimate won't be perfect - there will be statistical fluctuations - but it doesn't need to be perfect: all we really care about is moving in a general direction that will help decrease $C$, and that means we don't need an exact computation of the gradient.

In practice, stochastic gradient descent is a commonly used and powerful technique for learning in neural networks, and it's the basis for most of the learning techniques we'll develop in this book.

That is, given a training input, $x$, we update our weights and biases according to the rules $w_k \rightarrow w_k' = w_k - \eta \partial C_x / \partial w_k$ and $b_l \rightarrow b_l' = b_l - \eta \partial C_x / \partial b_l$.

Name one advantage and one disadvantage of online learning, compared to stochastic gradient descent with a mini-batch size of, say, $20$.

In neural networks the cost $C$ is, of course, a function of many variables - all the weights and biases - and so in some sense defines a surface in a very high-dimensional space.

I won't go into more detail here, but if you're interested then you may enjoy reading this discussion of some of the techniques professional mathematicians use to think in high dimensions.

We'll leave the test images as is, but split the 60,000-image MNIST training set into two parts: a set of 50,000 images, which we'll use to train our neural network, and a separate 10,000 image validation set.

We won't use the validation data in this chapter, but later in the book we'll find it useful in figuring out how to set certain hyper-parameters of the neural network - things like the learning rate, and so on, which aren't directly selected by our learning algorithm.

When I refer to the 'MNIST training data' from now on, I'll be referring to our 50,000 image data set, not the original 60,000 image data set* *As noted earlier, the MNIST data set is based on two data sets collected by NIST, the United States' National Institute of Standards and Technology.

for x, y in zip(sizes[:-1], sizes[1:])]

So, for example, if we want to create a Network object with 2 neurons in the first layer, 3 neurons in the second layer, and 1 neuron in the final layer, we'd do this with the code: net = Network([2, 3, 1])

The biases and weights in the Network object are all initialized randomly, using the Numpy np.random.randn function to generate Gaussian distributions with mean $0$ and standard deviation $1$.

Note that the Network initialization code assumes that the first layer of neurons is an input layer, and omits to set any biases for those neurons, since biases are only ever used in computing the outputs from later layers.

The big advantage of using this ordering is that it means that the vector of activations of the third layer of neurons is: \begin{eqnarray} a' = \sigma(w a + b).

(This is called vectorizing the function $\sigma$.) It's easy to verify that Equation (22)\begin{eqnarray} a' = \sigma(w a + b) \nonumber\end{eqnarray}$('#margin_552886241220_reveal').click(function() {$('#margin_552886241220').toggle('slow', function() {});});

gives the same result as our earlier rule, Equation (4)\begin{eqnarray} \frac{1}{1+\exp(-\sum_j w_j x_j-b)} \nonumber\end{eqnarray}$('#margin_7421600236_reveal').click(function() {$('#margin_7421600236').toggle('slow', function() {});});, for computing the output of a sigmoid neuron.

in component form, and verify that it gives the same result as the rule (4)\begin{eqnarray} \frac{1}{1+\exp(-\sum_j w_j x_j-b)} \nonumber\end{eqnarray}$('#margin_347257101140_reveal').click(function() {$('#margin_347257101140').toggle('slow', function() {});});

We then add a feedforward method to the Network class, which, given an input a for the network, returns the corresponding output* *It is assumed that the input a is an (n, 1) Numpy ndarray, not a (n,) vector.

Although using an (n,) vector appears the more natural choice, using an (n, 1) ndarray makes it particularly easy to modify the code to feedforward multiple inputs at once, and that is sometimes convenient.

All the method does is applies Equation (22)\begin{eqnarray} a' = \sigma(w a + b) \nonumber\end{eqnarray}$('#margin_335258165235_reveal').click(function() {$('#margin_335258165235').toggle('slow', function() {});});

training_data[k:k+mini_batch_size]

for k in xrange(0, n, mini_batch_size)]

self.update_mini_batch(mini_batch, eta)

print "Epoch {0}: {1} / {2}".format(

j, self.evaluate(test_data), n_test)

print "Epoch {0} complete".format(j)

This is done by the code self.update_mini_batch(mini_batch, eta), which updates the network weights and biases according to a single iteration of gradient descent, using just the training data in mini_batch.

delta_nabla_b, delta_nabla_w = self.backprop(x, y)

nabla_b = [nb+dnb for nb, dnb in zip(nabla_b, delta_nabla_b)]

nabla_w = [nw+dnw for nw, dnw in zip(nabla_w, delta_nabla_w)]

for w, nw in zip(self.weights, nabla_w)]

for b, nb in zip(self.biases, nabla_b)]

Most of the work is done by the line delta_nabla_b, delta_nabla_w = self.backprop(x, y)

The self.backprop method makes use of a few extra functions to help in computing the gradient, namely sigmoid_prime, which computes the derivative of the $\sigma$ function, and self.cost_derivative, which I won't describe here.

for x, y in zip(sizes[:-1], sizes[1:])]

training_data[k:k+mini_batch_size]

for k in xrange(0, n, mini_batch_size)]

self.update_mini_batch(mini_batch, eta)

print "Epoch {0}: {1} / {2}".format(

j, self.evaluate(test_data), n_test)

print "Epoch {0} complete".format(j)

delta_nabla_b, delta_nabla_w = self.backprop(x, y)

nabla_b = [nb+dnb for nb, dnb in zip(nabla_b, delta_nabla_b)]

nabla_w = [nw+dnw for nw, dnw in zip(nabla_w, delta_nabla_w)]

for w, nw in zip(self.weights, nabla_w)]

for b, nb in zip(self.biases, nabla_b)]

activations = [x] # list to store all the activations, layer by layer

zs = [] # list to store all the z vectors, layer by layer

# l = 1 means the last layer of neurons, l = 2 is the

delta = np.dot(self.weights[-l+1].transpose(), delta) * sp

for (x, y) in test_data]

Finally, we'll use stochastic gradient descent to learn from the MNIST training_data over 30 epochs, with a mini-batch size of 10, and a learning rate of $\eta = 3.0$, >>>

As was the case earlier, if you're running the code as you read along, you should be warned that it takes quite a while to execute (on my machine this experiment takes tens of seconds for each training epoch), so it's wise to continue reading in parallel while the code executes.

At least in this case, using more hidden neurons helps us get better results* *Reader feedback indicates quite some variation in results for this experiment, and some training runs give results quite a bit worse.

Using the techniques introduced in chapter 3 will greatly reduce the variation in performance across different training runs for our networks..

(If making a change improves things, try doing more!) If we do that several times over, we'll end up with a learning rate of something like $\eta = 1.0$ (and perhaps fine tune to $3.0$), which is close to our earlier experiments.

Exercise Try creating a network with just two layers - an input and an output layer, no hidden layer - with 784 and 10 neurons, respectively.

The data structures used to store the MNIST data are described in the documentation strings - it's straightforward stuff, tuples and lists of Numpy ndarray objects (think of them as vectors if you're not familiar with ndarrays): """mnist_loader~~~~~~~~~~~~A library to load the MNIST image data.

In some sense, the moral of both our results and those in more sophisticated papers, is that for some problems: sophisticated algorithm $\leq$ simple learning algorithm + good training data.

We could attack this problem the same way we attacked handwriting recognition - by using the pixels in the image as input to a neural network, with the output from the network a single neuron indicating either 'Yes, it's a face' or 'No, it's not a face'.

The end result is a network which breaks down a very complicated question - does this image show a face or not - into very simple questions answerable at the level of single pixels.

It does this through a series of many layers, with early layers answering very simple and specific questions about the input image, and later layers building up a hierarchy of ever more complex and abstract concepts.

Comparing a deep network to a shallow network is a bit like comparing a programming language with the ability to make function calls to a stripped down language with no ability to make such calls.

Build your First Deep Learning Neural Network Model using Keras in Python

I have chosen my today’s topic as Neural Network because it is most the fascinating learning model in the world of data science and starters in Data Science think that Neural Network is difficult and its understanding requires knowledge of neurons, perceptron and blaahhhh…There is nothing like that, I have been working on Neural Network for quite a few month now and realized that it is so easy.

The fact of the matter is Keras is built on top of Tensorflow and Theano so this 2 insane library will be running in back-end whenever you run the program in Keras.

Deep understanding of NN(you can skip this if you don’t want to learn in depth) Now you can see that Country names are replaced by 0,1 and 2 while male and female are replaced by 0 and 1.

Dummy variable is difficult concept if you read in depth but don’t take tension, I have found this simple resource which will help you in understanding.

In Machine Learning, we always divide our data into training and testing part meaning that we train our model on training data and then we check the accuracy of a model on testing data.

Here we are using rectifier(relu) function in our hidden layer and Sigmoid function in our output layer as we want binary result from output layer but if the number of categories in output layer is more than 2 then use SoftMax function.

First argument is Optimizer, this is nothing but the algorithm you wanna use to find optimal set of weights(Note that in step 9 we just initialized weights now we are applying some sort of algorithm which will optimize weights in turn making out neural network more powerful.

Since out dependent variable is binary, we will have to use logarithmic loss function called ‘binary_crossentropy’, if our dependent variable has more than 2 categories in output then use ‘categorical_crossentropy’.

Dynamically Expandable Neural Networks — Explained

Neural networks can learn complicated representations fairly easily.

This article talks about a very recent technique that attempts to constantly adapt to new data at a fraction of the cost of retraining entire models.

Usually, techniques like transfer learning are used, where the model is trained on previous data, and some features are used from that model to learn new data.

However, if the new task is very different from the old tasks, the model will not be able to perform well on that new task, as features from the old task are not useful, e.g.

Another problem is that after fine-tuning, the model may begin to perform the original task poorly (in this example, predicting animals).

If a new task arrives that is vastly different from an existing task, extract whatever useful information you can from the old model and train a new model.The authors used these logical ideas and developed techniques to make such a construct possible.

When the next task needs to be learned, a sparse linear classifier is fit on the last layer of the model, then the network is trained using: The notation: This means the weights of all the layers except the last layer.

The new weight matrix for that layer (and the previous layer ) will look have dimensions: 𝒩 is the total number of neurons after adding the k neurons.

won’t go into detail here, but using that on a layer gives such results (Group Lasso is the technique that was used): The authors used a layer basis (only on the newly added k neurons) instead of the entire network.

There is a common problem in transfer learning called semantic drift, or catastrophic forgetting, where the model slowly shifts its weights so much that it forgets about the original tasks.

Although it is possible to add L2 regularization, which ensures that the weights don’t shift dramatically, it won’t help if the new tasks are very different (the model will just fail to learn after a certain point).

If the value of a neuron changes beyond a certain value, a copy of the neuron is made, and a split occurs, and that duplicate unit is added as a copy to that same layer.

Artificial neural network

Artificial neural networks (ANN) or connectionist systems are computing systems vaguely inspired by the biological neural networks that constitute animal brains.[1]

For example, in image recognition, they might learn to identify images that contain cats by analyzing example images that have been manually labeled as 'cat' or 'no cat' and using the results to identify cats in other images.

An ANN is based on a collection of connected units or nodes called artificial neurons which loosely model the neurons in a biological brain.

In common ANN implementations, the signal at a connection between artificial neurons is a real number, and the output of each artificial neuron is computed by some non-linear function of the sum of its inputs.

Signals travel from the first layer (the input layer), to the last layer (the output layer), possibly after traversing the layers multiple times.

Artificial neural networks have been used on a variety of tasks, including computer vision, speech recognition, machine translation, social network filtering, playing board and video games and medical diagnosis.

(1943) created a computational model for neural networks based on mathematics and algorithms called threshold logic.

With mathematical notation, Rosenblatt described circuitry not in the basic perceptron, such as the exclusive-or circuit that could not be processed by neural networks at the time.[8]

In 1959, a biological model proposed by Nobel laureates Hubel and Wiesel was based on their discovery of two types of cells in the primary visual cortex: simple cells and complex cells.[9]

Much of artificial intelligence had focused on high-level (symbolic) models that are processed by using algorithms, characterized for example by expert systems with knowledge embodied in if-then rules, until in the late 1980s research expanded to low-level (sub-symbolic) machine learning, characterized by knowledge embodied in the parameters of a cognitive model.[citation needed]

key trigger for renewed interest in neural networks and learning was Werbos's (1975) backpropagation algorithm that effectively solved the exclusive-or problem and more generally accelerated the training of multi-layer networks.

Support vector machines and other, much simpler methods such as linear classifiers gradually overtook neural networks in machine learning popularity.

The vanishing gradient problem affects many-layered feedforward networks that used backpropagation and also recurrent neural networks (RNNs).[21][22]

As errors propagate from layer to layer, they shrink exponentially with the number of layers, impeding the tuning of neuron weights that is based on those errors, particularly affecting deep networks.

To overcome this problem, Schmidhuber adopted a multi-level hierarchy of networks (1992) pre-trained one level at a time by unsupervised learning and fine-tuned by backpropagation.[23]

(2006) proposed learning a high-level representation using successive layers of binary or real-valued latent variables with a restricted Boltzmann machine[25]

Once sufficiently many layers have been learned, the deep architecture may be used as a generative model by reproducing the data when sampling down the model (an 'ancestral pass') from the top level feature activations.[26][27]

In 2012, Ng and Dean created a network that learned to recognize higher-level concepts, such as cats, only from watching unlabeled images taken from YouTube videos.[28]

Earlier challenges in training deep neural networks were successfully addressed with methods such as unsupervised pre-training, while available computing power increased through the use of GPUs and distributed computing.

for very large scale principal components analyses and convolution may create a new class of neural computing because they are fundamentally analog rather than digital (even though the first implementations may use digital devices).[30]

in Schmidhuber's group showed that despite the vanishing gradient problem, GPUs makes back-propagation feasible for many-layered feedforward neural networks.

Between 2009 and 2012, recurrent neural networks and deep feedforward neural networks developed in Schmidhuber's research group won eight international competitions in pattern recognition and machine learning.[32][33]

Researchers demonstrated (2010) that deep neural networks interfaced to a hidden Markov model with context-dependent states that define the neural network output layer can drastically reduce errors in large-vocabulary speech recognition tasks such as voice search.

A team from his lab won a 2012 contest sponsored by Merck to design software to help find molecules that might identify new drugs.[46]

As of 2011, the state of the art in deep learning feedforward networks alternated between convolutional layers and max-pooling layers,[41][47]

Artificial neural networks were able to guarantee shift invariance to deal with small and large natural objects in large cluttered scenes, only when invariance extended beyond shift, to all ANN-learned concepts, such as location, type (object class label), scale, lighting and others.

An artificial neural network is a network of simple elements called artificial neurons, which receive input, change their internal state (activation) according to that input, and produce output depending on the input and activation.

The network forms by connecting the output of certain neurons to the input of other neurons forming a directed, weighted graph.

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The learning rule is a rule or an algorithm which modifies the parameters of the neural network, in order for a given input to the network to produce a favored output.

A common use of the phrase 'ANN model' is really the definition of a class of such functions (where members of the class are obtained by varying parameters, connection weights, or specifics of the architecture such as the number of neurons or their connectivity).

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(commonly referred to as the activation function[52]) is some predefined function, such as the hyperbolic tangent or sigmoid function or softmax function or rectifier function.

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is an important concept in learning, as it is a measure of how far away a particular solution is from an optimal solution to the problem to be solved.

For applications where the solution is data dependent, the cost must necessarily be a function of the observations, otherwise the model would not relate to the data.

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While it is possible to define an ad hoc cost function, frequently a particular cost (function) is used, either because it has desirable properties (such as convexity) or because it arises naturally from a particular formulation of the problem (e.g., in a probabilistic formulation the posterior probability of the model can be used as an inverse cost).

In 1970, Linnainmaa finally published the general method for automatic differentiation (AD) of discrete connected networks of nested differentiable functions.[61][62]

In 1986, Rumelhart, Hinton and Williams noted that this method can generate useful internal representations of incoming data in hidden layers of neural networks.[68]

The choice of the cost function depends on factors such as the learning type (supervised, unsupervised, reinforcement, etc.) and the activation function.

For example, when performing supervised learning on a multiclass classification problem, common choices for the activation function and cost function are the softmax function and cross entropy function, respectively.

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The network is trained to minimize L2 error for predicting the mask ranging over the entire training set containing bounding boxes represented as masks.

the cost function is related to the mismatch between our mapping and the data and it implicitly contains prior knowledge about the problem domain.[76]

commonly used cost is the mean-squared error, which tries to minimize the average squared error between the network's output,

Minimizing this cost using gradient descent for the class of neural networks called multilayer perceptrons (MLP), produces the backpropagation algorithm for training neural networks.

Tasks that fall within the paradigm of supervised learning are pattern recognition (also known as classification) and regression (also known as function approximation).

The supervised learning paradigm is also applicable to sequential data (e.g., for hand writing, speech and gesture recognition).

This can be thought of as learning with a 'teacher', in the form of a function that provides continuous feedback on the quality of solutions obtained thus far.

The cost function is dependent on the task (the model domain) and any a priori assumptions (the implicit properties of the model, its parameters and the observed variables).

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whereas in statistical modeling, it could be related to the posterior probability of the model given the data (note that in both of those examples those quantities would be maximized rather than minimized).

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The aim is to discover a policy for selecting actions that minimizes some measure of a long-term cost, e.g., the expected cumulative cost.

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because of the ability of Artificial neural networks to mitigate losses of accuracy even when reducing the discretization grid density for numerically approximating the solution of the original control problems.

Tasks that fall within the paradigm of reinforcement learning are control problems, games and other sequential decision making tasks.

Training a neural network model essentially means selecting one model from the set of allowed models (or, in a Bayesian framework, determining a distribution over the set of allowed models) that minimizes the cost.

This is done by simply taking the derivative of the cost function with respect to the network parameters and then changing those parameters in a gradient-related direction.

convolutional neural network (CNN) is a class of deep, feed-forward networks, composed of one or more convolutional layers with fully connected layers (matching those in typical Artificial neural networks) on top.

can find an RNN weight matrix that maximizes the probability of the label sequences in a training set, given the corresponding input sequences.

provide a framework for efficiently trained models for hierarchical processing of temporal data, while enabling the investigation of the inherent role of RNN layered composition.[clarification needed]

This is particularly helpful when training data are limited, because poorly initialized weights can significantly hinder model performance.

that integrate the various and usually different filters (preprocessing functions) into its many layers and to dynamically rank the significance of the various layers and functions relative to a given learning task.

This grossly imitates biological learning which integrates various preprocessors (cochlea, retina, etc.) and cortexes (auditory, visual, etc.) and their various regions.

Its deep learning capability is further enhanced by using inhibition, correlation and its ability to cope with incomplete data, or 'lost' neurons or layers even amidst a task.

The link-weights allow dynamic determination of innovation and redundancy, and facilitate the ranking of layers, of filters or of individual neurons relative to a task.

LAMSTAR had a much faster learning speed and somewhat lower error rate than a CNN based on ReLU-function filters and max pooling, in 20 comparative studies.[136]

These applications demonstrate delving into aspects of the data that are hidden from shallow learning networks and the human senses, such as in the cases of predicting onset of sleep apnea events,[128]

The whole process of auto encoding is to compare this reconstructed input to the original and try to minimize the error to make the reconstructed value as close as possible to the original.

with a specific approach to good representation, a good representation is one that can be obtained robustly from a corrupted input and that will be useful for recovering the corresponding clean input.

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of the first denoising auto encoder is learned and used to uncorrupt the input (corrupted input), the second level can be trained.[142]

Once the stacked auto encoder is trained, its output can be used as the input to a supervised learning algorithm such as support vector machine classifier or a multi-class logistic regression.[142]

It formulates the learning as a convex optimization problem with a closed-form solution, emphasizing the mechanism's similarity to stacked generalization.[146]

Each block estimates the same final label class y, and its estimate is concatenated with original input X to form the expanded input for the next block.

Thus, the input to the first block contains the original data only, while downstream blocks' input adds the output of preceding blocks.

It offers two important improvements: it uses higher-order information from covariance statistics, and it transforms the non-convex problem of a lower-layer to a convex sub-problem of an upper-layer.[148]

TDSNs use covariance statistics in a bilinear mapping from each of two distinct sets of hidden units in the same layer to predictions, via a third-order tensor.

The need for deep learning with real-valued inputs, as in Gaussian restricted Boltzmann machines, led to the spike-and-slab RBM (ssRBM), which models continuous-valued inputs with strictly binary latent variables.[152]

One of these terms enables the model to form a conditional distribution of the spike variables by marginalizing out the slab variables given an observation.

However, these architectures are poor at learning novel classes with few examples, because all network units are involved in representing the input (a distributed representation) and must be adjusted together (high degree of freedom).

It is a full generative model, generalized from abstract concepts flowing through the layers of the model, which is able to synthesize new examples in novel classes that look 'reasonably' natural.

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deep predictive coding network (DPCN) is a predictive coding scheme that uses top-down information to empirically adjust the priors needed for a bottom-up inference procedure by means of a deep, locally connected, generative model.

DPCNs predict the representation of the layer, by using a top-down approach using the information in upper layer and temporal dependencies from previous states.[170]

For example, in sparse distributed memory or hierarchical temporal memory, the patterns encoded by neural networks are used as addresses for content-addressable memory, with 'neurons' essentially serving as address encoders and decoders.

Preliminary results demonstrate that neural Turing machines can infer simple algorithms such as copying, sorting and associative recall from input and output examples.

Approaches that represent previous experiences directly and use a similar experience to form a local model are often called nearest neighbour or k-nearest neighbors methods.[185]

Unlike sparse distributed memory that operates on 1000-bit addresses, semantic hashing works on 32 or 64-bit addresses found in a conventional computer architecture.

These models have been applied in the context of question answering (QA) where the long-term memory effectively acts as a (dynamic) knowledge base and the output is a textual response.[190]

While training extremely deep (e.g., 1 million layers) neural networks might not be practical, CPU-like architectures such as pointer networks[192]

overcome this limitation by using external random-access memory and other components that typically belong to a computer architecture such as registers, ALU and pointers.

The key characteristic of these models is that their depth, the size of their short-term memory, and the number of parameters can be altered independently – unlike models like LSTM, whose number of parameters grows quadratically with memory size.

In that work, an LSTM RNN or CNN was used as an encoder to summarize a source sentence, and the summary was decoded using a conditional RNN language model to produce the translation.[197]

For the sake of dimensionality reduction of the updated representation in each layer, a supervised strategy selects the best informative features among features extracted by KPCA.

The main idea is to use a kernel machine to approximate a shallow neural net with an infinite number of hidden units, then use stacking to splice the output of the kernel machine and the raw input in building the next, higher level of the kernel machine.

The basic search algorithm is to propose a candidate model, evaluate it against a dataset and use the results as feedback to teach the NAS network.[201]

game-playing and decision making (backgammon, chess, poker), pattern recognition (radar systems, face identification, signal classification,[204]

object recognition and more), sequence recognition (gesture, speech, handwritten and printed text recognition), medical diagnosis, finance[205]

models of how the dynamics of neural circuitry arise from interactions between individual neurons and finally to models of how behavior can arise from abstract neural modules that represent complete subsystems.

These include models of the long-term, and short-term plasticity, of neural systems and their relations to learning and memory from the individual neuron to the system level.

specific recurrent architecture with rational valued weights (as opposed to full precision real number-valued weights) has the full power of a universal Turing machine,[219]

but also in statistical learning theory, where the goal is to minimize over two quantities: the 'empirical risk' and the 'structural risk', which roughly corresponds to the error over the training set and the predicted error in unseen data due to overfitting.

Supervised neural networks that use a mean squared error (MSE) cost function can use formal statistical methods to determine the confidence of the trained model.

A confidence analysis made this way is statistically valid as long as the output probability distribution stays the same and the network is not modified.

By assigning a softmax activation function, a generalization of the logistic function, on the output layer of the neural network (or a softmax component in a component-based neural network) for categorical target variables, the outputs can be interpreted as posterior probabilities.

Potential solutions include randomly shuffling training examples, by using a numerical optimization algorithm that does not take too large steps when changing the network connections following an example and by grouping examples in so-called mini-batches.

No neural network has solved computationally difficult problems such as the n-Queens problem, the travelling salesman problem, or the problem of factoring large integers.

Sensor neurons fire action potentials more frequently with sensor activation and muscle cells pull more strongly when their associated motor neurons receive action potentials more frequently.[222]

Other than the case of relaying information from a sensor neuron to a motor neuron, almost nothing of the principles of how information is handled by biological neural networks is known.

The motivation behind Artificial neural networks is not necessarily to strictly replicate neural function, but to use biological neural networks as an inspiration.

Alexander Dewdney commented that, as a result, artificial neural networks have a 'something-for-nothing quality, one that imparts a peculiar aura of laziness and a distinct lack of curiosity about just how good these computing systems are.

argued that the brain self-wires largely according to signal statistics and therefore, a serial cascade cannot catch all major statistical dependencies.

While the brain has hardware tailored to the task of processing signals through a graph of neurons, simulating even a simplified neuron on von Neumann architecture may compel a neural network designer to fill many millions of database rows for its connections – 

Schmidhuber notes that the resurgence of neural networks in the twenty-first century is largely attributable to advances in hardware: from 1991 to 2015, computing power, especially as delivered by GPGPUs (on GPUs), has increased around a million-fold, making the standard backpropagation algorithm feasible for training networks that are several layers deeper than before.[227]

Arguments against Dewdney's position are that neural networks have been successfully used to solve many complex and diverse tasks, ranging from autonomously flying aircraft[229]

Neural networks, for instance, are in the dock not only because they have been hyped to high heaven, (what hasn't?) but also because you could create a successful net without understanding how it worked: the bunch of numbers that captures its behaviour would in all probability be 'an opaque, unreadable table...valueless as a scientific resource'.

In spite of his emphatic declaration that science is not technology, Dewdney seems here to pillory neural nets as bad science when most of those devising them are just trying to be good engineers.

Although it is true that analyzing what has been learned by an artificial neural network is difficult, it is much easier to do so than to analyze what has been learned by a biological neural network.

Furthermore, researchers involved in exploring learning algorithms for neural networks are gradually uncovering general principles that allow a learning machine to be successful.

Advocates of hybrid models (combining neural networks and symbolic approaches), claim that such a mixture can better capture the mechanisms of the human mind.[232][233]

The simplest, static types have one or more static components, including number of units, number of layers, unit weights and topology.

IRIS Flower data set tutorial in artificial neural network in matlab

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