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where vij is calculated using equation (162) or (166) The value of Vmax,j is very important, since it controls the granularity of the search by clamping escalating velocities Large values of Vmax,j facilitate global exploration, while smaller values encourage local exploitation If Vmax,j is too small, the swarm may not explore su ciently beyond locally good regions Also, too small values for Vmax,j increase the number of time steps to reach an optimum Furthermore, the swarm may become trapped in a local optimum, with no means of escape On the other hand, too large values of Vmax,j risk the possibility of missing a good region The particles may jump over good solutions, and continue to search in fruitless regions of the search space While large values do have the disadvantage that particles may jump over optima, particles are moving faster This leaves the problem of nding a good value for each Vmax,j in order to balance between (1) moving too fast or too slow, and (2) exploration and exploitation Usually, the Vmax,j values are selected to be a fraction of the domain of each dimension of the search space That is, (1618) Vmax,j = (xmax,j xmin,j ) where xmax,j and xmin,j are respectively the maximum and minimum values of the domain of x in dimension j, and (0, 1] The value of is problem-dependent, as was found in a number of empirical studies [638, 781] The best value should be found for each di erent problem using empirical techniques such as cross-validation There are two important aspects of the velocity clamping approach above that the reader should be aware of: 1 Velocity clamping does not con ne the positions of particles, only the step sizes as determined from the particle velocity 2 In the above equations, explicit reference is made to the dimension, j A maximum velocity is associated with each dimension, proportional to the domain of that dimension For the sake of the argument, assume that all dimensions are clamped with the same constant Vmax Therefore if a dimension, j, exists such that xmax,j xmin,j << Vmax , particles may still overshoot an optimum in dimension j While velocity clamping has the advantage that explosion of velocity is controlled, it also has disadvantages that the user should be aware of (that is in addition to.

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approximation to (z) is found from the training set DT , memorization is determined from DV (more about this later), and the generalization accuracy is estimated from the test set DG (more about this later) Since prior knowledge about (D) is usually not known, a nonparametric regression approach is used by the NN learner to search through its hypothesis space H for a function fN N (DT , W) which gives a good estimation of the unknown function (z), where fN N (DT , W) H For multilayer NNs, the hypothesis space consists of all functions realizable from the given network architecture as described by the weight vector W During learning, the function fN N : RI RK is found which minimizes the empirical error P 1 T ET (DT ; W) = (FN N (zp , W) tp )2 (325) PT p=1 where PT is the total number of training patterns The hope is that a small empirical (training) error will also give a small true error, or generalization error, de ned as EG ( ; W) = (fN N (z, W) t)2 d (z, t) (326)

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54 ACTIONS Here is an example of smart data and smart data strategy at work From an interview focused on data strategy with Michael E Krieger, director of information policy for the Department of Defense, by Ben Bradley for CIO magazine, we learned that he is responsible for providing policy and guidance for implementing the DoD s net-centric data strategy and enabling the transition to an enterprise service-oriented architecture (SOA) The DoD is separating data from applications, decoupling data, because systems that are too tightly coupled to the data result in high maintenance expense and other problems The data strategy gives priority to visibility, accessibility, and understandability over standardization When asked by Bradley what the DoD s Achilles heel is Krieger replied: the transformation and change required to get communities together to address

For the purpose of NN learning, the empirical error in equation (325) is referred to as the objective function to be optimized by the optimization method Several optimization algorithms for training NNs have been developed [51, 57, 221] These algorithms are grouped into two classes: Local optimization, where the algorithm may get stuck in a local optimum without nding a global optimum Gradient descent and scaled conjugate gradient are examples of local optimizers Global optimization, where the algorithm searches for the global optimum by employing mechanisms to search larger parts of the search space Global optimizers include LeapFrog, simulated annealing, evolutionary algorithms and swarm optimization Local and global optimization techniques can be combined to form hybrid training algorithms Learning consists of adjusting weights until an acceptable empirical error has been reached Two types of supervised learning algorithms exist, based on when weights are updated: Stochastic/online learning, where weights are adjusted after each pattern presentation In this case the next input pattern is selected randomly from the training set, to prevent any bias that may occur due to the order in which patterns occur in the training set Batch/offline learning, where weight changes are accumulated and used to adjust weights only after all training patterns have been presented

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Gradient descent (GD) optimization has led to one of the most popular learning algorithms, namely backpropagation, popularized by Werbos [897] Learning iterations (one learning iteration is referred to as an epoch) consists of two phases: 1 Feedforward pass, which simply calculates the output value(s) of the NN for each training pattern (as discussed in Section 31) 2 Backward propagation, which propagates an error signal back from the output layer toward the input layer Weights are adjusted as functions of the backpropagated error signal

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