The Backpropagation algorithm was first proposed by Paul Werbos in the 1970's. Explorations in parallel distributed processing: A handbook of models, programs, and exercises. Cambridge, MA. Cambridge, MA: MIT Press. This installation program includes evaluation versions of three products for neural network design and development: NeuroSolutions, NeuroSolutions for Excel and the Custom Solution Wizard. Utilizing backpropagation algorithm It can be used in almost all research fields such as nonlinear regression, forecasting, curve fit, pattern.
![Program Program](http://staff.itee.uq.edu.au/janetw/cmc/chapters/BackProp/AndExample.gif)
A hyperplane (the slanted line) separating the blue data points (class -1) from the red data points (class +1) As we saw last time, the Perceptron model is particularly bad at learning data. More accurately, the Perceptron model is very good at learning linearly separable data, but most kinds of data just happen to more complicated. Even with those disappointing results, there are two interesting generalizations of the Perceptron model that have exploded into huge fields of research. The two generalizations can roughly be described as • Use a number of Perceptron models in some sort of conjunction. • Use the Perceptron model on some non-linear transformation of the data.
![Backpropagation Backpropagation](http://slideplayer.it/1/552682/big_thumb.jpg)
The point of both of these is to introduce some sort of non-linearity into the decision boundary. The first generalization leads to the neural network, and the second leads to the support vector machine. Obviously this post will focus entirely on the first idea, but we plan to cover support vector machines in the near future. Recall further that the separating hyperplane was itself defined by a single vector (a normal vector to the plane). To “decide” what class the new point is in, we check the sign of an inner product with an added constant shifting term: The class of a point is just the value of this function, and as we saw with the Perceptron this corresponds geometrically to which side of the hyperplane the point lies on. Now we can design a “neuron” based on this same formula.
We consider a point to be an input to the neuron, and the output will be the sign of the above sum for some coefficients. In picture form it would look like this: It is quite useful to literally think of this picture as a directed graph (see this blog’s if you don’t know what a graph is).
The edges corresponding to the coordinates of the input vector have weights, and the output edge corresponds to the sign of the linear combination. If we further enforce the inputs to be binary (that is, ), then we get a very nice biological interpretation of the system.
If we think of the unit as a neuron, then the input edges correspond to nerve impulses, which can either be on or off (identically to an electrical circuit: there is high current or low current). The weights correspond to the strength of the neuronal connection. The neuron transmits or does not transmit a pulse as output depending on whether the inputs are strong enough.
We’re not quite done, though, because in this interpretation the output of the neuron will either fire or not fire. However, neurons in real life are somewhat more complicated. Specifically, neurons do not fire signals according to a discontinuous function. In addition, we want to use the usual tools from classical calculus to analyze our neuron, but we cannot do that unless the activation function is differentiable, and a prerequisite for that is to be continuous. In plain words, we need to allow our neurons to be able to “partially fire.” We need a small range at which the neuron ramps up quickly from not firing to firing, so that the activation function as a whole is differentiable. This raises the obvious question: what function should we pick?
It turns out that there are we could use, ranging from polynomial to exponential in nature. But before we pick one in particular, let’s outline the qualities we want such a function to have.
Definition: A function is an activation function if it satisfies the following properties: • It has a first derivative. • is non-decreasing, that is for all • has horizontal asymptotes at both 0 and 1 (and as a consequence,, and ).
• and are both computable functions. With appropriate shifting and normalizing, there are a few reasonable (and time-tested) activation functions. Unity 4 pro serial number crack photoshop. The two main ones are the hyperbolic tangent and the sigmoid curve. They both look (more or less) like this. A sigmoid function (source: Wikipedia) And it is easy to see visually that this is what we want. As a side note, the sigmoid function is actually not used very often in practice for a good reason: it gets too “flat” when the function value approaches 0 or 1. The reason this is bad is because how “flat” the function is (the gradient) will guide the learning process.
If the function is very flat, then the network won’t learn as quickly. This will manifest itself in our test later in this post, when we see that a neural network struggles to learn the sine function.