Gradient descent

The Back-Propagation Algorithm (BPA) is a supervised learning method for training ANNs, and is one of the most common forms of training techniques. It uses a gradient-descent optimization method, also referred to as the delta rule when applied to feedforward networks. A feedforward network that has employed the delta rule for training, is called a Multi-Layer Perceptron (MLP). 

We mentioned above that a typical problem for a Boltzman Machine is to obtain a set of weights such that the states of the visible neurons take on some desired probability distribution. For example, the task may he to teach the net to learn that the first component of an Ai-component input vector has value +1 40% of the time. To accompli.sh this, a Boltzman Machine uses the familiar gradient-descent technique, but not on the energy of the net instead, it maximizes the relative entropy of the system. 

We see that our task is none other than to minimize 5, which we can do by using gradient descent ... 

The calculations are slightly different between the two cases where the weights are between (i) hidden and output layers and (ii) input and hidden layers. For the hidden-to-output connections, the gradient descent algorithm gives... 

The calibration phase focuses on the determination of the planarization length itself. This is a crucial characterization phase since once the planarization length is determined, the effective density, and thus the thickness evolution, can be determined for any layout of interest polished under similar process conditions. The determination of planarization length is an iterative process. First, an initial approximate length is chosen. This is used to determine the effective density as detailed in the previous subsection. The calculated effective density is then used in the model to compute predicted oxide thicknesses, which are then compared to measured thickness data. A sum of square error minimization scheme is used to determine when an acceptably small error is achieved by gradient descent on the choice of planarization length. 

Backpropagation is currently the most popular method of fitting the m(p + n + 1) + n parameters in Equation 9.6. Backpropagation is a gradient descent-based procedure that addresses the problem... 

Backpropagation by gradient-descent is generally a reliable procedure nevertheless, it has its limitations it is not a fast training method and it can be trapped in local minima. To avoid the latter, a variant of the above algorithm called gradient-descent with momentum (GDM) introduces a third term, / ... 

The above-related situation can also be improved by using a different algorithm for training. One of the most efficient minimization algorithms is the Levenberg-Marquardt (LM) [56,59]. It is between 10 and 100 times faster than gradient-descent, given it employs a second-derivative approach, while GDM employs only first-derivative terms. As the calculation of the Hessian matrix (matrix of the second derivatives of the error in... 

Bayesian regularization (BR) This technique searches for the simplest network which adjusts itself to the function to be approximated, but which also is able to predict most efficiently the points that did not participate in the training [63], In contrast to gradient-descent, in this case not only the global error of the ANN is taken... 

A neural network is typically trained by variations of gradient descent-based algorithms, trying to minimize an error function [77]. It is important that additional validation data be left untouched during ANN training, so as to have an objective measure of the model s generalization ability [78],... 

Owing to its gradient-descent nature, back-propagation is very sensitive to initial conditions. The choice of initial weights will influence whether the net reaches a global (or only a local) minimum of the error and, if so, how quickly it converges. In practice, the weights are usually initialized to small zero-mean random values between -0.5 and 0.5 (or between -1 and 1 or some other suitable interval). 

Darwen, P. J., Bourne, G. T., Nielson, J., Tran, T. T., and Smythe, M. L. (2003) A gradient descent algorithm for minimizing the number of steps required for synthesis of cyclic-peptide libraries. Personal communication. 

The learning procedure consists of two stages. In the forward pass training input data go forward the ANFIS architecture, and in the backward pass the error rates propagate backward, being the both the consequent and the membership parameters updated by gradient descent. 

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