The linear system of equations is solved for calculating the weights of the output layer, using the modified Gram-Schmidt algorithm [17, 18]. The hidden layers are trained using EBP, delta rule and the other proposed techniques. The major benefit of the new approach presented in this paper is its ability to train MLPs much faster than the conventional error backpropagation algorithm and results are comparable.
The remainder of this paper is organised as follows. In Section III we propose techniques for determining the weights of hidden layers and the output layer. In Section IV we present a description of the proposed methods in detail. In Section V computer simulations for performance evaluation and comparison are presented.
Finally, Section VI concludes the paper. Single neuron model. Figure 2. Multilayer Perceptron with single hidden layer. Figure 1 shows a single neuron model.
In the output layer for each neuron, the value net can be calculated as shown in Figure 1. We use this value to establish the linear system of equations for the output layer, which is described in Section III. Figure 2 shows a multilayer perceptron with single hidden layer suitable for training with the proposed approach. As shown in Figure 2, the input signal is simply passed through to the weights on their outputs. Each neuron in subsequent layers produces net and ok signals as shown in Equations 1 and 2.
Training the weights of the hidden layer. The technique of unsupervised learning is often used to perform clustering as the unsupervised classification of objects without providing information about the actual classes.
In our approach this kind of learning is used to adjust the weights of the hidden layer of MLP. For this purpose we propose to apply the following techniques: 1. Delta Rule DR The delta rule [2,3] is only valid for continuous activation functions and in supervised training mode.
The mapping of the input space into hidden units is shown in [12]. The input space consists of the set of all input vectors and hidden unit space consists of the set of the hidden units.
The extra hidden units are adjusted to 0. Random Weights RW The weights are adjusted at small random values. It is assumed that they fulfil the following conditions [12]. Small weights are recommended for good generalisation ability. The weights cannot be zero because if the weights are zero then we will get the same input vector for the output layer and the possibility of finding the weights of the output layer will be minimal.
If for different input vectors, we have the same output values in the hidden layer and the desired values targets are different, then it is difficult to calculate the weights of the output layer. Minimum Bit Distance MBD The basic idea of MBD is to compute the measures of similarity in vector space for classification of the input vectors using unsupervised learning. The weights of the ith neuron of the hidden layer are input vectors of the ith pair.
But as we can see they are not the same. Training the weights of the output layer In supervised learning we assume that at each instance of time when the input is applied, the desired response d of the system is known. An important fact used in implementing a MLP is the relation between the desired response d and net. It is easy to calculate the net from 8 by replacing the desired value dk with the ok value. The weights are easily trained using direct solution methods for linear equations.
In real world problems, p is usually greater than h; that is, there are more training patterns than hidden units. Therefore, we need to solve an overdetermined system of equations.
In this study, many techniques have been tested and we have found that the Modified Gram-Schmidt MGS algorithm is very stable and needs less computer memory than other existing algorithms [17, 18]. In our approach, we use MGS to solve linear systems of equations This method can be applied to MLPs with any number of layers; however only two layers are needed to demonstrate the method and solve any real world problem.
This method uses a gradient search technique and direct solution methods least square techniques to minimise the cost function equal to the mean square difference between the desired and the actual net outputs. The desired output of all nodes is typically low 0.
The multilayer perceptron is trained by initially selecting small random weights and internal thresholds and then presenting all training data repeatedly.
Weights are adjusted after each iteration and after some iteration or even when the training process is trapped at a local minimum; training of the MLPs can be stopped and the weights of the output layer can be calculated using direct solution methods.
Consider a two layer perceptron with a single hidden layer as shown in Figure 2. Present input vectors x1, x2, Initialise all weights of the two layer perceptron at small random values. Calculate actual outputs of the MLP.
Use Equations 1 and 2. Train the weights of the hidden layer to output. Use an error backpropagation technique for this purpose. After a certain number of iterations stop the iterative training process. Develop a linear system of equations for the output layer.
Use Equation 10 and convert the output nonlinear activation function into linear function. Use Equation 11 and develop a linear system of equations as shown in Equation Calculate the weights of the output layer.
Use the modified Gram-Schmidt algorithm to solve Repeat Step 7 through 8 for each neuron in the hidden layer. The delta rule is used to adjust the weights of the hidden layer. This layer is used to classify input data. For this classification we used two techniques, binary coding described in the previous section and the symmetric matrix coding which can be used in rare cases where the number of the hidden units is equal to the number of training pairs.
In this technique, each input pair has a neuron in the hidden layer which has a desired value 0. Back Propagation Explained It is straightforward to show that if we take infinitesimal steps down the gradient vector, running a new training epoch to recompute the gradient after each step, we will eventually reach.
This and the summary of all triangles of a leaf image are the representation of the tokens of a leaf from which we can start the neural network calculations.
An analogy for understanding gradient descent[edit] Further information: Gradient descent The basic intuition behind gradient descent can be illustrated by a hypothetical scenario. Among the many possible ways of measuring the size or complexity of a computational task, the predicate order defined by Minsky and Papert , provides the most useful and important measure. In: Christmas Answer it!
Journal of Mathematical Analysis and Applications, 5 1 , Let the function F w be the input-output mapping realized by this network; w denotes the vector of all synaptic weights including biases contained in the network. The network layout satisfies the following structural requirements, as illustrated in Fig.
Yet batch learning typically yields a faster, more stable descent to a local minima, since each update is performed in the direction of the average error of the batch samples.
Save your draft before refreshing this page. The target output for o1 is 0. Our goal with back propagation algorithm is to update each weight in the network so that the actual output is closer to the target output, thereby minimizing the error for each output neuron and the network as a whole.
Since we are propagating backward, the first thing we need to do is to calculate the change in total errors w. Now, we will propagate further backward and calculate the change in the output o1 w. We perform the actual updates in the neural network after we have the new weights leading into the hidden layer neurons. Thus, we need to take Eo1 and Eo2 into consideration.
We will calculate the partial derivative of the total net input of h1 w. It might not seem like much, but after repeating this process 10, times, for example, the error plummets to 0. At this point, when we feedforward 0. Prepare yourself for the Artificial Intelligence Interview questions and answers Now! In batch gradient descent, we use the complete dataset available to compute the gradient of the cost function.
Batch gradient descent is very slow because we need to calculate the gradient on the complete dataset to perform just one update, and if the dataset is large then it will be a difficult task. It is a widely used algorithm that makes faster and accurate results.
It is faster because it does not use the complete dataset. It reduces the variance of the parameter updates, which can lead to more stable convergence. It can also make use of a highly optimized matrix that makes computing of the gradient very efficient. We use stochastic gradient descent for faster computation. The first step is to randomize the complete dataset. Then, we use only one training example in every iteration to calculate the gradient of the cost function for updating every parameter.
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