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the next training period. This process repeats until the NN is trained to satisfaction. The main difference between these two approaches to active learning is that no examples are discarded by incremental learning. In the limit, all examples in the candidate set will be used for training. With selective learning, training starts on all candidate examples, and uninformative examples are discarded as training progresses.
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Not much research has been done in selective learning. Hunt and Deller developed Selective Updating, where training starts on an initial candidate training set [Hunt and Deller 1995]. Patterns that exhibit a high influence on weights, i.e. patterns that cause the largest changes in weight values, are selected from the candidate set and added to the training set. Patterns that have a high influence on weights are selected at each epoch by calculating the effect that patterns have on weight estimates. These calculations are based on matrix perturbation theory, where an input pattern is viewed as a perturbation of previous patterns. If the perturbation is expected to cause large changes to weights, the corresponding pattern is included in the training set. The learning algorithm does use current knowledge to select the next training subset, and training subsets may differ from epoch to epoch. Selective Updating has the drawback of assuming uncorrelated input units, which is often not the case for practical applications. Another approach to selective learning is simply to discard those patterns that have been classified correctly [Barnard 1991]. The effect of such an approach is that the training set will include those patterns that lie close to decision boundaries. If the candidate set contains outlier patterns, these patterns will, however, also be selected. This error selection approach therefore requires a robust estimator (objective function) to be used in the case of outliers. Engelbrecht et al. developed a selective learning approach for classification problems where sensitivity analysis is used to locate patterns close to decision boundaries [Engelbrecht and Cloete 1998a, Engelbrecht and Cloete 1998b, Engelbrecht 2001b]. Only those patterns that are close to a decision boundary are selected for training. The algorithm resulted in substantial reductions in the number of learning calculations due to reductions in the training set size, while either maintaining performance as obtained from learning from all the training data, or improving performance.
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Incremental learning Research on incremental learning is more abundant than for selective learning. Most current incremental learning techniques have their roots in information theory, adapting Fedorov's optimal experiment design for NN learning [Cohn 1994, Fukumizu 1996, MacKay 1992, Plutowski and White 1993, Sung and Niyogi 1996]. The different information theoretic incremental learning algorithms are very similar, and differ only in whether they consider only bias, only variance, or both bias and variance terms in their selection criteria. Cohn developed neural network Optimal Experiment Design (OED), where the objective is to select at each iteration a new pattern from a candidate set which minimizes the expectation of the mean squared error (MSE) [Cohn 1994]. This is achieved by minimizing output variance as estimated from the Fisher information matrix [Cohn 1994, Cohn et al. 1996]. The model assumes an unbiased estimator and considers only the minimization of variance. OED is computationally very expensive because it requires the calculation of the inverse of the information matrix. MacKay proposed similar Information-Based Objective Functions for active learning, where the aim is to maximize the expected information gain by maximizing the change in Shannon entropy when new patterns are added to the actual training set, or by maximizing cross-entropy gain [MacKay 1992]. Similar to OED, the maximization of information gain is achieved by selecting patterns that minimize the expected MSE. Information-Based Objective Functions also ignore bias, by minimizing only variance. The required inversion of the Hessian matrix makes this approach computationally expensive. Plutowski and White proposed selecting patterns that minimize the Integrated Squared Bias (ISB) [Plutowski and White 1993]. At each iteration, a new pattern is selected from a candidate set that maximizes the change, AISB, in the ISB. In effect, the patterns with error gradient most highly correlated with the error gradient of the entire set of patterns is selected. A noise-free environment is assumed and variance is ignored. Drawbacks of this method are the need to calculate the inverse of a Hessian matrix, and the assumption that the target function is known. Sung and Niyogi proposed an information theoretic approach to active learning that considers both bias and variance [Sung and Niyogi 1996]. The learning goal is to minimize the expected misfit between the target function and the approximated function. The patterns that minimizes the expected squared difference between the target and approximated function are selected to be included in the actual training set. In effect, the net amount of information gained with each new pattern is then maximized. No assumption is made about the target function. This technique is computationally expensive, since it requires computations over two expectations, i.e. the a-posteriori distribution over function space, and the a-posteriori distribution over the space of targets one would expect given a candidate sample location.
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