Weight Initialization in Java

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Gradient-based optimization methods, for example gradient descent, is very sensitive to the initial weight vectors. If the initial position is close to a local minimum, convergence will be fast. However, if the initial weight vector is on a flat area in the error surface, convergence is slow. Furthermore, large initial weight values have been shown to prematurely saturate units due to extreme output values with associated zero derivatives [Hush et al. 1991]. In the case of optimization algorithms such as
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PSO and GAs, initialization should be uniformly over the entire search space to ensure that all parts of the search space are covered. A sensible weight initialization strategy is to choose small random weights centered around 0. This will cause net input signals to be close to zero. Activation functions then output midrange values regardless of the values of input units. Hence, there is no bias toward any solution. Wessels and Barnard showed that random weights in the range [-==i=, //* nin ] is a good choice, where fanin is the number of connections leading to a unit [Wessels and Barnard 1992]. Why don't we just initialize all the weights to zero in the case of gradient-based optimization This strategy will work only if the NN has just one hidden unit. For more than one hidden unit, all the units produce the same output, and thus make the same contribution to the approximation error. All the weights are therefore adjusted with the same value. Weights will remain the same irrespective of training time - hence, no learning takes place. Initial weight values of zero for PSO will also fail, since no velocity changes are made; therefore no weight changes. GAs, on the other hand, will work with initial zero weights if mutation is implemented.
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Learning Rate and Momentum
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The convergence speed of NNs is directly proportional to the learning rate 77. Considering stochastic GD, the momentum term added to the weight updates also has as its objective improving convergence time. Learning Rate The learning rate controls the size of each step toward the minimum of the objective function. If the learning rate is too small, the weight adjustments are correspondingly small. More learning iterations are then required to reach a local minimum. However, the search path will closely approximate the gradient path. Figure 7.5(a) illustrates the effect of small rj. On the other hand, large n will have large weight updates. Convergence will initially be fast, but the algorithm will eventually oscillate without reaching the minimum. It is also possible that too large a learning rate will cause "jumping" over a good local minimum proceeding toward a bad local minimum. Figure 7.5(b) illustrates the oscillating behavior, while Figure 7.5(c) illustrates how large learning rates may cause the network to overshoot a good minimum and get trapped in a bad local minimum. Small learning rates also have the disadvantage of being trapped in a bad local minimum as illustrated in Figure 7.5(d). The search path goes down the first local minimum, with no mechanism to move out of it toward the next, better minimum. Of course, all depends on the initial starting position. If the second initial point is used, the NN will converge to the better local minimum.
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(a) Small 77
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(b) Large n gets stuck
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(c) Large 77 overshoots
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(d) Small 77 gets stuck
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Figure 7.5: Effect of learning rate
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But how do we choose the value of the learning rate One approach is to find the optimal value of the learning rate through cross-validation, which is a lengthy process. An alternative is to select a small value (e.g. 0.1) and to increase the value if convergence is too slow, or to decrease it if the error does not decrease fast enough. Plaut et al. proposed that the learning rate should be inversely proportional to the fanin of a neuron [Plaut et al. 1986]. This approach has been theoretically justified through an analysis of the eigenvalue distribution of the Hessian matrix of the objective function [Le Gun et al. 1991]. Several heuristics have been developed to dynamically adjust the learning rate during training. One of the simplest approaches is to assume that each weight has a different learning rate rjkj. The following rule is then applied to each weight before that weight is updated: if the direction in which the error decreases at this weight change is the same as the direction in which it has been decreasing recently, then nkj is increased; if not, nkj is decreased [Jacobs 1988]. The direction in which the error decreases is determined by the sign of the partial derivative of the objective function with respect to the weight. Usually, the average change over a number of pattern presentations is considered and not just the previous adjustment. An alternative is to use an annealing schedule to gradually reduce a large learning rate to a smaller value (refer to equation 4.22). This allows for large initial steps, and ensures small steps in the region of the minimum. Of course more complex adaptive learning rate techniques have been developed, with elaborate theoretical analysis. The interested reader is referred to [Darken and Moody, Magoulas et al. 1997, Salomon and Van Hemmen 1996, Vogl et al. 1988],
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